On the measurement problem for a two-level quantum
system
Alexey A. Kryukov ∗
November 12, 2018
A geometric approach to quantum mechanics with unitary evolution and
non-unitary collapse processes is developed. In this approach the Schrödinger
evolution of a quantum system is a geodesic motion on the space of states of
the system furnished with an appropriate Riemannian metric. The measuring
device is modeled by a perturbation of the metric. The process of measurement
is identified with a geodesic motion of state of the system in the perturbed
metric. Under the assumption of random fluctuations of the perturbed metric,
the Born rule for probabilities of collapse is derived. The approach is applied
to a two-level quantum system to obtain a simple geometric interpretation of
quantum commutators, the uncertainty principle and Planck’s constant. In
light of this, a lucid analysis of the double-slit experiment with collapse and an
experiment on a pair of entangled particles is presented.
KEY WORDS: measurement problem - Born rule - Berry’s phase - EPR-paradox
1 GEOMETRY AND QUANTUM MECHANICS
Geometric ideas have played a well recognized role in modern physics, especially in
general relativity (GR) and gauge theories (GT). They also found a well established
position in quantum mechanics (QM) in considerations related to Berry’s phase
Ref. [1]. However, whereas in GR and GT geometry (i.e., the metric or connection)
defines the dynamics of the theory, the geometric methods pertaining to Berry’s
phase do not enjoy such a sweeping significance. The reason for this difference is
quite obvious. Indeed, the geometry underlying GR and GT is directly related to
∗Department of Mathematics, University of Wisconsin Colleges
E-mail: alexey.kryukov@uwc.edu, aakrioukov@facstaff.wisc.edu
http://arxiv.org/abs/0704.1934v1
2 Alexey A. Kryukov
the physical fields (gravitational or gauge) in the theory. At the same time, the
Fubini-Study metric in the geometric interpretation of Berry’s phase (Refs. [2],[3]
amongst many others) depends only on the geometry of the Hilbert space of states of
quantum system. The latter geometry (i.e. Hilbert metric) is insensitive to changes
in the Hamiltonian of the system and, consequently, is not dynamical.
At the same time, it turns out to be easy to make the metric on Hilbert space
of states of a closed quantum system dynamical Refs. [4],[5]. For this, notice first
of all that the Schrödinger equation
= − i
ĥϕt (1.1)
is the equation for integral curves of the vector field hϕ : H −→ TH, hϕ = − ih̄ ĥϕ
associated with the Hamiltonian ĥ of the system. Here H is the Hilbert space
of states of the system and TH is the tangent bundle over H. Assume that the
Hilbert space H is a space of functions that are square-integrable with respect to
an appropriate measure. Because the evolution governed by Eq. (1.1) is unitary,
the integral curve through initial point ϕ0 on the unit sphere S
H in H will stay on
the sphere. Since this holds true for any initial point (modulo the domain issues),
one concludes that the restriction of the vector field hϕ to the sphere S
H is a vector
field on the sphere.
In the ordinary QM spaces TϕH tangent to H at ϕ ∈ H are identified with
the space H itself. Similarly, spaces TϕS
H tangent to the sphere SH at ϕ ∈ SH
are identified with (real) affine subspaces of H. In particular, the metric on SH ,
whenever used, is assumed to be induced by the embedding of SH into H.
However, the sphere SH is a manifold and thus, can be defined independently of
the ambient space H. As such, SH is a Banach manifold which means that it can
be obtained by “gluing together” open sets in a Banach space. The Hilbert metric
Gϕ : TϕH × TϕH −→ C on tangent spaces TϕH can be also defined independently
of the metric on H as an Hermitian tensor field on H. Such a tensor field gives rise
to a Riemannian metric GRϕ : TϕS
H ×TϕSH −→ R on SH , defined at each ϕ ∈ SH
GRϕ(X,Y ) = 2ReGϕ(ξ, η). (1.2)
Here X = (ξ, ξ), Y = (η, η) ∈ TϕSH are considered as vectors in the realization
of the tangent space TϕH. The manifold S
H , furnished with the (2, 0)-tensor field
GRϕ, is then a Riemannian manifold. In the following, the manifold S
H with the
metric GRϕ will be denoted by S
The final step in making the metric Gϕ on the sphere of states S
G dynamical is
to ensure that the integral curves of hϕ (i.e. the solutions to Schrödinger equation
Eq. (1.1)) are geodesics on SG. For this it turns out to be sufficient to define
Gϕ : TϕH × TϕH −→ C by
Gϕ(ξ, η) = h̄
(ĥĥ∗)−1ξ, η
. (1.3)
On the measurement problem 3
Here ĥ∗ is the adjoint of ĥ (normally equal to ĥ) and the Hamiltonian ĥ is assumed
to be invertible. Incidentally, even if the Hamiltonian ĥ is not bounded on H, it
becomes bounded as an operator mapping points ϕ ∈ H into tangent spaces TϕH
with the metric Eq. (1.3) (see Ref. [5]).
Further general results concerning QM on Hilbert manifolds can be found in Ref.
[4]. These results demonstrate that QM can be formulated in terms of geometry of
the space of states. The goal of the Letter is to provide such a geometric formulation
in case of a simple two-level system and to establish its advantages. Namely, the
point of view that the space of states represents a new arena for physical processes
and the evolution of state is a motion along geodesic is shown to be effective in
addressing the major conceptual difficulties of quantum mechanics. Although the
discussion deals primarily with a simple model, the most important results can be
shown to be quite general. Some of these generalizations are described in the Letter
while others are left for the upcoming publications.
2 ELECTRON IN A HOMOGENEOUS MAGNETIC
FIELD
Consider a free non-relativistic electron propagating in the direction of the X-axis
in a homogeneous magnetic field B. The evolution equation (the Pauli equation)
for the electron is
Ψ− µσ̂ ·BΨ, (2.1)
where Ψ = Ψ(s, x, t), s = 1, 2 is a two-component state function of the electron, µ
is the electron’s magnetic moment and σ̂ = (σ̂x, σ̂y, σ̂z) is the vector made of Pauli
matrices. The substitution Ψ(s, x, t) = ψt(x)ϕt(s) separates variables and produces
two independent evolution equations. The first describes the evolution governed by
the free Hamiltonian
ψt. (2.2)
The second equation describes the evolution in the space C2 of spinors ϕ:
= −µσ̂ ·Bϕt. (2.3)
It follows that in the case of the product states Ψ(s, x, t) = ψt(x)ϕt(s), one can
analyze the evolution of spin state ϕt in the space of states C
2 without needing to
involve the infinite-dimensional Hilbert space of states Ψ.
2.1 Quantum Mechanics on the Space of States S3
Let us proceed to reformulation of quantum mechanics of the system in geometrical
terms. In this, the fact that the sphere S3 of unit normalized spin states can be
4 Alexey A. Kryukov
furnished with the group structure of the group SU(2) will be helpful. The group
structure will allow us to exploit simple results from differential geometry of Lie
groups which will make the resulting picture more transparent and complete.
First of all, the Hamiltonian ĥ = −µσ̂ ·B defines the vector field hϕ = ih̄µσ̂ ·Bϕ
on the sphere S3 in the space C2 with the metric (ξ, η)C2 =
k ξkηk. The integral
curve of hϕ (i.e. the solution of Eq. (2.3)) through ϕ0 ∈ S3 is given by
ϕt = e
µσ̂·Btϕ0. (2.4)
Since ϕt is a path in C
2, it is natural to call the vector
the velocity of evolution
of the system. The speed of evolution in the C2 is the norm of dϕt
in C2 metric.
Using
(σ̂ ·A)(σ̂ ·B) = A ·B+ iσ̂ ·A×B, (2.5)
one has
(σ̂ ·B)2 = B2. (2.6)
Therefore, by Hermicity of the matrix σ̂ ·B, one obtains
µσ̂ ·Bϕt,
µσ̂ ·Bϕt
, (2.7)
where B is the norm of B. In particular, the speed of evolution of the system
depends only on the magnitude of the field.
To make the evolution of the system a motion along a geodesic, the metric Gϕ
on S3 will be defined by Eq. (1.3). Since ĥ is self-adjoint, one obtains ĥĥ∗ = ĥ2 =
µ2(σ̂ ·B)2 = µ2B2I, where I is the identity operator on C2 and Eq. (2.6) has been
used at the last step. Therefore, up to the constant factor (h̄/µB)2, the metric
Gϕ coincides with the one induced by the embedding of S
3 into C2. That means
that the carriers of the geodesics on S3 are the intersections of S3 with the planes
through the origin. The fact that the found Riemannian metric is so simple is due
to an especially simple form of the Hamiltonian in the model.
If S3 is identified with the group manifold SU(2), the obtained metric is the
Killing metric on SU(2). To see this, let us identify in the standard way the space
C2 of complex vectors ϕ =
with the space Mat of 2× 2 matrices
z1 z2
−z2 z1
. (2.8)
The map ω : ϕ −→ ϕ̂ is an isomorphism of (real) vector spaces C2 and Mat. The
sphere S3 of unit states in C2 is identified via ω with the subset of matrices with
unit determinant. The latter subset is the group SU(2) under matrix multiplication.
On the measurement problem 5
The Killing metric on the Lie algebra su(2) can be defined by
(X̂, Ŷ )K = cTr(X̂Ŷ
+), (2.9)
where c 6= 0 is an arbitrary constant and Tr stands for the trace. The Killing metric
on the group SU(2) is defined for all ϕ̂ ∈ SU(2) and all left-invariant vector fields
(ϕ̂) = ϕ̂X̂, L
(ϕ̂) = ϕ̂Ŷ , X̂, Ŷ ∈ su(2) by
(ϕ̂), L
(ϕ̂))K = (LX̂(e), LŶ (e))K = (X̂, Ŷ )K , (2.10)
where e is the identity element in SU(2).
By direct substitution one verifies that (L
(e), L
(e))K = 2cRe(X,Y )C2 when-
ever X̂ = dω(X) and Ŷ = dω(Y ) with dω being the differential of the map ω. In
other words, the Killing metric is proportional to the metric induced by the em-
bedding of S3 into the Euclidean space C2 = R4. This verifies that the metric
Gϕ = (h̄/µB)
2I obtained earlier, is the Killing metric.
For any two left invariant vector fields L
on SU(2) the connection ∇ on
SU(2) can be defined by
[X̂,Ŷ ]
. (2.11)
Notice that the left invariant vector fields form a basis on the tangent space T
SU(2)
for all ϕ̂ ∈ SU(2). In particular, Eq. (2.11) is sufficient to define a connection on
SU(2). This connection is symmetric, as the torsion tensor vanishes:
) = ∇L
[X̂,Ŷ ]
[Ŷ ,X̂]
[X̂,Ŷ ]
= 0. (2.12)
The connection Eq. (2.11) is also compatible with the Killing metric, that is, for
any vector fields ξ, η, ζ on SU(2) the following is true:
∇ξ(η, ζ)K = (∇ξη, ζ)K + (η,∇ξζ)K . (2.13)
Indeed, assuming that ξ = L
, η = L
, ζ = L
are left invariant, one has
(ϕ̂), L
(ϕ̂)K = (LŶ (e), LẐ(e))K = const (2.14)
and therefore the left hand side of Eq. (2.13) vanishes. For the right hand side, by
definition Eq. (2.11) one obtains:
)K + (LŶ ,∇LX̂LẐ)K =
([X̂, Ŷ ], Ẑ)K +
(Ŷ , [X̂, Ẑ])K . (2.15)
From the anti-Hermicity of elements of su(2) one also has:
([X̂, Ŷ ], Ẑ)K = −cTr(X̂Ŷ Ẑ) + cTr(Ŷ X̂Ẑ) (2.16)
6 Alexey A. Kryukov
([Ŷ , X̂ ], Ẑ)K = −cTr(Ŷ X̂Ẑ) + cTr(X̂Ŷ Ẑ). (2.17)
As a result, the sum on the right hand side of Eq. (2.13) is also zero which verifies
that the connection Eq. (2.11) is compatible with the metric. In other words, the
connection ∇ is the Levi-Civita connection of the Killing metric.
For any magnetic fieldB in the model the one parameter subgroup ϕ̂t = e
µσ̂·Bt
is a geodesic through the identity e ∈ SU(2). Indeed, since dϕ̂t
= −ϕ̂t ih̄µσ̂ ·B, the
path ϕ̂t is the integral curve of the left invariant vector field Lĥϕ̂ = −ϕ̂
µσ̂ · B.
Using the definition Eq. (2.11) one then has:
[̂h,̂h]
= 0. (2.18)
Geodesics through an arbitrary point ϕ̂0 ∈ SU(2) can be then written in the form
ϕ̂t = ϕ̂0e
µσ̂·Bt. Considered as paths with values in C2, these geodesics take the
form ϕt = e
µσ̂·Btϕ0.
The curvature tensor of ∇ can be obtained directly from the definition
]LẐ . (2.19)
In particular,
[[X̂,Ŷ ],Ẑ]
. (2.20)
and (
[X̂, Ŷ ], [Ẑ, Ŵ ]
. (2.21)
The sectional curvature in the plane through L
is defined by
. (2.22)
With the help of Eqs. (2.21) and (2.10) this becomes
[X̂, Ŷ ], [X̂, Ŷ ]
X̂, X̂
Ŷ , Ŷ
X̂, Ŷ
. (2.23)
Suppose for example that X̂ , Ŷ , Ẑ and Ŵ correspond to the spin observables.
Recall that in the Planck system of units the operator of spin ŝ has eigenvalues ±1/2
and can be expressed in terms of the Pauli matrices σ̂1, σ̂2, σ̂3 as
σ̂, (2.24)
On the measurement problem 7
where σ̂ = (σ̂1, σ̂2, σ̂3). The corresponding anti-Hermitian generators êk =
k = 1, 2, 3, form a basis of the Lie algebra su(2) and satisfy the commutator relations
[êk, êl] = ǫklmêm, (2.25)
where ǫklm denotes the completely antisymmetric tensor of rank three. In the basis
{êk} the curvature tensor Eqs. (2.20), (2.21) takes the form
Rik,lm =
(δilδkm − δimδkl), (2.26)
Rik,lm =
(δilδkm − δimδkl) (2.27)
where δik is the Kronecker delta. The symmetry property
Rik,lm +R
l,mk +R
m,kl = 0 (2.28)
of the curvature tensor coincides in the model with the Jacobi identity
[[X̂, Ŷ ], Ẑ] + [[Ŷ , Ẑ], X̂] + [[Ẑ, X̂ ], Ŷ ] = 0 (2.29)
for the Lie algebra elements X̂, Ŷ , Ẑ.
From the isomorphism given by Eq. (2.8) it follows that any vector x =
(x1, x2, x3) in the Euclidean space R3 can be identified with the element
kiσ̂k =∑
k êk of the Lie algebra su(2). Under such an identification the Euclidean norm
‖x‖R3 of x is equal to detx and rotations in R3 are represented by transformations
x −→ ÛxÛ+ with Û ∈ SU(2). One can make this identification into an isometry
by assuming the equality of Euclidean and Killing norms
∥∥∥∥∥
2xkêk
∥∥∥∥∥
= ‖x‖R3 . (2.30)
This will fix the constant factor in front of the Killing metric.
Note that the Euclidean space R3 can be identified with the space of all possible
classical angular momenta of a particle. The electron’s possible angular momenta
form a sphere S2 in R3. By equating the norms according to Eq. (2.30), the spaces
tangent to S2 are identified with affine subspaces of spaces tangent to the sphere
of states S3 with the induced metric. In particular, the sphere S2 can be identified
with a submanifold of the space of states S3 with the induced metric. Let us remark
that this identification is analogous to the identification of the classical space with
the submanifold of point supported states in an infinite-dimensional Hilbert space
of states, considered in Refs. [4], [6].
In the Killing metric Eq. (2.10) on S3 one has
kêk,
k,m x
kxmδkm. To satisfy Eq. (2.30) the constant c must be 1/2, that is, the
needed metric in Planck units has the form
(X̂, Ŷ )K =
Tr(X̂Ŷ +). (2.31)
8 Alexey A. Kryukov
Using the formula Eq. (2.23), one obtains the following expression for the sectional
curvature R(p) in the plane p through orthogonal vectors L
([ê1, ê2], [ê1, ê2])K
(ê1, ê1)K (ê2, ê2)K
= 4 (ê3, ê3)K = 1. (2.32)
So the sectional curvature of S3 in Planck units is equal to 1. Note that in an
arbitrary system of units the Killing metric would be multiplied by h̄2 and the
sectional curvature would be equal to 1/h̄2. The dimension of sectional curvature
is consistent with the fact that the tangent space su(2) is spanned by the spin
operators having the dimension of angular momentum.
In this approach the Planck’s constant and the commutators of spin observables
acquire a transparent geometric interpretation. According to Eq. (2.23) the commu-
tator of two observables is directly related to the sectional curvature of the sphere
S3. Indeed, assume for simplicity that the vector fields L
are orthogonal and
unit normalized in the Killing metric. Then from Eqs. (2.22), (2.23) one has the
following expression for the norm of the commutator of X̂ and Ŷ :
∥∥∥[X̂, Ŷ ]
= 4R(p). (2.33)
Here R(p) is the sectional curvature of S3 in the plane p = L(L
) which for the
considered Riemannian metric was found to be a constant equal to 1/h̄2.
Note that for L
which are orthogonal but not unit, the equation Eq. (2.23)
takes the form
∥∥∥[X̂, Ŷ ]
= 4R(p)
∥∥∥X̂
∥∥∥Ŷ
. In particular, if the norms of
X̂ and Ŷ are of order h̄ (e.g., X̂, Ŷ are the spin observables), then the norm of
the commutator [X̂, Ŷ ] is of order h̄ as well. Note that despite the fact that the
commutator [X̂, Ŷ ] is small in these units, it is of the order of the radius of the
sphere of states, making quantum effects on the sphere quite transparent.
The results obtained so far in this section were model specific. It is then im-
portant to know whether they can be generalized to the case of higher dimensional
spaces of states and of arbitrary observables. Also, what if the Hamiltonian of the
system is time-dependent? Here is a sketch of what can be done in these cases.
For any n the sphere of states S2n−1 in the space Cn is a homogeneous space
U(n)/U(n − 1), where U(n) denotes the unitary group on Cn. The Killing metric
on U(n) can be used to induce a Riemannian metric on S2n−1 via the embedding.
Namely, the 2n − 1 linearly independent generators in the Lie algebra u(n), which
belong to the orthogonal complement of a (fixed) subalgebra u(n − 1), form a sub-
space V ⊂ u(n). The one-parameter subgroups eX̂τ with X̂ in V sweep a sphere
S2n−1 and yield geodesics in the induced metric. The curvature of S2n−1 can be
then computed via equations Eqs. (2.20), (2.21) with generators in V . Furthermore,
the commutators of generators in V are related to the sectional curvature of S2n−1
by the same formula Eq. (2.33).
On the measurement problem 9
Note that there exist anti-Hermitian observables that are not in V . For example,
if n = 2 so that S3 = U(2)/U(1) = SU(2), then V is the Lie subalgebra su(2) ⊂ u(2)
and so V consists of the traceless elements of u(2). If X̂ ∈ u(2) is not traceless,
the one-parameter subgroup eX̂τ is still a geodesic in U(2). However, this geodesic
does not “stay” on the subgroup SU(2) = S3. Of course, one could still consider
the curves on S3 given by ϕt = e
X̂τϕ0, for some point ϕ0 in S
3 ⊂ C2. For any given
X̂ ∈ u(2) and all initial points ϕ0 these curves are still geodesics in the appropriate
Riemannian metric (see section 1 and Ref. [4]). However, the algebraic features of
the model change and the formulas connecting the commutators with the curvature
are different.
For the time-dependent Hamiltonians ĥ in the Hilbert space Cn the approach can
be generalized as follows. The sphere of states S2n−1 is replaced with the manifold
M = S2n−1 ×R, where R is the time line. Then, there exists a Riemannian metric
on M in which the paths (ϕt, t) with ϕt = e
−îhtϕ0 are geodesics for all ϕ0 ∈ S2n−1.
Finally, as already mentioned, in the infinite-dimensional case there still exists
a Riemannian metric for which all solutions to the Schrödinger equation with an
invertible (time-independent) Hamiltonian are geodesics. However, the algebraic
properties of the model require further investigation in this case.
2.2 Quantum Mechanics on the Projective Space of States CP 1
In physical experiments one can only determine the state of a system up to a com-
plex non-zero factor. That means that the space of physical states is the complex
projective space CPH of complex lines in the space of states. In the considered
example it is the one dimensional complex projective space CP 1. By definition,
CP 1 is the quotient C2∗/C∗, were ∗ means “take away zero”. In other words, CP 1
is the base manifold of the fibre bundle π : C2∗ −→ CP 1 with the natural projection
along the fibres C∗. By considering unit normalized states only, one obtains CP
as a quotient S3/S1. It is then the base of the fibre bundle π : S3 −→ CP 1, which
is a sub-bundle of the previous fibre bundle.
If ϕt is a path of the electron’s state on S
3 and π : S3 −→ CP 1 is the bundle
projector, then π(ϕt) is a path on the base CP
1. Since this latter path represents
what can be measured in experiments, it is important to obtain an explicit formula
for π. For this consider a point ϕ =
on S3 and let {ϕ} be the complex
line formed by vectors λϕ, λ ∈ C. Provided ϕ1 6= 0, there is a unique point of
intersection of the line with the affine plane in C2 formed by vectors
, ξ ∈ C.
Namely, by setting
, (2.34)
10 Alexey A. Kryukov
one obtains
. (2.35)
The map ρ = ϕ −→ ξ provides a coordinate chart on CP 1 which identifies CP 1
without a point (complex line through
) with the set C of complex numbers.
The affine plane of vectors
form a subspace in the Lie algebra su(2). The
algebra su(2) itself has been identified earlier with the Euclidean space R3 of vectors
kiσ̂k. The stereographic projection then identifies the unit sphere S
the origin of R3 with the above plane C plus a point, i.e., with CP 1 itself.
The relationship of the coordinate ξ in the plane C with coordinates (x1 =
x, x2 = y, x3 = z) of the corresponding point on the sphere S2 is given by
x+ iy
. (2.36)
Solving this for x, y and z and using Eq. (2.35), one obtains:
x = ϕ1ϕ2 + ϕ1ϕ2, (2.37)
y = i(ϕ1ϕ2 − ϕ1ϕ2), (2.38)
z = ϕ2ϕ2 − ϕ1ϕ1. (2.39)
The resulting map π : S3 −→ S2 given by (ϕ1, ϕ2) −→ (x, y, z) is the needed
projection on the space of physical states.
The equation for the integral curve Eq. (2.4) can be simplified by choosing the
coordinate axes properly. In particular, one can always assume that the Z-axis is
parallel to the magnetic field B. In this case σ̂ ·B = σ̂3B and Eq. (2.4) simplifies to
eiωtϕ0+
e−iωtϕ0−
, (2.40)
where ω = µB
and the initial state ϕ0 is equal to
. Recall that the speed
of evolution of the electron along S3 was given by Eq. (2.7). Let us find the speed
of the projection of this evolution on the space of physical states S2 = CP 1.
For this recall that the Killing metric on su(2) coincides with the Euclidean metric.
The embedding of S2 into su(2) = R3 induces the familiar metric on S2. Such a
metric also coincides with the famous Fubini-Study metric on CP 1 identified with
S2. Since this metric is just a restriction of the Euclidean metric on R3, one has(
. Using Eqs. (2.37)-(2.40), one obtains then
= 4ω|ϕ0+||ϕ0−| =
|ϕ0+||ϕ0−|. (2.41)
On the measurement problem 11
If θ is the angle between the Z-axis and the vector (x, y, z), then sin2 θ = 1− z2 and
one finds with the help of Eq. (2.39):
sin θ. (2.42)
In particular, if the initial state is an eigenstate of σz so that θ = 0 or π, then
the speed ds
vanishes and the evolution of the electron is projectively trivial. In
other words, the eigenstates of the Hamiltonian ĥ = −µσ̂ ·B are zeros of the “push-
forward” vector field dπ(hϕ) on CP
1. Here dπ is the differential of the map π and
µσ̂ ·Bϕ as before.
Recall that the integral curves ϕt given by Eq. (2.4) are geodesics on S
3. At
the same time, as one can see from Eqs. (2.37)-(2.40), the projected curves {ϕt}
are not in general geodesics in the induced metric. Moreover, there does not exist
a Riemannian metric on CP 1 = S2 in which the projection by π of an arbitrary
geodesic on S3 yields a geodesic on S2. Indeed, the great circles of S2 (parametrized
by arc length) are geodesics in the Fubini-Study metric. At the same time, they are
projections of geodesics on S3. The latter fact is obvious for the equator of S2 if
one chooses |ϕ0+| = |ϕ0−| in Eqs. (2.37)-(2.40). For any other great circle this fact
is verified by a change in direction of the Z-axis. But the condition that the great
circles are geodesics fixes the metric on S2 up to a constant factor. This, together
with the above observation that the projection {ϕt} of a geodesic may not be a
geodesic itself, proves the claim.
Suppose, as it is advocated here, that evolution by geodesics is a valid principle
of quantum dynamics. Then the obtained result suggests that the space of states
S3 may be “more physical” than the projective space CP 1 of “physical states”. In
general terms, the actual, “easy to describe” physics may be happening on the space
of states of a quantum system. However, our experiments can so far access only the
projection of physical processes into the space of physical states. The projection of
a process may look much less “natural”, then the original process on the space of
states.
It was verified earlier that the commutators of spin observables are related to
geometry of the space of states S3. Later on in the paper the geometry of a measure-
ment process will be discussed. It is then important to know that the probability
of transition from state ϕ to state ψ in a measurement also has ties to geometry.
Namely, this probability depends only on the distance between {ϕ} and {ψ} in the
Fubini-Study metric on the projective space of states (see Ref. [2]).
This result is immediate in the model under consideration. In fact, assume that
the state ψ is the spin-up eigenstate of σ̂z (one can always assure this by a proper
choice of coordinate axes). Then, according to Eq. (2.37), one has
z = |ϕ1|2 − |ϕ2|2. (2.43)
12 Alexey A. Kryukov
Using the normalization condition |ϕ1|2 + |ϕ2|2 = 1, one obtains
|ϕ1|2 =
. (2.44)
If θ ∈ [0, π] is the angle between the vectors representing {ϕ} and {ψ} on S2, then
z = cos θ and the equation Eq. (2.44) takes the form
P (θ) = cos2
. (2.45)
Here P (θ) = |(ψ,ϕ)|2 = |ϕ1|2 denotes the probability of transition. Notice that θ
is the length of the arc of a great circle through {ϕ} and {ψ}. In other words, it
is a geodesic distance between the points {ϕ} and {ψ} on S2 in the Fubini-Study
metric. In particular, the probability P (θ) of transition between two states depends
only on the distance between them.
Furthermore, the uncertainty principle for the spin observables also has a simple
geometrical interpretation. Since uncertainties of observables do not depend on the
overall phase of a state, the geometry underlying the uncertainty principle is, once
again, projective. The uncertainty principle for non-commuting observables X̂, Ŷ
can be written in the form
∆X∆Y ≥
∣∣∣(ϕ, [X̂, Ŷ ]ϕ)
∣∣∣ , (2.46)
where ∆X2 = (ϕ, X̂2ϕ) − (ϕ, X̂ϕ)2 and similarly for ∆Y 2. Assuming X̂, Ŷ are the
components ŝx, ŝy of the spin observable ŝ =
σ̂, one has
∆sx∆sy ≥
|(ϕ, ŝzϕ)| . (2.47)
Let us connect this inequality with geometry of the projective space S2 = CP 1.
In light of the geometric interpretation of the commutators of observables Eq. (2.33)
and the equation Eq. (2.46), the existence of such a connection is not surprising.
Indeed, from the geometry of the sphere S2 furnished with the Fubini-Study (i.e.,
the usual!) metric, for any point (x, y, z) on the sphere one has:
(y2 + z2)(x2 + z2) ≥ z2. (2.48)
The inequality simply says that the product of distances from a point (x, y, z) on the
sphere S2 ⊂ R3 to the X and Y -axes is at least |z|. This equation is the geometric
form of the uncertainty principle. Indeed, from Eqs. (2.37)-(2.39) it follows that
(ϕ, σ̂xϕ) = x, (ϕ, σ̂yϕ) = y, (ϕ, σ̂zϕ) = z, (2.49)
where x, y, z are coordinates of the point of S2 representing the state ϕ. In addittion,
σ̂2x = σ̂
y = σ̂
z = I and so
∆σ2x = 1−x2 = y2+z2, ∆σ2y = 1−y2 = x2+z2, ∆σ2z = 1−z2 = x2+y2. (2.50)
On the measurement problem 13
With the use of Eqs. (2.49), (2.50), the inequality Eq. (2.48) is now equivalent to
the uncertainty principle Eq. (2.47).
Note that using the angles θx, θy, θz between the coordinate axes and the vector
x = (x, y, z) in R3, one can write the equations Eqs. (2.50) in the form
∆σx = sin θx ∆σy = sin θy, ∆σz = sin θz. (2.51)
In particular, the uncertainty ∆E in energy for the electron in the state ϕ in the
model takes the form
∆E = µB sin θ, (2.52)
where θ is the angle between the vectors B and x. Recall that θ is a geodesic
distance between the points {ϕ} and {ψ} on S2. Therefore, the uncertainty ∆E is
the largest, when {ϕ} is furthest away from the eigenstates, i.e. when θ = π
the other hand, ∆E vanishes at the eigenstates of ĥ.
The latter statement can be generalized as follows. Let H = Cn and let
Â : H −→ H be an observable with a simple spectrum λ1 < λ2 < ... < λn and
eigenfunctions ϕ1, ϕ2, ..., ϕn. Let {ϕt} be a geodesic through the states {ϕk}, {ϕl}
in the Fubini-Study metric on the projective space of states CPH . Then for any
point {ϕ} on the geodesic the variance ∆A2 = (ϕ, Â2ϕ)− (ϕ, Âϕ)2 is an increasing
function of the distance from {ϕ} to the pair of eigenstates {ϕk}, {ϕl} in CPH . The
latter distance is simply the shortest of the distances from {ϕ} to the states {ϕk},
{ϕl}.
In fact, because the eigenstates are orthogonal, the probability of transition from
a state {ϕ} on the geodesic to a state {ϕi} with i 6= k, l is equal to zero. In particular,
{ϕ} = {ckϕk + clϕl} for some coefficients ck, cl. Therefore,
∆A2 = |ck|2λ2k +
1− |ck|2
λ2l −
|ck|2λk + (1− |ck|2)λl
. (2.53)
If {ϕ} coincides with {ϕk} so that |ck| = 1, then ∆A vanishes. On the other hand,
as |ck| decreases, ∆A increases until |ck| becomes equal to 1√2 . By Eq. (2.45) this
means that ∆A increases with the distance from {ϕ} to {ϕk} until {ϕ} becomes
equally distant from {ϕk} and {ϕl}.
It is important to note that the “geometric” probability of transition formula Eq.
(2.45) is valid in an arbitrary space of states H = Cn. Also, as already discussed,
the relationship Eq. (2.33) between the commutators and the curvature holds true
on the sphere of states S2n−1 for observables in the subspace V ⊂ u(n) (see the
end of section 2.1). Because of that, the uncertainty principle Eq. (2.46) can be
still interpreted geometrically. The infinite-dimensional case requires a different
approach and needs further analysis.
14 Alexey A. Kryukov
3 TENSOR PROPERTIES OF EQUATIONS IN THE
MODEL
In the developed geometrical formulation of the model, the space of states is a
manifold furnished with a Riemannian metric which defines quantum dynamics on
the space. It is well known that differential geometry of manifolds admits two
equivalent formulations: local coordinate and coordinate free. Let us discuss the
role played by both formulations in the model.
Consider a pair X̂, Ŷ of elements of the Lie algebra su(2) and the corresponding
pair Xϕ = X̂ϕ, Yϕ = Ŷ ϕ of the associated vector fields. By direct computation one
sees that
[Xϕ, Yϕ] = −[X̂, Ŷ ]ϕ, (3.1)
where [Xϕ, Yϕ] is the Lie bracket of the vector fields. Recall that the integral curves
of non-commuting vector fields cannot form a coordinate grid on the manifold. In
particular, the integral curves of vector fields sxϕ, syϕ, szϕ associated with the spin
observables ŝx, ŝy, ŝz do not form a coordinate grid on S
3. By the above, these
integral curves are geodesics in the Killing metric on S3. The fact that they do not
form a coordinate grid is then a direct consequence of the curvature of S3. Instead,
the fields sxϕ, syϕ, szϕ form a local (non-coordinate) basis at every point of S
There are, of course, many ways of choosing coordinates on S3. One natural
choice is the normal coordinate system given on a neighborhood of any point by
the exponential map. If {ek} is a basis on the tangent space TϕS3 at ϕ ∈ S3 and
kek is a tangent vector, then the equation of geodesic ϕt through ϕ in the
direction of A in the normal coordinates ϕ1, ϕ2, ϕ3 is linear:
ϕkt = a
kt, k = 1, 2, 3. (3.2)
The evolution of spin state ϕ in time could be thought of as a motion along the
manifold S3×R with R being the time axis. The direct product of the Killing metric
on SU(2) and the usual Euclidean metric on R makes S3 × R into a Riemannian
manifold. If ϕt is a geodesic on S
3, then (ϕt, t) will be a geodesic on S
3 × R.
Notice that for any evolution ϕt of the state along S
3 ×R one can find a co-moving
coordinate system on S3×R in which (ϕt, t) = (ϕ0, t), i.e., the state is at rest. Such
a co-moving system is directly related to the well known Heisenberg representation.
The coordinates typically used on the projective space of physical states CP 1
are homogeneous (ϕ1, ϕ2) and inhomogeneous ξ =
, η =
coordinates. For
example, the Fubini-Study metric is usually expressed in terms of these coordinates.
Other coordinate systems may be useful in applications. In particular, according
to Eq. (2.49), the expectation values of Pauli matrices for a system in a state ϕ
coincide with the x, y and z coordinates of the point {ϕ} in S2. Therefore, these
expectation values can be identified with local coordinates on CP 1. This fact was
used in section 2 to describe the motion of state along CP 1.
On the measurement problem 15
The group SU(2) acting on the space C2 is the symmetry group of the theory.
In particular, the group SU(2) “extends” to act on tensor algebra over C2 and the
Schrödinger equation is a tensor equation. For any fixed time t the Schrödinger
evolution operator Û(t, 0) given by Eq. (2.4) is an active realization of SU(2)-
transformations on C2. The corresponding passive realization consists in a unitary
change of basis on C2.
One could consider instead the sphere S3 ⊂ C2 as a base manifold for tensor
bundles and make the group SU(2) act locally on tensor products of spaces tangent
and cotangent to S3. For each tensor type this gives a subbundle of the correspond-
ing tensor bundle over C2. The Schrödinger equation is then a tensor equation with
terms which are vector fields on S3. In fact, it is the equation for integral curves of
the vector field hϕ = − ih̄ ĥϕ on S
3. Alternatively, it is the equation of geodesics in
the Killing metric on S3.
The notion of symmetry in QM is usually understood as an invariance of the
Hamiltonian of the system under a symmetry transformation. In this case the
Hamiltonian commutes with the transformation and the generator of transformation
becomes a constant of motion. Although this is certainly true for rotations about
the field direction in the model, such a restricted understanding of symmetry is not
suitable for this Letter.
Mathematically, the Schrödinger equation hϕ = − ih̄ ĥϕ in the model is written in
a specific basis on the space of states C2. Under a unitary transformation Û ∈ SU(2)
of the basis the Schrödinger equation behaves as a vector equation. In particular,
the right hand side of the equation becomes equal to − i
Û−1ĥϕ. Since coordinates
of points ϕ also change to become ψ = Û−1ϕ, the right hand side of the equation
takes the form
Û−1ĥÛψ. (3.3)
Note that Û−1 and Û in Eq. (3.3) act on different spaces! Namely, Û−1 acts on
the tangent space TϕC
2, while Û acts on C2 itself. It is common, however, to
identify vector spaces with the spaces tangent to them. By following this practice,
one sees that under a change of basis the Hamiltonian is transformed by ĥ −→
Û−1ĥÛ . In other words, it transforms as a tensor of rank (1, 1). If in addition ĥ
and Û commute, then (and only then!) the Hamiltonian is invariant and the usual
conserved quantities exist in accordance with the Noether’s theorem.
A particular choice of an orthonormal basis on the space of states C2 (alterna-
tively, on S3 or CP 1) has an affect on results of observations expressed in the basis.
The reason for that is clear: which state is a spin-up state, for example, depends
on the basis in C2. This is quite analogous to dependency of the state of rest in
classical mechanics on a choice of reference system.
Alternatively, under the identification Eq. (2.8) of space C2 with a space of 2×2-
matrices the choice of an orthonormal basis in C2 dictates the choice of basis in the
Lie algebra su(2), i.e., the choice of sigma-matrices. A unitary transformation of
16 Alexey A. Kryukov
sigma-matrices changes their eigenvectors, thus affecting the results of observations
expressed in terms of these eigenvectors.
The above can be also rephrased in terms of the geometry of the projective
space CP 1 = S2 ⊂ R3. Namely, a unitary transformation of basis in C2 induces an
orthogonal transformation of basis on R3. But the choice of a basis in R3 determines
the result of measurement of the Z-component of electron’s spin.
One can, therefore, conclude that a choice of coordinates on manifolds C2, S3
and CP 1 has a physical meaning similar to the choice of a reference system on the
classical space and should not be neglected.
On the other hand, physical laws must be, of course, coordinate independent.
In the considered case this is assured by the geometric nature of the model. In
particular, the Schrödinger equation Eq. (1.1) can be written in a form independent
of an orthonormal basis on the space of spin states C2, i.e., in a vector form. In this
case a state is just a vector Φ of C2 rather than a column
of components in a
particular orthonormal basis on C2. The Hamiltonian is an operator ĥ rather than
a matrix. The equation Eq. (1.1) then becomes a vector equation on the manifold
S3 written in a coordinate-free form:
= − i
ĥΦt. (3.4)
Note that the eigenvalue problems for observables in the model are also tensor
equations. In particular, the eigenvalue problem for the Hamiltonian can be written
in a coordinate free form:
ĥΦ = λΦ. (3.5)
The tensor character of equations of quantum theory in the model signifies that
the principle of relativity holds true on the space of states. This means, first of
all, that both, the active transformations on the space of states and the passive
transformations of coordinates on the space are available. In particular, there exist
various physically distinguishable reference systems on the space of states (say, dif-
ferent bases on C2). Most importantly, the validity of principle of relativity in the
theory means that the equations of the theory are the same when written in any
such reference system, that is, they are tensor equations.
One may doubt the significance of such a principle in the example. After all,
the SU(2) symmetry in the model has been known for years. Why would such
a “relativistic” view of this symmetry be useful? Note however, that the quantum
dynamics in the advocated approach takes place on the space of states. In particular,
the evolution of a quantum system is the motion along a geodesic on the (curved)
space of states. Because of that the notions of a reference system, of passive and
active transformations, of tensor equations, as well as other differential-geometric
notions on the space of states, become physically meaningful. This meaning is very
On the measurement problem 17
much in line with the meaning of similar notions in special and general relativity or
in the theory of gauge fields.
For instance, a change in direction of the magnetic field B induces an active
transformation on the space of states. This transformation has physically measur-
able consequences: it moves geodesics on the space of states and hence changes the
evolution of electron’s state. At the same time, the change in B can be compensated
by a passive transformation. Indeed, one can choose an orthonormal basis on C2 so
that the components of B in the corresponding basis in R3 remain the same. The
equation of the new electron’s path in this basis coincides then with the original
Most certainly, the above principle is different from the principle of relativity in
space-time. In fact, it deals with tensor properties of equations on a Hilbert space
of states, rather than on space-time. At the same time, it has the same kind of
underlying mathematics and the same spirit as the ordinary principle of relativity.
The principle is in fact a particular instance of the principle of functional relativity
introduced in Ref. [4].
As discussed, the Hamiltonian in the model is not in general invariant under
unitary transformations. In particular, an active unitary transformation associated
with a change in direction of the magnetic field B produces a new Hamiltonian. In
other words, the Hamiltonian is not a scalar in the tensor approach to the model.
On the other hand, due to geometric nature of the model, one can easily identify
the most important scalars (or invariants), forming the “bone structure” of the
theory. One such invariant is the distance between any two points on spaces of
states C2, S3 and CP 1 furnished with the above discussed metrics. Another one
is the speed of quantum evolution in S3 in a given magnetic field given by Eq.
(2.7). Yet another one is the scalar curvature of S3 or CP 1. This curvature can be
expressed in terms of the sectional curvature of S3 which was found to be 1/h̄2.
Let us remark that tensor character of the theory allows one to extend the origi-
nal symmetry group SU(2) to the group GL(2, C) of general linear transformations
acting on fibers of the tangent bundle over S3. Moreover, Schrödinger dynamics on
the space of states S3 can be formulated in a way invariant under general coordi-
nate transformations on S3. This follows at once from the fact that the Schrödinger
equation in the model is the equation of geodesics on the Riemannian manifold S3.
As such, this equation is meaningful in arbitrary coordinates on the manifold.
Finally, let us comment on a possible argument against the advocated geometric
approach. In the model considered here the Riemannian metric on the sphere S3 has
turned the sphere into a manifold of constant sectional curvature. This allowed us
to relate the curvature of the metric with the Planck’s constant. However, in general
the sectional curvature of the Riemannian metric defined by Eq. (1.3) will not be
a constant. Even when it is, there is no reason for this constant to be the same as
in the model under discussion. It seems therefore that by making the Riemannian
metric depend on the Hamiltonian of the system, one shall in general loose the
18 Alexey A. Kryukov
relationship between the curvature and the Planck’s constant!
To reply to this argument recall that the model considered here is non-relativistic
(in the usual sense). In particular, the equation Eq. (2.1) is a special case of the
Pauli equation. The latter equation is well known to be the non-relativistic limit of
the Dirac equation for electron in electromagnetic field. The Dirac equation can be
written in the form
= cσ̂ ·
p̂− e
, (3.6)
where ϕ̃, χ̃ are two-component spinors, (φ,A) is the 4-potential of the field, p̂ is the
momentum operator and e is the electron’s charge (see, for example, Ref. [8]). The
largest term in Eq. (3.6) is the one containing the mc2 factor. By substituting
(3.7)
into Eq. (3.6) one recovers in the standard way the Pauli equation for the spin state
ϕ with values in C2.
Assume that the 4-potential in Eq. (3.6) describes a weak homogeneous magnetic
field B and let ĥD be the corresponding Hamiltonian. Consider the metric GD given
by Eq. (1.3) with the Hamiltonian ĥD. Then solutions to the Dirac equation for
electron in the field B are geodesics in this metric. Note that the Hamiltonian ĥD
has in the non-relativistic limit the form mc2 + ĥ, where ĥ is the non-relativistic
Hamiltonian used in Eq. (2.1). Accordingly, the metric GD can be written in
the form (h̄/mc2)2(I + ǫ), where ǫ is a small correction due to the Hamiltonian ĥ
and I is the identity. It follows that the sectional curvature of GD consists of the
main term of the order (mc2/h̄)2 and a small correction due to ĥ. Since the main
term is constant (i.e., it does not depend on the fields), the advocated geometric
interpretation of the curvature remains possible. However, a more careful analysis
of the situation requires a “functional relativistic” formulation of the problem and
will be discussed elsewhere.
4 THE PROCESS OF MEASUREMENT
In the model under consideration the Schrödinger equation is the equation of geodesics
on the space of states S3 furnished with the Killing metric. That means that the
dynamics in the theory takes place on the Hilbert space of states rather than on the
classical space. In this and the following sections it will be argued that the space
of states is also the most appropriate background for tackling problems related to
quantum measurement. In particular, the process of collapse of a state can be re-
garded as a geodesic motion in the space of states with the metric “skewed” by the
measuring device.
On the measurement problem 19
Consider a pair of spin-1/2 particles. In QM the most general spin state of such
a pair has a form
i,j=±
cijϕiψj , (4.1)
where ϕ+, ψ+ are spin-up, and ϕ−, ψ− spin-down states of the particles. The Hilbert
space of states having such a form is the tensor product C2⊗C2. The unit normalized
states form a sphere S7 in this four-dimensional complex space.
Whenever Ψ is not a product of states of the particles, the state of the pair is
called entangled. It is well known that, when the particles are microscopic (i.e., suf-
ficiently small in mass and size), the entangled states do indeed exist. By assuming
the universal validity of QM, one concludes that the entangled states can be also
prepared when one of the particles is replaced with a macroscopic measuring device,
designed to measure spin of the second particle. In this case the total state of the
system has the form Ψ = aϕ+ψ+ + bϕ−ψ− where ψ± represent states of the device,
corresponding to spin-up and spin-down outcomes of measurement. However, unlike
the case of microscopic objects, the entangled states with macroscopic objects have
never been observed in experiments. The phenomenon of decoherence does not help
resolve this problem because the mixtures of states of macroscopic objects have not
been observed in experiments either.
Recall that in the classical physics the motion of a pair of interacting particles
on a manifold can be thought of as a motion of a point in a higher dimensional
configuration space. Suppose in particular that particles of massesm andM interact
gravitationally and move in the space R3 in accordance with the Newton’s Second
Law. Then the motion of the pair is represented by a trajectory in configuration
space R6. However, if M ≫ m, the motion simplifies and can be thought of as a
motion of the particle of mass m in the field created by the particle of mass M . In
this case the configuration space R6 of the pair is “effectively reduced” to the space
R3 and the field on R3 created by the heavier particle.
An analogous “reduction” of configuration space is implicitly present in the
unitary QM whenever the influence of a “macroscopic surrounding” of a quantum
system is accounted for by an appropriate choice of potential in the Schrödinger
equation.
Let us explore the idea that a similar approach can be applied to the process
of measurement in the model. Namely, assume that the motion of the total state
function of the electron and the measuring device during their interaction can be
effectively replaced with the motion of electron’s state function in S3 under the
influence of a physical field on S3 created by the measuring device. This means
that, in some sense, the state function of the device does not change much as a
result of interaction.
One immediate objection is that the observed states of the device are orthogonal
and so the state cannot change “just a little”. Without addressing this problem in
detail, let us point out that the metric on the space of states of the device may be
20 Alexey A. Kryukov
“skewed”. As a result, two different position eigenstates of a pointer may become
very close in this metric. For instance, let H be the Hilbert space obtained by
completion of a space of ordinary functions on R3 with respect to the inner product
(ϕ,ψ)H =
e−(x−y)
ϕ(x)ψ(y)d3xd3y. (4.2)
Such a space contains in general the eigenstates of position operator, i.e., the delta-
functions δ(x − a). Moreover, two different position eigenstates δ(x − a), δ(x − b)
with ‖a− b‖R3 ≪ 1 are close in the metric Eq. (4.2) (see Ref. [4]). If, in particular,
δ(x − a), δ(x − b) are the eigenstates of a pointer, then the fact that theses states
are close in H can imply that the state of the pointer does not change much in the
process of interaction with a measured microscopic system.
This argument does not prove, of course, that the proposed “reduction formal-
ism” can be consistently implemented into the theory. To validate the formalism
one must demonstrate that all imaginable measurements in QM can be modeled (at
least in terms of their outcomes) by a field on the Hilbert space of states of the mea-
sured system. In what follows such a demonstration will be presented in the case of
a finite dimensional space of states and a time-independent observable. Namely, in
this case a specific working model of measurement based on a perturbation of the
metric on the space of states of the system will be constructed. Moreover, the effec-
tiveness and the scope of the proposed method suggest that it can be successfully
applied in general.
For guidance in modeling the process of measurement let us return to the ex-
ample under consideration. Suppose that the device in the example measures the
component of electron’s spin in the direction of magnetic field. Equivalently, since
the Hamiltonian is given by ĥ = −µσ̂ · B, the device can be designed to measure
the electron’s energy. Without loss of generality one may assume that the field is
directed along the Z-axis. Then the two eigenstates {ψ1}, {ψ2} of ĥ in CP 1 are
positioned on the Z-axis at the poles of the sphere S2 = CP 1. Recall that these
eigenstates are zeros of the vector field hϕ = − ih̄ ĥϕ projected on CP
1. In particu-
lar, the Schrödinger evolution of ψ1, ψ2 is projectively trivial. So, as a result of the
interaction between the electron and the device, the original circular motion of the
electron’s state along a parallel on S2 = CP 1 is changed to the state of rest at one
of the poles on the sphere. The following hypothesis, which will be clarified and
exemplified later on, seems to be in order:
(H1) The measuring device creates a physical field on the sphere of states with
sources at the eigenstates of the measured observable. This field is capable of
driving the electron’s state toward one of the eigenstates.
Note that in the position measuring experiments a measuring device is a system
of counters distributed in space at the eigenstates of the position observable. The
counters are indeed sources of interaction between the particle and the apparatus.
On the measurement problem 21
The points in the classical space where the measured particle can be found, can be
identified with the position eigenstates of the particle (see Refs. [4], [6]). Since the
counters are also located at these points, one can identify them with sources in the
space of states of the particle. Finally, because the sources are capable of catching
the particles, the (loosely stated) hypothesis (H1) is satisfied. This gives one the
hope that a specific form of the hypothesis can be, in fact, realized.
What could be the nature of the field postulated in the hypothesis? Recall that
the Riemannian metric in the model is dynamical, i.e., it drives the evolution of the
electron’s state. Suppose that the field is nothing but a perturbation of the metric
on the sphere of states S3, induced by the presence of the measuring device. The
evolution of the electron’s state during the measurement is then the motion along a
geodesic in the perturbed metric. Note that the metric on the total space S3 and not
only on the base space CP 1 = S2 must be perturbed. In fact, as already discussed,
the projection {ϕt} of a Schrödinger evolution ϕt is not in general a geodesic.
The above approach is certainly attractive, in particular, it does not require any
ad hoc features in the theory. Furthermore, the approach can be easily implemented
by an appropriate “denting” of the sphere of states of the system. Namely, as shown
below, by perturbing the metric on the sphere S3 one can “redirect” the evolution
of electron’s state so that the state would become stationary.
For a greater generality, assume that the Hilbert space H of states of the system
is n-dimensional and let (ϕ1, ..., ϕn) be the usual coordinates on H = Cn. Let the
Riemannian metric on H to be of the form
gik = η
2δik, (4.3)
where η = η(ϕ) is a function on H. The equation of geodesics in this metric can be
obtained in the usual way by variation of the length functional on paths ϕt. Let s
be the arc length parameter and let τ be a parameter defined by dτ = ds
. Then the
equation of geodesics in the metric Eq. (4.3) on H can be written in the form
∇η2, (4.4)
where ϕτ is identified with ϕt(τ). A similar equation in R
3 is well known in geomet-
rical optics where it describes propagation of rays in a media with refractive index
η. The equation Eq. (4.4) is also similar to the Newton equation of motion for a
unit mass in the field U = −η2/2.
The form of equation Eq. (4.4) makes it easy to see that for any sufficiently
smooth path ϕτ there exists a function η such that the equation is satisfied, at
least on a neighborhood of ϕ0 = ϕτ |τ=0. That is, ϕτ is a geodesic through ϕ0 in
the Riemannian metric Eq. (4.3) on H, at least for small values of τ > 0. That
also means that an arbitrary sufficiently smooth path ϕτ with values on the sphere
SH ⊂ H is a geodesic through ϕ0 = ϕτ |τ=0 in an appropriate Riemannian metric
on SH , at least for small τ > 0. As a side remark, various global results of this kind
22 Alexey A. Kryukov
can be obtained by applying the methods of geometrical optics to Eq. (4.4) (see
Ref. [7] for a review of geometrical optics).
Consider now an arbitrary non-stationary Schrödinger evolution ϕt on S
H driven
by an invertible time-independent Hamiltonian ĥ. Let SH be furnished with a
Riemannian metric GRϕ in which the evolution is a motion along a geodesic. It
is known that such a Riemannian metric on SH exists. Pick a moment of time
t = a and an eigenstate ψ ∈ SH of the Hamiltonian ĥ. Consider the geodesic χt
connecting ϕa and ψ. By Eq. (4.4), there exists a perturbation of the metric GRϕ
on a small neighborhood of ϕa which transforms the geodesic ϕt into the geodesic
χt. Note that one could similarly perturb the metric on a small neighborhood of ψ
to transform χt into the stationary geodesics through ψ.
The provided perturbation of the metric takes place on the sphere of state SH =
S2n−1. What could one say about the metric and the motion on the projective space
CPn−1 = S2n−1/S1? To answer, consider the Riemannian metric on Cn∗ (i.e., C
without the origin), defined for all ϕ ∈ Cn∗ and all pairs of vectors ξ, η ∈ TϕCn∗ by
GRϕ(ξ, η) =
ĥ−2ξ, η
||ϕ||2Cn
. (4.5)
This metric, being restricted to the sphere S2n−1, is a particular case of the metric
Eq. (1.3). Solutions to the Schrödinger equation are geodesics in the metric Eq.
(4.5) on Cn∗ (and in the induced metric on the sphere, see Ref. [4]).
Notice that the multiplication map λ : ϕ −→ λϕ with λ ∈ C∗ is an isometry of
the metric Eq. (4.5), that is, GRλϕ(dλξ, dλη) = GRϕ(ξ, η). This is clear because
dλ = λ by linearity of the map and Re(λξ, λη)Cn/||λϕ||2Cn = Re(ξ, η)Cn/||ϕ||2Cn .
Because of that, the metric Eq. (4.5) “projects down” to CPn, giving a metric
on the projective space. More precisely, the metric Eq. (4.5) is induced by the
projection of Cn∗ onto CP
n−1, furnished with a Riemannian metric. Provided ĥ2 is
proportional to the identity operator, the latter metric coincides with the Fubini-
Studi metric on CPn−1 (see Ref. [4]).
Note however that the multiplication by a complex number may not remain an
isometry of the perturbed metric on Cn∗ . That is, the metric on C
∗ , needed to
“redirect” the Schrödinger evolution to account for the process of measurement,
may not originate in a Riemannian metric on CPn−1. In mathematical terms, the
projection of Cn∗ with a perturbed metric onto CP
n−1 is not in general a Riemannian
submersion. In particular, such is the case for the discussed local perturbation of
the metric on Cn∗ . This suggests once again that the metric on the sphere S
the approach under investigation has a greater significance than the metric on the
projective space CPH .
This analysis demonstrates that by an appropriate “denting” of the sphere of
states SH , dimH <∞, one can locally affect geodesics on the sphere in a desirable
fashion. In particular, by perturbing the Riemannian metric GRϕ on S
H one can
alter the Schrödinger evolution of the state and drive the state toward one of the
On the measurement problem 23
eigenstates of the measured observable. It follows that the physical field in the
hypothesis (H1) can be indeed identified with a perturbation of the metric GRϕ.
Note that the resulting metric is time-independent. The electron’s state in the
construction propagates along a geodesic on the sphere of states, runs into a region
with perturbed metric and collapses.
It is important to remark that the above demonstration is only an existence
proof; it does not provide a realistic model of interaction between the system and
the device. Moreover, a particular nature of the field in the hypothesis (H1) will
not be essential in the following. The thorough analysis of this nature requires the
equations of the field and is left for the upcoming publications. The mere existence
of the field satisfying the needed properties will be sufficient for the purpose of this
Letter.
The postulated physical field may be able to drive the state to one of the eigen-
states, i.e., it may be responsible for the collapse itself. However, in such a scenario
collapse seems to be a deterministic process and the probabilistic nature of collapse
to a particular eigenstate is not explained. Recall now that in accordance with
Eq. (2.45), the probability of collapse of a given state ϕ to an eigenstate ψk of an
observable depends only on the distance θ between the states in the Fubini-Study
metric on the projective space of states. This crucial property allows one to resolve
the remaining difficulty in creating a working probabilistic model of collapse.
Indeed, suppose that the field sources in the hypothesis (H1) are not fixed at
the eigenstates ψk but fluctuate randomly about the eigenstates. In particular,
the projections of the sources fluctuate randomly about the points {ψk} on CPH .
Suppose further that fluctuations with projections of a small (in the Fubini-Study
metric) amplitude are more likely to occur. Suppose finally that if a source reaches a
small neighborhood of the state ϕ, it alters the evolution of the state and diverts it to
the corresponding eigenstate (say, by perturbing the metric on the neighborhood).
Then, the closer the state {ϕ} is to a particular eigenstate {ψk} (consequently, the
larger the modulus of the coefficient ck in the decomposition ϕ =
i ciψi is), the
more likely it becomes for the source fluctuating about ψk to reach (and collapse)
the state. At the same time, the further {ϕ} and {ψm} are (and hence, the smaller
the modulus of cm is), the less likely it becomes for the source fluctuating about ψm
to reach the state. In such a way, the competition between the sources can lead to
the standard Born rule for the probability of collapse.
To prove the latter claim, let us first of all make the above assumptions precise.
To keep the analysis simple, the assumptions will refer to the fibre bundle π : S3 −→
S2 corresponding to the model under discussion. However, it will be clear that the
measuring process on any fibre bundle π : S2n−1 −→ CPn−1 and for any time-
independent observable on the corresponding space of states can be treated in the
same way.
Let θ ∈ (−π, π] and α ∈ (−π/2, π/2] be the angular coordinates on the projective
space CP 1 = S2. Here the coordinate curves α = α0 yield great circles (pairs of
24 Alexey A. Kryukov
meridians) through the poles θ = 0, θ = π of S2 and the curves θ = θ0 yield half
the parallels on S2. As before, the eigenstates {ψ1}, {ψ2} are located at the poles
of S2. Let β ∈ (−π, π] be the phase of a state on the sphere S3. Then the triple
(θ, α, β) form a coordinate system on S3. In terms of these coordinates the following
hypothesis is now accepted:
(H2a) Fluctuations of each source along the sphere of states S3 can be described by
a three-dimensional stochastic process (θt, αt, βt). For instance, consider the
source associated with the eigenstate at θ = 0. For any t = t0 the random
variables θt0 , αt0 , βt0 describing the source are independent. For any t = t0
the probability density of the random variable θt0 ∈ (−π, π] is equal to 1πcos
The random variables αt0 , βt0 are uniformly distributed. The mean function
of each process is zero. For any two times t1, t2, t1 6= t2, the random variables
θt1 , θt2 are practically statistically independent, so that the stochastic process
θt is uncorrelated in time. In other words, θt is ideally a white noise process.
The same is true about the processes αt, βt. The stochastic processes describing
different sources are independent.
(H2b) A source at a point ϕ ∈ S3 with coordinates (θ, α, β) may be identified
with a perturbation of the metric on a small neighborhood U ⊂ S3 of the
point. If at some time t the U -neighborhood of a particular source contains the
electron’s state, the perturbation of the metric alters the evolution of the state
and collapses it to the corresponding eigenstate.
Is there a realization of the hypothesis? It was already verified that (H2b) can be
realized for any Schrödinger evolution ϕt by a “lensing” effect, i.e., by redirecting ϕt
toward the eigenstate. Also, the white noise process postulated in (H2a) certainly
exists as a mathematical idealization. Moreover, the processes of this kind are
common in physics. Probably the most appropriate example is the thermal noise,
i.e., the random process describing the electric current created by the thermal motion
of electrons inside a conductor.
Could the random fluctuations of the sources in (H2a) be of a similar origin?
During a measurement the measuring device interacts with the measured system. At
the same time, the molecules (atoms, particles) of the device experience a random
thermal motion. In a general (non-stationary) case fluctuations of molecules result
in fluctuations of their states on the space of states. So the main new assumption
made is that the interaction between the measured system and the device also takes
place on the space of states of the system rather than on the classical space alone.
Fluctuations of states of the molecules are then associated with fluctuations of the
field sources along the space of states leading to the postulated stochastic process.
What is the probability dP1 of collapse of a state ϕt to a particular eigenstate
ψ1 at some specific time t = t0 in the hypothesis? Such a probability is equal to the
probability for the U -neighborhood of the corresponding source to contain the state
On the measurement problem 25
at this time. Let (θ0, α0, β0) be coordinates of ϕt0 in the chosen coordinate system
on S3. If U is sufficiently small, the change in the probability density of θt0 across
U can be neglected, and, therefore,
dP1 =
dV. (4.6)
Here dV is the volume of U which in this simplest case is identified with dθdαdβ for
some fixed values of the differentials. 1 According to the hypothesis, the random
variables describing the positions of different sources at t = t0 are independent.
Therefore, the probability dP2 of collapse of the state ϕt0 to another eigenstate ψ2
can be computed in the same way. Finally, since the stochastic processes describing
the sources are uncorrelated in time, the probability for the U -neighborhood of a
source to contain any particular point ϕ0 is not affected by the previous history of
the source. In particular, the probability rule Eq. (4.6) is universally valid.
On the other hand, according to Eq. (2.45), the expression
|c1|2 = cos2
(4.7)
represents the standard probability of transition from the state {ϕt0} to the state
{ψ1}, provided θ0 is the distance between the states in the Fubini-Study metric and
c1 is the coefficient of ψ1 in the decomposition of ϕt0 . Clearly, the distance θ0 in
the formula Eq. (4.7) can be replaced with the the angle θ0 between the states,
explaining the chosen notation. Of course, a similar formula holds true for the
coefficient c2 of ψ2. It follows that the ratio dP1/dP2 coincides with |c1|2/|c2|2. The
conclusion is that the postulated hypothesis yields the Born rule for the probability
of collapse as was claimed.
The “single-push” process of collapse of the state ϕt to an eigenstate can be
replaced with a more elaborate stochastic process. Each encounter with a source in
this process results in a decrease in the distance θ ∈ [0, π] between the state {ϕt}
and the corresponding eigenstate {ψk} by a certain value δ. Between the encounters
the state undergoes the ordinary Schrödinger evolution. Assume for simplicity that
the frequency of encounters is sufficiently high. In this case one can neglect the
Schrödinger evolution of the state during the measurement. The stochastic process
of collapse can be then defined as a finite, time-homogeneous Markov chain with
absorbing boundaries θ = 0 and θ = π and with the number of states equal to
π/δ + 1. The transition matrix for the process can be found via simple formulas
1The volume element for the sphere S3 with the usual metric in the chosen coordinates is
dSV =
sin θ
cos θ
dθdαdβ. It would be more appropriate to associate dSV with the volume of
U . Moreover, the expression sin θ
cos θ
is the derivative of cos2 θ
. This leads one to interesting
models in which at any t = t0 the random variable θt0 is uniformly distributed on (−π, π] and
the coefficient cos2 θ
appears in the (complementary) cumulative distribution function due to the
factor sin θ
cos θ
in the volume element dSV . However, the element dV will be sufficient for the
purpose of this Letter.
26 Alexey A. Kryukov
from the condition that the steady-state transition matrix has the right transition
probabilities from any state θi to the absorbing states {ψ1}, {ψ2} (namely, cos2 θi2
and cos2 π−θi
= sin2 θi
). The resulting process is a generalization of the (biased)
random walk with absorbing boundaries (also known as the gambler’s ruin), in which
the transition probabilities vary with the state. Namely, the transition probabilities
for a step toward an absorbing state {ψi} increase as the electron’s state moves
closer to {ψi}.
Various stochastic processes have been extensively used in modeling collapse
(see reviews Refs. [9] and [10]). In general words, the existing models are based
on adding an external random noise term and a term containing the measured
observable to the Schrödinger equation. The term with the observable provides the
“choices” for observations, while the random noise term is a “chooser” (see Ref.
[9]). The probability density for a particular noise in the models is given by yet
another equation. This equation makes it more probable for the noise to fluctuate
around values associated with the eigenstates of the observable and in such a way
that the probability of the noise also depends on the initial state ϕ0 of the system.
When applied to the process of measurement, the models of this kind explain the
probabilistic results of observations by relating them to the random noise, selected
by the mentioned probability rule. At the same time, the physical reason for a
particular random noise remains unexplained (see Ref. [9]).
Even without analyzing the existing stochastic models of collapse in detail, one
can pinpoint the essential difference of the model considered here. Namely, the
noise in the advocated approach is a process on the space of states which does not
depend on a particular state ϕ of the measured system. In particular, the noise in
the model does not change when the state ϕ changes. This independence of the
noise from the state of the measured system opens a way for associating it with the
measuring device itself. For example, as already discussed, the noise may originate
in the thermal motion of molecules of the device, considered as a process on the
space of states.
Another important observation is that the process of collapse in the model is
a deterministic process on the space of states. In fact, by associating the random
noise with a physical process, one should be able, in principle, to provide a specific
functional form of the noise. In this case it becomes possible to predict the time
and the outcome of collapse for an arbitrary evolution ϕt of the system.
Note that the proposed mechanism of collapse, although particularly simple, is
not the only one satisfying the hypotheses (H1), (H2). Furthermore, as already
mentioned, the proposed mechanism is far from being realistic at this stage. The
ultimate choice of a physically valid scenario of collapse depends crucially on the
field equations on the space of states and cannot be provided at this time. Instead,
let us demonstrate that under the above assumptions even such a simple mechanism
sheds new light on the quantum measurement problem.
On the measurement problem 27
5 THE MEASUREMENT PROBLEM
The observations made so far, combined with the results of Ref. [4] suggest the
following statements about objects and interactions in QM:
(S1) Physical objects in QM are most adequately represented by points of a Hilbert
manifold of states. In this sense, they have a functional nature.
(S2) Physical interactions involving microscopic objects (in particular, the process
of measurement) are most adequately described as processes on the manifold
of states, rather than on the classical space alone. In other words, the manifold
of states represents a new arena for description of physical processes.
(S3) The interactions can be described in terms of the Riemannian metric on the
manifold of states. In particular, the states of microscopic particles move along
geodesics on the sphere of states furnished with a Riemannian metric. In this
sense, the interactions may have a geometric origin.
Let us investigate how these statements together with the hypotheses (H1),(H2)
in the previous section help provide an understanding of the measurement process
in QM. First of all, a particular measuring device can be modeled by a metric field
with sources at the eigenstates of the measured observable. That is, the kind of
measurement performed on the system determines a specific field created by the
device on the manifold of states of the system. Provided the model based on the
hypotheses can be developed into a consistent physical theory, the latter result would
resolve the so-called preferred basis problem in QM. The problem can be formulated
as follows:
(P1) How could the electron’s state ϕ “know”, which basis {ek} to use to associate
the right probabilities to the coefficients ck in decomposition ϕ =
k ckek?
The constructed model suggests the following answer:
(R1) The coefficients of state of the system in the basis of eigenvectors of the ob-
servable describe position of the state relative to the sources of the field created
by the device. As already discussed, this position determines the probability
for the state to be “pushed” by sources to a particular eigenstate point on the
projective space of states. In other words, by creating a surrounding field in
the space of states, the device itself defines the “preferred” basis.
Next, the process of collapse in the model is an objective process driving the state
of the system to an eigenstate of the measured observable. The stochastic nature of
the process is due in the model to random fluctuations of sources associated with
measuring “parts” of the device. These fluctuations could be directly related to the
usual chaotic oscillations of the “parts” extended to the space of states of the system.
28 Alexey A. Kryukov
The “classical world” in the approach is represented by eigenstates of observ-
ables. The set of all eigenstates of an observable Â for a quantum system will be
called the set of Â-classical states (or points) for the system. So the set of Â-classical
states is a subset in the Hilbert space of states (or the corresponding space of phys-
ical states) of the system. Let us point out that there is nothing special about the
classical states as what is “classical” with respect to one observable is “quantum”
with respect to another one.
In the model under investigation, the integral curves of vector fields associated
with observables are geodesics in the Killing metric on the sphere of states. More
generally, the integral curves of vector fields associated with any reasonable set
of physical observables of a quantum system can be shown to be geodesics in an
appropriate Riemannian metric on the sphere of states of the system. The non-
commutativity of observables is then tied to the curvature of the metric.
Let us investigate in this light the “mother of quantum mechanics”, i.e., the
double-slit experiment. There are two main paradoxes associated with the experi-
ment:
(P2) How could the electron pass trough both slits at once?
(P3) How could a measuring device inserted after the screen with the slits instanta-
neously change the way in which the electron has passed through the screen?
The Hilbert space of states in the double-slit experiment is infinite-dimensional.
It would be helpful to consider at the same time a version of the experiment with
a finite dimensional space of states. For this let us return to the motion of electron
in a homogeneous magnetic field. Recall that the spin state of the electron evolves
in accordance with equation Eq. (2.3). If the field is directed along the Y -axis and
the initial state of the electron is
, the solution of Eq. (2.3) is given by
cos µB0
sin µB0
. (5.1)
If t changes, say, between 0 and π
, then the process of passing through the field
results in a “splitting” of the original spin-up eigenstate of the operator σ̂z into a
superposition of spin-up and spin-down states. In this respect the experiment is a
finite dimensional version of the double-slit experiment in which a localized electron
wave packet gets transformed by the screen with the slits into a superposition of
two wave packets.
With this in hand, let us address the above mentioned paradoxes (P2) and
(P3) of the double-slit experiment. Let us call the original double-slit experiment
and the experiment with an electron in a homogeneous magnetic field the E1 and
E2 experiments respectively. The electron in the experiment E2 evolves from the
On the measurement problem 29
original “classical” state
, into superposition
of two eigenstates of
σ̂z. During this evolution the Z-component of the electron’s spin is unknown. The
reason for that is clear: for 0 < t ≤ π
, the trajectory of electron’s state on
S2 = CP 1 does not pass through the classical states, i.e., through the eigenstates
of σ̂z. Classically speaking, one has a paradox here: the electron’s intrinsic angular
momentum is not defined. Instead, the electron is in a superposition of states of
two different angular momenta. In a way, the electron’s spin is up and down at the
same time.
Note however that the state function ϕt is defined for t = 0 as well as for t > 0.
In other words, it describes the classical and the non-classical states equally well.
Moreover, any (physical) state of the electron is just a point on S2. The evolution
of the electron’s state is just a path {ϕt} with values in S2. The classical way of
thinking tells us that the electron somehow splits into two parts that evolve along
different paths. However, the actual evolution of the electron is most adequately
described by a single path ϕt, thereby confirming the statements (S1), (S2).
The situation in experiment E1 is almost identical, although the paradox here
is more dramatic as our classical intuition of position is very strong. Again, the
intuition tells us that the electron splits into two parts which are passing through
different slits. However, the electron’s evolution is most adequately described by a
path in the space of states. Of course, such a path does not “split” and it describes
the evolution of electron before and after the screen with the slits equally well. The
resolution of the paradox (P2) is then as follows:
(R2) The electron in the experiment E2 is not in the spin-up and spin-down states
at once. Rather, it is in the state that is neither a spin-up, nor a spin-down
state. Similarly, the electron in the experiment E1 does not pass through two
slits at once. Rather, it does not pass through the slits at all! Indeed, for
a state to be a spin-up state, for example, it must be at the north pole of
the sphere S2 of states, which is not the case for the electron’s state in the
experiment E2 for t > 0. Similarly, to pass through a slit is to have a state
localized at that slit. But the state of the electron after its interaction with
the screen in the experiment E1 is not localized. In other words, the electron
(i.e., the electron’s state) is located at a point on the space of states that is
different from the point at which an electron passing through the slit would
have been. To put it figuratively, the electron passes over rather than through
the slits.
One can see that the paradox (P2) is resolved by considering the motion of
electron in the experiments E1, E2 as happening in the functional space of states
rather than on the classical space or on the space of angular momenta. Vaguely
speaking, the “functional” (i.e., consistent with (S1) and (S2)) way of thinking
makes the paradox disappear. In light of this, the resolution of the paradox (P3)
30 Alexey A. Kryukov
is now immediate:
(R3) How could a measuring device inserted after the screen change the way the
electron has passed through the screen? The answer is: it does not! If a counter
is inserted behind the screen (and sufficiently far from the screen), the process
of “passing through” the screen is not affected by it. In particular, the counter
can be placed after the electron has already passed “through” the screen and
this will not change the history. Indeed, the evolution of electron is described
by a path ϕt. If only one slit is open, this path passes through a point in the
space of states which is represented by a state function localized at the slit.
If, however, both slits are open, the path does not pass through such a point.
This is true independently of any measurement done behind the screen. What
the counter does is to change the path ϕt for larger values of the parameter t
so as to produce a state localized at one of the slits in a way discussed in the
previous section. As a result, the final state is as if the electron had passed
through only one of the slits. However, no reality should be attached in this
case to the event of passing though the slits. Once again, the electron in the
experiment E1 does not pass through the slits. Likewise, the state function ϕt
does not describe the probability of passing through one of the slits (but only
the probability to be found by one of the slits). Rather, ϕt itself represents a
new “functional” reality of the world which is more adequate in QM than the
familiar classical reality.
To summarize, the paradox (P3) is resolved by accepting the statements (S1)
and (S2), i.e., by recognizing the evolution of electron in the space of states as
physical (i.e., real) and by allowing a “deformation” of such an evolution in the
presence of a measuring device.
Let us finally analyze a measurement performed on a pair of spin-1/2 particles.
This will give a hint as to how to proceed in more general cases. As already discussed
in section 4, the total quantum mechanical state of the pair is a point in the tensor
product of Hilbert spaces of each particle. In particular, the spin state of a pair of
electrons is an element of C2 ⊗ C2. A unit normalized state is a point on the unit
sphere S7 in this four dimensional complex space. Physical spin states of the pair
are then points in the complex projective space CP 3. Note that there may be points
in CP 3 that do not represent a physical state of the pair. In particular, if the total
angular momentum of the pair vanishes, the state of the pair can only be of the
form aϕ+ ⊗ ψ− + bϕ− ⊗ ψ+ with a, b ∈ C. Moreover, if the particles are identical,
one must have a = −b.
The σ̂z-classical points on CP
3 are the points where both particles have a specific
value of the Z-component of spin. These points are represented by the products of
ϕ± and ψ±. In the case when the total angular momentum of the pair vanishes, the
points are represented by ϕ+ ⊗ ψ− and ϕ− ⊗ ψ+. The evolution of the pair is now
a path with values in S7. This path projects down to a path with values on the
underlying space CP 3.
On the measurement problem 31
With these standard ingredients in place, one can analyze now a version of the
famous Einstein-Podolsky-Rosen (EPR) paradox in QM:
(P4) Given a pair of spin-1/2 particles in entangled state atϕ+ ⊗ψ− + btϕ− ⊗ψ+,
how could it be, that by measuring the Z-component of spin of one of them
one fixes the Z-component of spin of the other one, even if the particles are
far apart?
Note that this paradox is similar to the paradox (P3), taking place on the space
of states of the pair. Indeed, the resolution of the paradox is almost identical:
(R4) Physical reality is described by the path ϕt with values in the space S
7 (or
CP 3) of states of the pair. Unless one of the coefficients at, bt in atϕ+ ⊗
ψ− + btϕ− ⊗ ψ+ is zero for some t, the path ϕt does not pass through the
σ̂z-classical points ϕ+ ⊗ ψ− or ϕ− ⊗ ψ+. That is, the particles do not have
any Z-component of spin. To measure the Z-component of spin of a particle
is to make the path ϕt pass through one of the σ̂z-classical points ϕ+ ⊗ψ− or
ϕ− ⊗ ψ+. In this case (and only in this case!) the Z-components of spin of
both particles are defined and take opposite values. Furthermore, as with a
single particle, the interaction with the measuring device is assumed to cause
a “deformation” of the path ϕt. The resulting path ends up then at one of the
classical points via a stochastic process on the space of states.
The full version of the experiment involving a spatial separation of the particles
is even more dramatic. How could the second particle at a point y “know” about
measurement of the Z-component of spin performed on the first particle at a distant
point x?
Again, physical reality of the pair is most adequately described by a path
ϕt = atϕx+ ⊗ ψy− + btϕx− ⊗ ψy+ in the space H of states of the pair. Here
ϕx+ is the spin-up state of the first particle located at x and similarly for the
other state functions in ϕt. The classical points in H have the form ϕx+⊗ψy−
and ϕx−⊗ψy+. If ϕt does not pass through these points the spin of individual
particles is not defined. Intuitively, we think that if a particle is “here” (at
a point x), then it ought to have all attributes of a “real” particle, including
spin. But before the measurement is performed, the particle in the experiment
is not really here! Indeed, it is somewhere else on the sphere of states SH in
H (or on the corresponding projective space CPH). So if reality is associated
with the state function of the pair, the paradox is resolved.
But what about this “spooky action at a distance”? Notice that the new “func-
tional” reality does not use it! Indeed, the equation of geodesics is “local” in the
space of states S3, because it is a differential equation of geodesics on S3. Of course,
this locality in the space of states does not preclude a non-locality in the classical
32 Alexey A. Kryukov
space. Indeed, what is a point in the space of states may represent a pair of well
separated particles in the classical space. Furthermore, what is close in the metric
on the space of functions does not have to be close in the metric on the classical
space (see Ref. [4]). A detailed analysis of this will be, however, a subject for a
different paper.
REFERENCES
1. M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984)
2. J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990)
3. B. Simon, Phys. Rev. Lett. 51, 2167 (1983)
4. A. Kryukov, Found. Phys. 36, 175 (2006)
5. A. Kryukov, Int. J. Math. & Math. Sci. 14, 2241 (2005)
6. A. Kryukov, Found. Phys. 34, 1225 (2004)
7. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous
Media (Springer, 1990)
8. J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964)
9. P. Pearle, “Collapse models,” in Open Systems and Measurement in Relativis-
tic Quantum Theory, F. Petruccione and H.P. Breuer, eds. (Springer, 1999)
10. A. Bassi & G. Ghirardi, Physics Reports 379, 257 (2003)
	GEOMETRY AND QUANTUM MECHANICS
	ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD
	Quantum Mechanics on the Space of States S3
	Quantum Mechanics on the Projective Space of States CP1
	TENSOR PROPERTIES OF EQUATIONS IN THE MODEL
	THE PROCESS OF MEASUREMENT
	THE MEASUREMENT PROBLEM

  A geometric approach to quantum mechanics with unitary evolution and
non-unitary collapse processes is developed. In this approach the Schrodinger
evolution of a quantum system is a geodesic motion on the space of states of
the system furnished with an appropriate Riemannian metric. The measuring
device is modeled by a perturbation of the metric. The process of measurement
is identified with a geodesic motion of state of the system in the perturbed
metric. Under the assumption of random fluctuations of the perturbed metric,
the Born rule for probabilities of collapse is derived. The approach is applied
to a two-level quantum system to obtain a simple geometric interpretation of
quantum commutators, the uncertainty principle and Planck's constant. In light
of this, a lucid analysis of the double-slit experiment with collapse and an
experiment on a pair of entangled particles is presented.

On the measurement problem for a two-level quantum
system
Alexey A. Kryukov ∗
November 12, 2018
A geometric approach to quantum mechanics with unitary evolution and
non-unitary collapse processes is developed. In this approach the Schrödinger
evolution of a quantum system is a geodesic motion on the space of states of
the system furnished with an appropriate Riemannian metric. The measuring
device is modeled by a perturbation of the metric. The process of measurement
is identified with a geodesic motion of state of the system in the perturbed
metric. Under the assumption of random fluctuations of the perturbed metric,
the Born rule for probabilities of collapse is derived. The approach is applied
to a two-level quantum system to obtain a simple geometric interpretation of
quantum commutators, the uncertainty principle and Planck’s constant. In
light of this, a lucid analysis of the double-slit experiment with collapse and an
experiment on a pair of entangled particles is presented.
KEY WORDS: measurement problem - Born rule - Berry’s phase - EPR-paradox
1 GEOMETRY AND QUANTUM MECHANICS
Geometric ideas have played a well recognized role in modern physics, especially in
general relativity (GR) and gauge theories (GT). They also found a well established
position in quantum mechanics (QM) in considerations related to Berry’s phase
Ref. [1]. However, whereas in GR and GT geometry (i.e., the metric or connection)
defines the dynamics of the theory, the geometric methods pertaining to Berry’s
phase do not enjoy such a sweeping significance. The reason for this difference is
quite obvious. Indeed, the geometry underlying GR and GT is directly related to
∗Department of Mathematics, University of Wisconsin Colleges
E-mail: alexey.kryukov@uwc.edu, aakrioukov@facstaff.wisc.edu
http://arxiv.org/abs/0704.1934v1
2 Alexey A. Kryukov
the physical fields (gravitational or gauge) in the theory. At the same time, the
Fubini-Study metric in the geometric interpretation of Berry’s phase (Refs. [2],[3]
amongst many others) depends only on the geometry of the Hilbert space of states of
quantum system. The latter geometry (i.e. Hilbert metric) is insensitive to changes
in the Hamiltonian of the system and, consequently, is not dynamical.
At the same time, it turns out to be easy to make the metric on Hilbert space
of states of a closed quantum system dynamical Refs. [4],[5]. For this, notice first
of all that the Schrödinger equation
= − i
ĥϕt (1.1)
is the equation for integral curves of the vector field hϕ : H −→ TH, hϕ = − ih̄ ĥϕ
associated with the Hamiltonian ĥ of the system. Here H is the Hilbert space
of states of the system and TH is the tangent bundle over H. Assume that the
Hilbert space H is a space of functions that are square-integrable with respect to
an appropriate measure. Because the evolution governed by Eq. (1.1) is unitary,
the integral curve through initial point ϕ0 on the unit sphere S
H in H will stay on
the sphere. Since this holds true for any initial point (modulo the domain issues),
one concludes that the restriction of the vector field hϕ to the sphere S
H is a vector
field on the sphere.
In the ordinary QM spaces TϕH tangent to H at ϕ ∈ H are identified with
the space H itself. Similarly, spaces TϕS
H tangent to the sphere SH at ϕ ∈ SH
are identified with (real) affine subspaces of H. In particular, the metric on SH ,
whenever used, is assumed to be induced by the embedding of SH into H.
However, the sphere SH is a manifold and thus, can be defined independently of
the ambient space H. As such, SH is a Banach manifold which means that it can
be obtained by “gluing together” open sets in a Banach space. The Hilbert metric
Gϕ : TϕH × TϕH −→ C on tangent spaces TϕH can be also defined independently
of the metric on H as an Hermitian tensor field on H. Such a tensor field gives rise
to a Riemannian metric GRϕ : TϕS
H ×TϕSH −→ R on SH , defined at each ϕ ∈ SH
GRϕ(X,Y ) = 2ReGϕ(ξ, η). (1.2)
Here X = (ξ, ξ), Y = (η, η) ∈ TϕSH are considered as vectors in the realization
of the tangent space TϕH. The manifold S
H , furnished with the (2, 0)-tensor field
GRϕ, is then a Riemannian manifold. In the following, the manifold S
H with the
metric GRϕ will be denoted by S
The final step in making the metric Gϕ on the sphere of states S
G dynamical is
to ensure that the integral curves of hϕ (i.e. the solutions to Schrödinger equation
Eq. (1.1)) are geodesics on SG. For this it turns out to be sufficient to define
Gϕ : TϕH × TϕH −→ C by
Gϕ(ξ, η) = h̄
(ĥĥ∗)−1ξ, η
. (1.3)
On the measurement problem 3
Here ĥ∗ is the adjoint of ĥ (normally equal to ĥ) and the Hamiltonian ĥ is assumed
to be invertible. Incidentally, even if the Hamiltonian ĥ is not bounded on H, it
becomes bounded as an operator mapping points ϕ ∈ H into tangent spaces TϕH
with the metric Eq. (1.3) (see Ref. [5]).
Further general results concerning QM on Hilbert manifolds can be found in Ref.
[4]. These results demonstrate that QM can be formulated in terms of geometry of
the space of states. The goal of the Letter is to provide such a geometric formulation
in case of a simple two-level system and to establish its advantages. Namely, the
point of view that the space of states represents a new arena for physical processes
and the evolution of state is a motion along geodesic is shown to be effective in
addressing the major conceptual difficulties of quantum mechanics. Although the
discussion deals primarily with a simple model, the most important results can be
shown to be quite general. Some of these generalizations are described in the Letter
while others are left for the upcoming publications.
2 ELECTRON IN A HOMOGENEOUS MAGNETIC
FIELD
Consider a free non-relativistic electron propagating in the direction of the X-axis
in a homogeneous magnetic field B. The evolution equation (the Pauli equation)
for the electron is
Ψ− µσ̂ ·BΨ, (2.1)
where Ψ = Ψ(s, x, t), s = 1, 2 is a two-component state function of the electron, µ
is the electron’s magnetic moment and σ̂ = (σ̂x, σ̂y, σ̂z) is the vector made of Pauli
matrices. The substitution Ψ(s, x, t) = ψt(x)ϕt(s) separates variables and produces
two independent evolution equations. The first describes the evolution governed by
the free Hamiltonian
ψt. (2.2)
The second equation describes the evolution in the space C2 of spinors ϕ:
= −µσ̂ ·Bϕt. (2.3)
It follows that in the case of the product states Ψ(s, x, t) = ψt(x)ϕt(s), one can
analyze the evolution of spin state ϕt in the space of states C
2 without needing to
involve the infinite-dimensional Hilbert space of states Ψ.
2.1 Quantum Mechanics on the Space of States S3
Let us proceed to reformulation of quantum mechanics of the system in geometrical
terms. In this, the fact that the sphere S3 of unit normalized spin states can be
4 Alexey A. Kryukov
furnished with the group structure of the group SU(2) will be helpful. The group
structure will allow us to exploit simple results from differential geometry of Lie
groups which will make the resulting picture more transparent and complete.
First of all, the Hamiltonian ĥ = −µσ̂ ·B defines the vector field hϕ = ih̄µσ̂ ·Bϕ
on the sphere S3 in the space C2 with the metric (ξ, η)C2 =
k ξkηk. The integral
curve of hϕ (i.e. the solution of Eq. (2.3)) through ϕ0 ∈ S3 is given by
ϕt = e
µσ̂·Btϕ0. (2.4)
Since ϕt is a path in C
2, it is natural to call the vector
the velocity of evolution
of the system. The speed of evolution in the C2 is the norm of dϕt
in C2 metric.
Using
(σ̂ ·A)(σ̂ ·B) = A ·B+ iσ̂ ·A×B, (2.5)
one has
(σ̂ ·B)2 = B2. (2.6)
Therefore, by Hermicity of the matrix σ̂ ·B, one obtains
µσ̂ ·Bϕt,
µσ̂ ·Bϕt
, (2.7)
where B is the norm of B. In particular, the speed of evolution of the system
depends only on the magnitude of the field.
To make the evolution of the system a motion along a geodesic, the metric Gϕ
on S3 will be defined by Eq. (1.3). Since ĥ is self-adjoint, one obtains ĥĥ∗ = ĥ2 =
µ2(σ̂ ·B)2 = µ2B2I, where I is the identity operator on C2 and Eq. (2.6) has been
used at the last step. Therefore, up to the constant factor (h̄/µB)2, the metric
Gϕ coincides with the one induced by the embedding of S
3 into C2. That means
that the carriers of the geodesics on S3 are the intersections of S3 with the planes
through the origin. The fact that the found Riemannian metric is so simple is due
to an especially simple form of the Hamiltonian in the model.
If S3 is identified with the group manifold SU(2), the obtained metric is the
Killing metric on SU(2). To see this, let us identify in the standard way the space
C2 of complex vectors ϕ =
with the space Mat of 2× 2 matrices
z1 z2
−z2 z1
. (2.8)
The map ω : ϕ −→ ϕ̂ is an isomorphism of (real) vector spaces C2 and Mat. The
sphere S3 of unit states in C2 is identified via ω with the subset of matrices with
unit determinant. The latter subset is the group SU(2) under matrix multiplication.
On the measurement problem 5
The Killing metric on the Lie algebra su(2) can be defined by
(X̂, Ŷ )K = cTr(X̂Ŷ
+), (2.9)
where c 6= 0 is an arbitrary constant and Tr stands for the trace. The Killing metric
on the group SU(2) is defined for all ϕ̂ ∈ SU(2) and all left-invariant vector fields
(ϕ̂) = ϕ̂X̂, L
(ϕ̂) = ϕ̂Ŷ , X̂, Ŷ ∈ su(2) by
(ϕ̂), L
(ϕ̂))K = (LX̂(e), LŶ (e))K = (X̂, Ŷ )K , (2.10)
where e is the identity element in SU(2).
By direct substitution one verifies that (L
(e), L
(e))K = 2cRe(X,Y )C2 when-
ever X̂ = dω(X) and Ŷ = dω(Y ) with dω being the differential of the map ω. In
other words, the Killing metric is proportional to the metric induced by the em-
bedding of S3 into the Euclidean space C2 = R4. This verifies that the metric
Gϕ = (h̄/µB)
2I obtained earlier, is the Killing metric.
For any two left invariant vector fields L
on SU(2) the connection ∇ on
SU(2) can be defined by
[X̂,Ŷ ]
. (2.11)
Notice that the left invariant vector fields form a basis on the tangent space T
SU(2)
for all ϕ̂ ∈ SU(2). In particular, Eq. (2.11) is sufficient to define a connection on
SU(2). This connection is symmetric, as the torsion tensor vanishes:
) = ∇L
[X̂,Ŷ ]
[Ŷ ,X̂]
[X̂,Ŷ ]
= 0. (2.12)
The connection Eq. (2.11) is also compatible with the Killing metric, that is, for
any vector fields ξ, η, ζ on SU(2) the following is true:
∇ξ(η, ζ)K = (∇ξη, ζ)K + (η,∇ξζ)K . (2.13)
Indeed, assuming that ξ = L
, η = L
, ζ = L
are left invariant, one has
(ϕ̂), L
(ϕ̂)K = (LŶ (e), LẐ(e))K = const (2.14)
and therefore the left hand side of Eq. (2.13) vanishes. For the right hand side, by
definition Eq. (2.11) one obtains:
)K + (LŶ ,∇LX̂LẐ)K =
([X̂, Ŷ ], Ẑ)K +
(Ŷ , [X̂, Ẑ])K . (2.15)
From the anti-Hermicity of elements of su(2) one also has:
([X̂, Ŷ ], Ẑ)K = −cTr(X̂Ŷ Ẑ) + cTr(Ŷ X̂Ẑ) (2.16)
6 Alexey A. Kryukov
([Ŷ , X̂ ], Ẑ)K = −cTr(Ŷ X̂Ẑ) + cTr(X̂Ŷ Ẑ). (2.17)
As a result, the sum on the right hand side of Eq. (2.13) is also zero which verifies
that the connection Eq. (2.11) is compatible with the metric. In other words, the
connection ∇ is the Levi-Civita connection of the Killing metric.
For any magnetic fieldB in the model the one parameter subgroup ϕ̂t = e
µσ̂·Bt
is a geodesic through the identity e ∈ SU(2). Indeed, since dϕ̂t
= −ϕ̂t ih̄µσ̂ ·B, the
path ϕ̂t is the integral curve of the left invariant vector field Lĥϕ̂ = −ϕ̂
µσ̂ · B.
Using the definition Eq. (2.11) one then has:
[̂h,̂h]
= 0. (2.18)
Geodesics through an arbitrary point ϕ̂0 ∈ SU(2) can be then written in the form
ϕ̂t = ϕ̂0e
µσ̂·Bt. Considered as paths with values in C2, these geodesics take the
form ϕt = e
µσ̂·Btϕ0.
The curvature tensor of ∇ can be obtained directly from the definition
]LẐ . (2.19)
In particular,
[[X̂,Ŷ ],Ẑ]
. (2.20)
and (
[X̂, Ŷ ], [Ẑ, Ŵ ]
. (2.21)
The sectional curvature in the plane through L
is defined by
. (2.22)
With the help of Eqs. (2.21) and (2.10) this becomes
[X̂, Ŷ ], [X̂, Ŷ ]
X̂, X̂
Ŷ , Ŷ
X̂, Ŷ
. (2.23)
Suppose for example that X̂ , Ŷ , Ẑ and Ŵ correspond to the spin observables.
Recall that in the Planck system of units the operator of spin ŝ has eigenvalues ±1/2
and can be expressed in terms of the Pauli matrices σ̂1, σ̂2, σ̂3 as
σ̂, (2.24)
On the measurement problem 7
where σ̂ = (σ̂1, σ̂2, σ̂3). The corresponding anti-Hermitian generators êk =
k = 1, 2, 3, form a basis of the Lie algebra su(2) and satisfy the commutator relations
[êk, êl] = ǫklmêm, (2.25)
where ǫklm denotes the completely antisymmetric tensor of rank three. In the basis
{êk} the curvature tensor Eqs. (2.20), (2.21) takes the form
Rik,lm =
(δilδkm − δimδkl), (2.26)
Rik,lm =
(δilδkm − δimδkl) (2.27)
where δik is the Kronecker delta. The symmetry property
Rik,lm +R
l,mk +R
m,kl = 0 (2.28)
of the curvature tensor coincides in the model with the Jacobi identity
[[X̂, Ŷ ], Ẑ] + [[Ŷ , Ẑ], X̂] + [[Ẑ, X̂ ], Ŷ ] = 0 (2.29)
for the Lie algebra elements X̂, Ŷ , Ẑ.
From the isomorphism given by Eq. (2.8) it follows that any vector x =
(x1, x2, x3) in the Euclidean space R3 can be identified with the element
kiσ̂k =∑
k êk of the Lie algebra su(2). Under such an identification the Euclidean norm
‖x‖R3 of x is equal to detx and rotations in R3 are represented by transformations
x −→ ÛxÛ+ with Û ∈ SU(2). One can make this identification into an isometry
by assuming the equality of Euclidean and Killing norms
∥∥∥∥∥
2xkêk
∥∥∥∥∥
= ‖x‖R3 . (2.30)
This will fix the constant factor in front of the Killing metric.
Note that the Euclidean space R3 can be identified with the space of all possible
classical angular momenta of a particle. The electron’s possible angular momenta
form a sphere S2 in R3. By equating the norms according to Eq. (2.30), the spaces
tangent to S2 are identified with affine subspaces of spaces tangent to the sphere
of states S3 with the induced metric. In particular, the sphere S2 can be identified
with a submanifold of the space of states S3 with the induced metric. Let us remark
that this identification is analogous to the identification of the classical space with
the submanifold of point supported states in an infinite-dimensional Hilbert space
of states, considered in Refs. [4], [6].
In the Killing metric Eq. (2.10) on S3 one has
kêk,
k,m x
kxmδkm. To satisfy Eq. (2.30) the constant c must be 1/2, that is, the
needed metric in Planck units has the form
(X̂, Ŷ )K =
Tr(X̂Ŷ +). (2.31)
8 Alexey A. Kryukov
Using the formula Eq. (2.23), one obtains the following expression for the sectional
curvature R(p) in the plane p through orthogonal vectors L
([ê1, ê2], [ê1, ê2])K
(ê1, ê1)K (ê2, ê2)K
= 4 (ê3, ê3)K = 1. (2.32)
So the sectional curvature of S3 in Planck units is equal to 1. Note that in an
arbitrary system of units the Killing metric would be multiplied by h̄2 and the
sectional curvature would be equal to 1/h̄2. The dimension of sectional curvature
is consistent with the fact that the tangent space su(2) is spanned by the spin
operators having the dimension of angular momentum.
In this approach the Planck’s constant and the commutators of spin observables
acquire a transparent geometric interpretation. According to Eq. (2.23) the commu-
tator of two observables is directly related to the sectional curvature of the sphere
S3. Indeed, assume for simplicity that the vector fields L
are orthogonal and
unit normalized in the Killing metric. Then from Eqs. (2.22), (2.23) one has the
following expression for the norm of the commutator of X̂ and Ŷ :
∥∥∥[X̂, Ŷ ]
= 4R(p). (2.33)
Here R(p) is the sectional curvature of S3 in the plane p = L(L
) which for the
considered Riemannian metric was found to be a constant equal to 1/h̄2.
Note that for L
which are orthogonal but not unit, the equation Eq. (2.23)
takes the form
∥∥∥[X̂, Ŷ ]
= 4R(p)
∥∥∥X̂
∥∥∥Ŷ
. In particular, if the norms of
X̂ and Ŷ are of order h̄ (e.g., X̂, Ŷ are the spin observables), then the norm of
the commutator [X̂, Ŷ ] is of order h̄ as well. Note that despite the fact that the
commutator [X̂, Ŷ ] is small in these units, it is of the order of the radius of the
sphere of states, making quantum effects on the sphere quite transparent.
The results obtained so far in this section were model specific. It is then im-
portant to know whether they can be generalized to the case of higher dimensional
spaces of states and of arbitrary observables. Also, what if the Hamiltonian of the
system is time-dependent? Here is a sketch of what can be done in these cases.
For any n the sphere of states S2n−1 in the space Cn is a homogeneous space
U(n)/U(n − 1), where U(n) denotes the unitary group on Cn. The Killing metric
on U(n) can be used to induce a Riemannian metric on S2n−1 via the embedding.
Namely, the 2n − 1 linearly independent generators in the Lie algebra u(n), which
belong to the orthogonal complement of a (fixed) subalgebra u(n − 1), form a sub-
space V ⊂ u(n). The one-parameter subgroups eX̂τ with X̂ in V sweep a sphere
S2n−1 and yield geodesics in the induced metric. The curvature of S2n−1 can be
then computed via equations Eqs. (2.20), (2.21) with generators in V . Furthermore,
the commutators of generators in V are related to the sectional curvature of S2n−1
by the same formula Eq. (2.33).
On the measurement problem 9
Note that there exist anti-Hermitian observables that are not in V . For example,
if n = 2 so that S3 = U(2)/U(1) = SU(2), then V is the Lie subalgebra su(2) ⊂ u(2)
and so V consists of the traceless elements of u(2). If X̂ ∈ u(2) is not traceless,
the one-parameter subgroup eX̂τ is still a geodesic in U(2). However, this geodesic
does not “stay” on the subgroup SU(2) = S3. Of course, one could still consider
the curves on S3 given by ϕt = e
X̂τϕ0, for some point ϕ0 in S
3 ⊂ C2. For any given
X̂ ∈ u(2) and all initial points ϕ0 these curves are still geodesics in the appropriate
Riemannian metric (see section 1 and Ref. [4]). However, the algebraic features of
the model change and the formulas connecting the commutators with the curvature
are different.
For the time-dependent Hamiltonians ĥ in the Hilbert space Cn the approach can
be generalized as follows. The sphere of states S2n−1 is replaced with the manifold
M = S2n−1 ×R, where R is the time line. Then, there exists a Riemannian metric
on M in which the paths (ϕt, t) with ϕt = e
−îhtϕ0 are geodesics for all ϕ0 ∈ S2n−1.
Finally, as already mentioned, in the infinite-dimensional case there still exists
a Riemannian metric for which all solutions to the Schrödinger equation with an
invertible (time-independent) Hamiltonian are geodesics. However, the algebraic
properties of the model require further investigation in this case.
2.2 Quantum Mechanics on the Projective Space of States CP 1
In physical experiments one can only determine the state of a system up to a com-
plex non-zero factor. That means that the space of physical states is the complex
projective space CPH of complex lines in the space of states. In the considered
example it is the one dimensional complex projective space CP 1. By definition,
CP 1 is the quotient C2∗/C∗, were ∗ means “take away zero”. In other words, CP 1
is the base manifold of the fibre bundle π : C2∗ −→ CP 1 with the natural projection
along the fibres C∗. By considering unit normalized states only, one obtains CP
as a quotient S3/S1. It is then the base of the fibre bundle π : S3 −→ CP 1, which
is a sub-bundle of the previous fibre bundle.
If ϕt is a path of the electron’s state on S
3 and π : S3 −→ CP 1 is the bundle
projector, then π(ϕt) is a path on the base CP
1. Since this latter path represents
what can be measured in experiments, it is important to obtain an explicit formula
for π. For this consider a point ϕ =
on S3 and let {ϕ} be the complex
line formed by vectors λϕ, λ ∈ C. Provided ϕ1 6= 0, there is a unique point of
intersection of the line with the affine plane in C2 formed by vectors
, ξ ∈ C.
Namely, by setting
, (2.34)
10 Alexey A. Kryukov
one obtains
. (2.35)
The map ρ = ϕ −→ ξ provides a coordinate chart on CP 1 which identifies CP 1
without a point (complex line through
) with the set C of complex numbers.
The affine plane of vectors
form a subspace in the Lie algebra su(2). The
algebra su(2) itself has been identified earlier with the Euclidean space R3 of vectors
kiσ̂k. The stereographic projection then identifies the unit sphere S
the origin of R3 with the above plane C plus a point, i.e., with CP 1 itself.
The relationship of the coordinate ξ in the plane C with coordinates (x1 =
x, x2 = y, x3 = z) of the corresponding point on the sphere S2 is given by
x+ iy
. (2.36)
Solving this for x, y and z and using Eq. (2.35), one obtains:
x = ϕ1ϕ2 + ϕ1ϕ2, (2.37)
y = i(ϕ1ϕ2 − ϕ1ϕ2), (2.38)
z = ϕ2ϕ2 − ϕ1ϕ1. (2.39)
The resulting map π : S3 −→ S2 given by (ϕ1, ϕ2) −→ (x, y, z) is the needed
projection on the space of physical states.
The equation for the integral curve Eq. (2.4) can be simplified by choosing the
coordinate axes properly. In particular, one can always assume that the Z-axis is
parallel to the magnetic field B. In this case σ̂ ·B = σ̂3B and Eq. (2.4) simplifies to
eiωtϕ0+
e−iωtϕ0−
, (2.40)
where ω = µB
and the initial state ϕ0 is equal to
. Recall that the speed
of evolution of the electron along S3 was given by Eq. (2.7). Let us find the speed
of the projection of this evolution on the space of physical states S2 = CP 1.
For this recall that the Killing metric on su(2) coincides with the Euclidean metric.
The embedding of S2 into su(2) = R3 induces the familiar metric on S2. Such a
metric also coincides with the famous Fubini-Study metric on CP 1 identified with
S2. Since this metric is just a restriction of the Euclidean metric on R3, one has(
. Using Eqs. (2.37)-(2.40), one obtains then
= 4ω|ϕ0+||ϕ0−| =
|ϕ0+||ϕ0−|. (2.41)
On the measurement problem 11
If θ is the angle between the Z-axis and the vector (x, y, z), then sin2 θ = 1− z2 and
one finds with the help of Eq. (2.39):
sin θ. (2.42)
In particular, if the initial state is an eigenstate of σz so that θ = 0 or π, then
the speed ds
vanishes and the evolution of the electron is projectively trivial. In
other words, the eigenstates of the Hamiltonian ĥ = −µσ̂ ·B are zeros of the “push-
forward” vector field dπ(hϕ) on CP
1. Here dπ is the differential of the map π and
µσ̂ ·Bϕ as before.
Recall that the integral curves ϕt given by Eq. (2.4) are geodesics on S
3. At
the same time, as one can see from Eqs. (2.37)-(2.40), the projected curves {ϕt}
are not in general geodesics in the induced metric. Moreover, there does not exist
a Riemannian metric on CP 1 = S2 in which the projection by π of an arbitrary
geodesic on S3 yields a geodesic on S2. Indeed, the great circles of S2 (parametrized
by arc length) are geodesics in the Fubini-Study metric. At the same time, they are
projections of geodesics on S3. The latter fact is obvious for the equator of S2 if
one chooses |ϕ0+| = |ϕ0−| in Eqs. (2.37)-(2.40). For any other great circle this fact
is verified by a change in direction of the Z-axis. But the condition that the great
circles are geodesics fixes the metric on S2 up to a constant factor. This, together
with the above observation that the projection {ϕt} of a geodesic may not be a
geodesic itself, proves the claim.
Suppose, as it is advocated here, that evolution by geodesics is a valid principle
of quantum dynamics. Then the obtained result suggests that the space of states
S3 may be “more physical” than the projective space CP 1 of “physical states”. In
general terms, the actual, “easy to describe” physics may be happening on the space
of states of a quantum system. However, our experiments can so far access only the
projection of physical processes into the space of physical states. The projection of
a process may look much less “natural”, then the original process on the space of
states.
It was verified earlier that the commutators of spin observables are related to
geometry of the space of states S3. Later on in the paper the geometry of a measure-
ment process will be discussed. It is then important to know that the probability
of transition from state ϕ to state ψ in a measurement also has ties to geometry.
Namely, this probability depends only on the distance between {ϕ} and {ψ} in the
Fubini-Study metric on the projective space of states (see Ref. [2]).
This result is immediate in the model under consideration. In fact, assume that
the state ψ is the spin-up eigenstate of σ̂z (one can always assure this by a proper
choice of coordinate axes). Then, according to Eq. (2.37), one has
z = |ϕ1|2 − |ϕ2|2. (2.43)
12 Alexey A. Kryukov
Using the normalization condition |ϕ1|2 + |ϕ2|2 = 1, one obtains
|ϕ1|2 =
. (2.44)
If θ ∈ [0, π] is the angle between the vectors representing {ϕ} and {ψ} on S2, then
z = cos θ and the equation Eq. (2.44) takes the form
P (θ) = cos2
. (2.45)
Here P (θ) = |(ψ,ϕ)|2 = |ϕ1|2 denotes the probability of transition. Notice that θ
is the length of the arc of a great circle through {ϕ} and {ψ}. In other words, it
is a geodesic distance between the points {ϕ} and {ψ} on S2 in the Fubini-Study
metric. In particular, the probability P (θ) of transition between two states depends
only on the distance between them.
Furthermore, the uncertainty principle for the spin observables also has a simple
geometrical interpretation. Since uncertainties of observables do not depend on the
overall phase of a state, the geometry underlying the uncertainty principle is, once
again, projective. The uncertainty principle for non-commuting observables X̂, Ŷ
can be written in the form
∆X∆Y ≥
∣∣∣(ϕ, [X̂, Ŷ ]ϕ)
∣∣∣ , (2.46)
where ∆X2 = (ϕ, X̂2ϕ) − (ϕ, X̂ϕ)2 and similarly for ∆Y 2. Assuming X̂, Ŷ are the
components ŝx, ŝy of the spin observable ŝ =
σ̂, one has
∆sx∆sy ≥
|(ϕ, ŝzϕ)| . (2.47)
Let us connect this inequality with geometry of the projective space S2 = CP 1.
In light of the geometric interpretation of the commutators of observables Eq. (2.33)
and the equation Eq. (2.46), the existence of such a connection is not surprising.
Indeed, from the geometry of the sphere S2 furnished with the Fubini-Study (i.e.,
the usual!) metric, for any point (x, y, z) on the sphere one has:
(y2 + z2)(x2 + z2) ≥ z2. (2.48)
The inequality simply says that the product of distances from a point (x, y, z) on the
sphere S2 ⊂ R3 to the X and Y -axes is at least |z|. This equation is the geometric
form of the uncertainty principle. Indeed, from Eqs. (2.37)-(2.39) it follows that
(ϕ, σ̂xϕ) = x, (ϕ, σ̂yϕ) = y, (ϕ, σ̂zϕ) = z, (2.49)
where x, y, z are coordinates of the point of S2 representing the state ϕ. In addittion,
σ̂2x = σ̂
y = σ̂
z = I and so
∆σ2x = 1−x2 = y2+z2, ∆σ2y = 1−y2 = x2+z2, ∆σ2z = 1−z2 = x2+y2. (2.50)
On the measurement problem 13
With the use of Eqs. (2.49), (2.50), the inequality Eq. (2.48) is now equivalent to
the uncertainty principle Eq. (2.47).
Note that using the angles θx, θy, θz between the coordinate axes and the vector
x = (x, y, z) in R3, one can write the equations Eqs. (2.50) in the form
∆σx = sin θx ∆σy = sin θy, ∆σz = sin θz. (2.51)
In particular, the uncertainty ∆E in energy for the electron in the state ϕ in the
model takes the form
∆E = µB sin θ, (2.52)
where θ is the angle between the vectors B and x. Recall that θ is a geodesic
distance between the points {ϕ} and {ψ} on S2. Therefore, the uncertainty ∆E is
the largest, when {ϕ} is furthest away from the eigenstates, i.e. when θ = π
the other hand, ∆E vanishes at the eigenstates of ĥ.
The latter statement can be generalized as follows. Let H = Cn and let
Â : H −→ H be an observable with a simple spectrum λ1 < λ2 < ... < λn and
eigenfunctions ϕ1, ϕ2, ..., ϕn. Let {ϕt} be a geodesic through the states {ϕk}, {ϕl}
in the Fubini-Study metric on the projective space of states CPH . Then for any
point {ϕ} on the geodesic the variance ∆A2 = (ϕ, Â2ϕ)− (ϕ, Âϕ)2 is an increasing
function of the distance from {ϕ} to the pair of eigenstates {ϕk}, {ϕl} in CPH . The
latter distance is simply the shortest of the distances from {ϕ} to the states {ϕk},
{ϕl}.
In fact, because the eigenstates are orthogonal, the probability of transition from
a state {ϕ} on the geodesic to a state {ϕi} with i 6= k, l is equal to zero. In particular,
{ϕ} = {ckϕk + clϕl} for some coefficients ck, cl. Therefore,
∆A2 = |ck|2λ2k +
1− |ck|2
λ2l −
|ck|2λk + (1− |ck|2)λl
. (2.53)
If {ϕ} coincides with {ϕk} so that |ck| = 1, then ∆A vanishes. On the other hand,
as |ck| decreases, ∆A increases until |ck| becomes equal to 1√2 . By Eq. (2.45) this
means that ∆A increases with the distance from {ϕ} to {ϕk} until {ϕ} becomes
equally distant from {ϕk} and {ϕl}.
It is important to note that the “geometric” probability of transition formula Eq.
(2.45) is valid in an arbitrary space of states H = Cn. Also, as already discussed,
the relationship Eq. (2.33) between the commutators and the curvature holds true
on the sphere of states S2n−1 for observables in the subspace V ⊂ u(n) (see the
end of section 2.1). Because of that, the uncertainty principle Eq. (2.46) can be
still interpreted geometrically. The infinite-dimensional case requires a different
approach and needs further analysis.
14 Alexey A. Kryukov
3 TENSOR PROPERTIES OF EQUATIONS IN THE
MODEL
In the developed geometrical formulation of the model, the space of states is a
manifold furnished with a Riemannian metric which defines quantum dynamics on
the space. It is well known that differential geometry of manifolds admits two
equivalent formulations: local coordinate and coordinate free. Let us discuss the
role played by both formulations in the model.
Consider a pair X̂, Ŷ of elements of the Lie algebra su(2) and the corresponding
pair Xϕ = X̂ϕ, Yϕ = Ŷ ϕ of the associated vector fields. By direct computation one
sees that
[Xϕ, Yϕ] = −[X̂, Ŷ ]ϕ, (3.1)
where [Xϕ, Yϕ] is the Lie bracket of the vector fields. Recall that the integral curves
of non-commuting vector fields cannot form a coordinate grid on the manifold. In
particular, the integral curves of vector fields sxϕ, syϕ, szϕ associated with the spin
observables ŝx, ŝy, ŝz do not form a coordinate grid on S
3. By the above, these
integral curves are geodesics in the Killing metric on S3. The fact that they do not
form a coordinate grid is then a direct consequence of the curvature of S3. Instead,
the fields sxϕ, syϕ, szϕ form a local (non-coordinate) basis at every point of S
There are, of course, many ways of choosing coordinates on S3. One natural
choice is the normal coordinate system given on a neighborhood of any point by
the exponential map. If {ek} is a basis on the tangent space TϕS3 at ϕ ∈ S3 and
kek is a tangent vector, then the equation of geodesic ϕt through ϕ in the
direction of A in the normal coordinates ϕ1, ϕ2, ϕ3 is linear:
ϕkt = a
kt, k = 1, 2, 3. (3.2)
The evolution of spin state ϕ in time could be thought of as a motion along the
manifold S3×R with R being the time axis. The direct product of the Killing metric
on SU(2) and the usual Euclidean metric on R makes S3 × R into a Riemannian
manifold. If ϕt is a geodesic on S
3, then (ϕt, t) will be a geodesic on S
3 × R.
Notice that for any evolution ϕt of the state along S
3 ×R one can find a co-moving
coordinate system on S3×R in which (ϕt, t) = (ϕ0, t), i.e., the state is at rest. Such
a co-moving system is directly related to the well known Heisenberg representation.
The coordinates typically used on the projective space of physical states CP 1
are homogeneous (ϕ1, ϕ2) and inhomogeneous ξ =
, η =
coordinates. For
example, the Fubini-Study metric is usually expressed in terms of these coordinates.
Other coordinate systems may be useful in applications. In particular, according
to Eq. (2.49), the expectation values of Pauli matrices for a system in a state ϕ
coincide with the x, y and z coordinates of the point {ϕ} in S2. Therefore, these
expectation values can be identified with local coordinates on CP 1. This fact was
used in section 2 to describe the motion of state along CP 1.
On the measurement problem 15
The group SU(2) acting on the space C2 is the symmetry group of the theory.
In particular, the group SU(2) “extends” to act on tensor algebra over C2 and the
Schrödinger equation is a tensor equation. For any fixed time t the Schrödinger
evolution operator Û(t, 0) given by Eq. (2.4) is an active realization of SU(2)-
transformations on C2. The corresponding passive realization consists in a unitary
change of basis on C2.
One could consider instead the sphere S3 ⊂ C2 as a base manifold for tensor
bundles and make the group SU(2) act locally on tensor products of spaces tangent
and cotangent to S3. For each tensor type this gives a subbundle of the correspond-
ing tensor bundle over C2. The Schrödinger equation is then a tensor equation with
terms which are vector fields on S3. In fact, it is the equation for integral curves of
the vector field hϕ = − ih̄ ĥϕ on S
3. Alternatively, it is the equation of geodesics in
the Killing metric on S3.
The notion of symmetry in QM is usually understood as an invariance of the
Hamiltonian of the system under a symmetry transformation. In this case the
Hamiltonian commutes with the transformation and the generator of transformation
becomes a constant of motion. Although this is certainly true for rotations about
the field direction in the model, such a restricted understanding of symmetry is not
suitable for this Letter.
Mathematically, the Schrödinger equation hϕ = − ih̄ ĥϕ in the model is written in
a specific basis on the space of states C2. Under a unitary transformation Û ∈ SU(2)
of the basis the Schrödinger equation behaves as a vector equation. In particular,
the right hand side of the equation becomes equal to − i
Û−1ĥϕ. Since coordinates
of points ϕ also change to become ψ = Û−1ϕ, the right hand side of the equation
takes the form
Û−1ĥÛψ. (3.3)
Note that Û−1 and Û in Eq. (3.3) act on different spaces! Namely, Û−1 acts on
the tangent space TϕC
2, while Û acts on C2 itself. It is common, however, to
identify vector spaces with the spaces tangent to them. By following this practice,
one sees that under a change of basis the Hamiltonian is transformed by ĥ −→
Û−1ĥÛ . In other words, it transforms as a tensor of rank (1, 1). If in addition ĥ
and Û commute, then (and only then!) the Hamiltonian is invariant and the usual
conserved quantities exist in accordance with the Noether’s theorem.
A particular choice of an orthonormal basis on the space of states C2 (alterna-
tively, on S3 or CP 1) has an affect on results of observations expressed in the basis.
The reason for that is clear: which state is a spin-up state, for example, depends
on the basis in C2. This is quite analogous to dependency of the state of rest in
classical mechanics on a choice of reference system.
Alternatively, under the identification Eq. (2.8) of space C2 with a space of 2×2-
matrices the choice of an orthonormal basis in C2 dictates the choice of basis in the
Lie algebra su(2), i.e., the choice of sigma-matrices. A unitary transformation of
16 Alexey A. Kryukov
sigma-matrices changes their eigenvectors, thus affecting the results of observations
expressed in terms of these eigenvectors.
The above can be also rephrased in terms of the geometry of the projective
space CP 1 = S2 ⊂ R3. Namely, a unitary transformation of basis in C2 induces an
orthogonal transformation of basis on R3. But the choice of a basis in R3 determines
the result of measurement of the Z-component of electron’s spin.
One can, therefore, conclude that a choice of coordinates on manifolds C2, S3
and CP 1 has a physical meaning similar to the choice of a reference system on the
classical space and should not be neglected.
On the other hand, physical laws must be, of course, coordinate independent.
In the considered case this is assured by the geometric nature of the model. In
particular, the Schrödinger equation Eq. (1.1) can be written in a form independent
of an orthonormal basis on the space of spin states C2, i.e., in a vector form. In this
case a state is just a vector Φ of C2 rather than a column
of components in a
particular orthonormal basis on C2. The Hamiltonian is an operator ĥ rather than
a matrix. The equation Eq. (1.1) then becomes a vector equation on the manifold
S3 written in a coordinate-free form:
= − i
ĥΦt. (3.4)
Note that the eigenvalue problems for observables in the model are also tensor
equations. In particular, the eigenvalue problem for the Hamiltonian can be written
in a coordinate free form:
ĥΦ = λΦ. (3.5)
The tensor character of equations of quantum theory in the model signifies that
the principle of relativity holds true on the space of states. This means, first of
all, that both, the active transformations on the space of states and the passive
transformations of coordinates on the space are available. In particular, there exist
various physically distinguishable reference systems on the space of states (say, dif-
ferent bases on C2). Most importantly, the validity of principle of relativity in the
theory means that the equations of the theory are the same when written in any
such reference system, that is, they are tensor equations.
One may doubt the significance of such a principle in the example. After all,
the SU(2) symmetry in the model has been known for years. Why would such
a “relativistic” view of this symmetry be useful? Note however, that the quantum
dynamics in the advocated approach takes place on the space of states. In particular,
the evolution of a quantum system is the motion along a geodesic on the (curved)
space of states. Because of that the notions of a reference system, of passive and
active transformations, of tensor equations, as well as other differential-geometric
notions on the space of states, become physically meaningful. This meaning is very
On the measurement problem 17
much in line with the meaning of similar notions in special and general relativity or
in the theory of gauge fields.
For instance, a change in direction of the magnetic field B induces an active
transformation on the space of states. This transformation has physically measur-
able consequences: it moves geodesics on the space of states and hence changes the
evolution of electron’s state. At the same time, the change in B can be compensated
by a passive transformation. Indeed, one can choose an orthonormal basis on C2 so
that the components of B in the corresponding basis in R3 remain the same. The
equation of the new electron’s path in this basis coincides then with the original
Most certainly, the above principle is different from the principle of relativity in
space-time. In fact, it deals with tensor properties of equations on a Hilbert space
of states, rather than on space-time. At the same time, it has the same kind of
underlying mathematics and the same spirit as the ordinary principle of relativity.
The principle is in fact a particular instance of the principle of functional relativity
introduced in Ref. [4].
As discussed, the Hamiltonian in the model is not in general invariant under
unitary transformations. In particular, an active unitary transformation associated
with a change in direction of the magnetic field B produces a new Hamiltonian. In
other words, the Hamiltonian is not a scalar in the tensor approach to the model.
On the other hand, due to geometric nature of the model, one can easily identify
the most important scalars (or invariants), forming the “bone structure” of the
theory. One such invariant is the distance between any two points on spaces of
states C2, S3 and CP 1 furnished with the above discussed metrics. Another one
is the speed of quantum evolution in S3 in a given magnetic field given by Eq.
(2.7). Yet another one is the scalar curvature of S3 or CP 1. This curvature can be
expressed in terms of the sectional curvature of S3 which was found to be 1/h̄2.
Let us remark that tensor character of the theory allows one to extend the origi-
nal symmetry group SU(2) to the group GL(2, C) of general linear transformations
acting on fibers of the tangent bundle over S3. Moreover, Schrödinger dynamics on
the space of states S3 can be formulated in a way invariant under general coordi-
nate transformations on S3. This follows at once from the fact that the Schrödinger
equation in the model is the equation of geodesics on the Riemannian manifold S3.
As such, this equation is meaningful in arbitrary coordinates on the manifold.
Finally, let us comment on a possible argument against the advocated geometric
approach. In the model considered here the Riemannian metric on the sphere S3 has
turned the sphere into a manifold of constant sectional curvature. This allowed us
to relate the curvature of the metric with the Planck’s constant. However, in general
the sectional curvature of the Riemannian metric defined by Eq. (1.3) will not be
a constant. Even when it is, there is no reason for this constant to be the same as
in the model under discussion. It seems therefore that by making the Riemannian
metric depend on the Hamiltonian of the system, one shall in general loose the
18 Alexey A. Kryukov
relationship between the curvature and the Planck’s constant!
To reply to this argument recall that the model considered here is non-relativistic
(in the usual sense). In particular, the equation Eq. (2.1) is a special case of the
Pauli equation. The latter equation is well known to be the non-relativistic limit of
the Dirac equation for electron in electromagnetic field. The Dirac equation can be
written in the form
= cσ̂ ·
p̂− e
, (3.6)
where ϕ̃, χ̃ are two-component spinors, (φ,A) is the 4-potential of the field, p̂ is the
momentum operator and e is the electron’s charge (see, for example, Ref. [8]). The
largest term in Eq. (3.6) is the one containing the mc2 factor. By substituting
(3.7)
into Eq. (3.6) one recovers in the standard way the Pauli equation for the spin state
ϕ with values in C2.
Assume that the 4-potential in Eq. (3.6) describes a weak homogeneous magnetic
field B and let ĥD be the corresponding Hamiltonian. Consider the metric GD given
by Eq. (1.3) with the Hamiltonian ĥD. Then solutions to the Dirac equation for
electron in the field B are geodesics in this metric. Note that the Hamiltonian ĥD
has in the non-relativistic limit the form mc2 + ĥ, where ĥ is the non-relativistic
Hamiltonian used in Eq. (2.1). Accordingly, the metric GD can be written in
the form (h̄/mc2)2(I + ǫ), where ǫ is a small correction due to the Hamiltonian ĥ
and I is the identity. It follows that the sectional curvature of GD consists of the
main term of the order (mc2/h̄)2 and a small correction due to ĥ. Since the main
term is constant (i.e., it does not depend on the fields), the advocated geometric
interpretation of the curvature remains possible. However, a more careful analysis
of the situation requires a “functional relativistic” formulation of the problem and
will be discussed elsewhere.
4 THE PROCESS OF MEASUREMENT
In the model under consideration the Schrödinger equation is the equation of geodesics
on the space of states S3 furnished with the Killing metric. That means that the
dynamics in the theory takes place on the Hilbert space of states rather than on the
classical space. In this and the following sections it will be argued that the space
of states is also the most appropriate background for tackling problems related to
quantum measurement. In particular, the process of collapse of a state can be re-
garded as a geodesic motion in the space of states with the metric “skewed” by the
measuring device.
On the measurement problem 19
Consider a pair of spin-1/2 particles. In QM the most general spin state of such
a pair has a form
i,j=±
cijϕiψj , (4.1)
where ϕ+, ψ+ are spin-up, and ϕ−, ψ− spin-down states of the particles. The Hilbert
space of states having such a form is the tensor product C2⊗C2. The unit normalized
states form a sphere S7 in this four-dimensional complex space.
Whenever Ψ is not a product of states of the particles, the state of the pair is
called entangled. It is well known that, when the particles are microscopic (i.e., suf-
ficiently small in mass and size), the entangled states do indeed exist. By assuming
the universal validity of QM, one concludes that the entangled states can be also
prepared when one of the particles is replaced with a macroscopic measuring device,
designed to measure spin of the second particle. In this case the total state of the
system has the form Ψ = aϕ+ψ+ + bϕ−ψ− where ψ± represent states of the device,
corresponding to spin-up and spin-down outcomes of measurement. However, unlike
the case of microscopic objects, the entangled states with macroscopic objects have
never been observed in experiments. The phenomenon of decoherence does not help
resolve this problem because the mixtures of states of macroscopic objects have not
been observed in experiments either.
Recall that in the classical physics the motion of a pair of interacting particles
on a manifold can be thought of as a motion of a point in a higher dimensional
configuration space. Suppose in particular that particles of massesm andM interact
gravitationally and move in the space R3 in accordance with the Newton’s Second
Law. Then the motion of the pair is represented by a trajectory in configuration
space R6. However, if M ≫ m, the motion simplifies and can be thought of as a
motion of the particle of mass m in the field created by the particle of mass M . In
this case the configuration space R6 of the pair is “effectively reduced” to the space
R3 and the field on R3 created by the heavier particle.
An analogous “reduction” of configuration space is implicitly present in the
unitary QM whenever the influence of a “macroscopic surrounding” of a quantum
system is accounted for by an appropriate choice of potential in the Schrödinger
equation.
Let us explore the idea that a similar approach can be applied to the process
of measurement in the model. Namely, assume that the motion of the total state
function of the electron and the measuring device during their interaction can be
effectively replaced with the motion of electron’s state function in S3 under the
influence of a physical field on S3 created by the measuring device. This means
that, in some sense, the state function of the device does not change much as a
result of interaction.
One immediate objection is that the observed states of the device are orthogonal
and so the state cannot change “just a little”. Without addressing this problem in
detail, let us point out that the metric on the space of states of the device may be
20 Alexey A. Kryukov
“skewed”. As a result, two different position eigenstates of a pointer may become
very close in this metric. For instance, let H be the Hilbert space obtained by
completion of a space of ordinary functions on R3 with respect to the inner product
(ϕ,ψ)H =
e−(x−y)
ϕ(x)ψ(y)d3xd3y. (4.2)
Such a space contains in general the eigenstates of position operator, i.e., the delta-
functions δ(x − a). Moreover, two different position eigenstates δ(x − a), δ(x − b)
with ‖a− b‖R3 ≪ 1 are close in the metric Eq. (4.2) (see Ref. [4]). If, in particular,
δ(x − a), δ(x − b) are the eigenstates of a pointer, then the fact that theses states
are close in H can imply that the state of the pointer does not change much in the
process of interaction with a measured microscopic system.
This argument does not prove, of course, that the proposed “reduction formal-
ism” can be consistently implemented into the theory. To validate the formalism
one must demonstrate that all imaginable measurements in QM can be modeled (at
least in terms of their outcomes) by a field on the Hilbert space of states of the mea-
sured system. In what follows such a demonstration will be presented in the case of
a finite dimensional space of states and a time-independent observable. Namely, in
this case a specific working model of measurement based on a perturbation of the
metric on the space of states of the system will be constructed. Moreover, the effec-
tiveness and the scope of the proposed method suggest that it can be successfully
applied in general.
For guidance in modeling the process of measurement let us return to the ex-
ample under consideration. Suppose that the device in the example measures the
component of electron’s spin in the direction of magnetic field. Equivalently, since
the Hamiltonian is given by ĥ = −µσ̂ · B, the device can be designed to measure
the electron’s energy. Without loss of generality one may assume that the field is
directed along the Z-axis. Then the two eigenstates {ψ1}, {ψ2} of ĥ in CP 1 are
positioned on the Z-axis at the poles of the sphere S2 = CP 1. Recall that these
eigenstates are zeros of the vector field hϕ = − ih̄ ĥϕ projected on CP
1. In particu-
lar, the Schrödinger evolution of ψ1, ψ2 is projectively trivial. So, as a result of the
interaction between the electron and the device, the original circular motion of the
electron’s state along a parallel on S2 = CP 1 is changed to the state of rest at one
of the poles on the sphere. The following hypothesis, which will be clarified and
exemplified later on, seems to be in order:
(H1) The measuring device creates a physical field on the sphere of states with
sources at the eigenstates of the measured observable. This field is capable of
driving the electron’s state toward one of the eigenstates.
Note that in the position measuring experiments a measuring device is a system
of counters distributed in space at the eigenstates of the position observable. The
counters are indeed sources of interaction between the particle and the apparatus.
On the measurement problem 21
The points in the classical space where the measured particle can be found, can be
identified with the position eigenstates of the particle (see Refs. [4], [6]). Since the
counters are also located at these points, one can identify them with sources in the
space of states of the particle. Finally, because the sources are capable of catching
the particles, the (loosely stated) hypothesis (H1) is satisfied. This gives one the
hope that a specific form of the hypothesis can be, in fact, realized.
What could be the nature of the field postulated in the hypothesis? Recall that
the Riemannian metric in the model is dynamical, i.e., it drives the evolution of the
electron’s state. Suppose that the field is nothing but a perturbation of the metric
on the sphere of states S3, induced by the presence of the measuring device. The
evolution of the electron’s state during the measurement is then the motion along a
geodesic in the perturbed metric. Note that the metric on the total space S3 and not
only on the base space CP 1 = S2 must be perturbed. In fact, as already discussed,
the projection {ϕt} of a Schrödinger evolution ϕt is not in general a geodesic.
The above approach is certainly attractive, in particular, it does not require any
ad hoc features in the theory. Furthermore, the approach can be easily implemented
by an appropriate “denting” of the sphere of states of the system. Namely, as shown
below, by perturbing the metric on the sphere S3 one can “redirect” the evolution
of electron’s state so that the state would become stationary.
For a greater generality, assume that the Hilbert space H of states of the system
is n-dimensional and let (ϕ1, ..., ϕn) be the usual coordinates on H = Cn. Let the
Riemannian metric on H to be of the form
gik = η
2δik, (4.3)
where η = η(ϕ) is a function on H. The equation of geodesics in this metric can be
obtained in the usual way by variation of the length functional on paths ϕt. Let s
be the arc length parameter and let τ be a parameter defined by dτ = ds
. Then the
equation of geodesics in the metric Eq. (4.3) on H can be written in the form
∇η2, (4.4)
where ϕτ is identified with ϕt(τ). A similar equation in R
3 is well known in geomet-
rical optics where it describes propagation of rays in a media with refractive index
η. The equation Eq. (4.4) is also similar to the Newton equation of motion for a
unit mass in the field U = −η2/2.
The form of equation Eq. (4.4) makes it easy to see that for any sufficiently
smooth path ϕτ there exists a function η such that the equation is satisfied, at
least on a neighborhood of ϕ0 = ϕτ |τ=0. That is, ϕτ is a geodesic through ϕ0 in
the Riemannian metric Eq. (4.3) on H, at least for small values of τ > 0. That
also means that an arbitrary sufficiently smooth path ϕτ with values on the sphere
SH ⊂ H is a geodesic through ϕ0 = ϕτ |τ=0 in an appropriate Riemannian metric
on SH , at least for small τ > 0. As a side remark, various global results of this kind
22 Alexey A. Kryukov
can be obtained by applying the methods of geometrical optics to Eq. (4.4) (see
Ref. [7] for a review of geometrical optics).
Consider now an arbitrary non-stationary Schrödinger evolution ϕt on S
H driven
by an invertible time-independent Hamiltonian ĥ. Let SH be furnished with a
Riemannian metric GRϕ in which the evolution is a motion along a geodesic. It
is known that such a Riemannian metric on SH exists. Pick a moment of time
t = a and an eigenstate ψ ∈ SH of the Hamiltonian ĥ. Consider the geodesic χt
connecting ϕa and ψ. By Eq. (4.4), there exists a perturbation of the metric GRϕ
on a small neighborhood of ϕa which transforms the geodesic ϕt into the geodesic
χt. Note that one could similarly perturb the metric on a small neighborhood of ψ
to transform χt into the stationary geodesics through ψ.
The provided perturbation of the metric takes place on the sphere of state SH =
S2n−1. What could one say about the metric and the motion on the projective space
CPn−1 = S2n−1/S1? To answer, consider the Riemannian metric on Cn∗ (i.e., C
without the origin), defined for all ϕ ∈ Cn∗ and all pairs of vectors ξ, η ∈ TϕCn∗ by
GRϕ(ξ, η) =
ĥ−2ξ, η
||ϕ||2Cn
. (4.5)
This metric, being restricted to the sphere S2n−1, is a particular case of the metric
Eq. (1.3). Solutions to the Schrödinger equation are geodesics in the metric Eq.
(4.5) on Cn∗ (and in the induced metric on the sphere, see Ref. [4]).
Notice that the multiplication map λ : ϕ −→ λϕ with λ ∈ C∗ is an isometry of
the metric Eq. (4.5), that is, GRλϕ(dλξ, dλη) = GRϕ(ξ, η). This is clear because
dλ = λ by linearity of the map and Re(λξ, λη)Cn/||λϕ||2Cn = Re(ξ, η)Cn/||ϕ||2Cn .
Because of that, the metric Eq. (4.5) “projects down” to CPn, giving a metric
on the projective space. More precisely, the metric Eq. (4.5) is induced by the
projection of Cn∗ onto CP
n−1, furnished with a Riemannian metric. Provided ĥ2 is
proportional to the identity operator, the latter metric coincides with the Fubini-
Studi metric on CPn−1 (see Ref. [4]).
Note however that the multiplication by a complex number may not remain an
isometry of the perturbed metric on Cn∗ . That is, the metric on C
∗ , needed to
“redirect” the Schrödinger evolution to account for the process of measurement,
may not originate in a Riemannian metric on CPn−1. In mathematical terms, the
projection of Cn∗ with a perturbed metric onto CP
n−1 is not in general a Riemannian
submersion. In particular, such is the case for the discussed local perturbation of
the metric on Cn∗ . This suggests once again that the metric on the sphere S
the approach under investigation has a greater significance than the metric on the
projective space CPH .
This analysis demonstrates that by an appropriate “denting” of the sphere of
states SH , dimH <∞, one can locally affect geodesics on the sphere in a desirable
fashion. In particular, by perturbing the Riemannian metric GRϕ on S
H one can
alter the Schrödinger evolution of the state and drive the state toward one of the
On the measurement problem 23
eigenstates of the measured observable. It follows that the physical field in the
hypothesis (H1) can be indeed identified with a perturbation of the metric GRϕ.
Note that the resulting metric is time-independent. The electron’s state in the
construction propagates along a geodesic on the sphere of states, runs into a region
with perturbed metric and collapses.
It is important to remark that the above demonstration is only an existence
proof; it does not provide a realistic model of interaction between the system and
the device. Moreover, a particular nature of the field in the hypothesis (H1) will
not be essential in the following. The thorough analysis of this nature requires the
equations of the field and is left for the upcoming publications. The mere existence
of the field satisfying the needed properties will be sufficient for the purpose of this
Letter.
The postulated physical field may be able to drive the state to one of the eigen-
states, i.e., it may be responsible for the collapse itself. However, in such a scenario
collapse seems to be a deterministic process and the probabilistic nature of collapse
to a particular eigenstate is not explained. Recall now that in accordance with
Eq. (2.45), the probability of collapse of a given state ϕ to an eigenstate ψk of an
observable depends only on the distance θ between the states in the Fubini-Study
metric on the projective space of states. This crucial property allows one to resolve
the remaining difficulty in creating a working probabilistic model of collapse.
Indeed, suppose that the field sources in the hypothesis (H1) are not fixed at
the eigenstates ψk but fluctuate randomly about the eigenstates. In particular,
the projections of the sources fluctuate randomly about the points {ψk} on CPH .
Suppose further that fluctuations with projections of a small (in the Fubini-Study
metric) amplitude are more likely to occur. Suppose finally that if a source reaches a
small neighborhood of the state ϕ, it alters the evolution of the state and diverts it to
the corresponding eigenstate (say, by perturbing the metric on the neighborhood).
Then, the closer the state {ϕ} is to a particular eigenstate {ψk} (consequently, the
larger the modulus of the coefficient ck in the decomposition ϕ =
i ciψi is), the
more likely it becomes for the source fluctuating about ψk to reach (and collapse)
the state. At the same time, the further {ϕ} and {ψm} are (and hence, the smaller
the modulus of cm is), the less likely it becomes for the source fluctuating about ψm
to reach the state. In such a way, the competition between the sources can lead to
the standard Born rule for the probability of collapse.
To prove the latter claim, let us first of all make the above assumptions precise.
To keep the analysis simple, the assumptions will refer to the fibre bundle π : S3 −→
S2 corresponding to the model under discussion. However, it will be clear that the
measuring process on any fibre bundle π : S2n−1 −→ CPn−1 and for any time-
independent observable on the corresponding space of states can be treated in the
same way.
Let θ ∈ (−π, π] and α ∈ (−π/2, π/2] be the angular coordinates on the projective
space CP 1 = S2. Here the coordinate curves α = α0 yield great circles (pairs of
24 Alexey A. Kryukov
meridians) through the poles θ = 0, θ = π of S2 and the curves θ = θ0 yield half
the parallels on S2. As before, the eigenstates {ψ1}, {ψ2} are located at the poles
of S2. Let β ∈ (−π, π] be the phase of a state on the sphere S3. Then the triple
(θ, α, β) form a coordinate system on S3. In terms of these coordinates the following
hypothesis is now accepted:
(H2a) Fluctuations of each source along the sphere of states S3 can be described by
a three-dimensional stochastic process (θt, αt, βt). For instance, consider the
source associated with the eigenstate at θ = 0. For any t = t0 the random
variables θt0 , αt0 , βt0 describing the source are independent. For any t = t0
the probability density of the random variable θt0 ∈ (−π, π] is equal to 1πcos
The random variables αt0 , βt0 are uniformly distributed. The mean function
of each process is zero. For any two times t1, t2, t1 6= t2, the random variables
θt1 , θt2 are practically statistically independent, so that the stochastic process
θt is uncorrelated in time. In other words, θt is ideally a white noise process.
The same is true about the processes αt, βt. The stochastic processes describing
different sources are independent.
(H2b) A source at a point ϕ ∈ S3 with coordinates (θ, α, β) may be identified
with a perturbation of the metric on a small neighborhood U ⊂ S3 of the
point. If at some time t the U -neighborhood of a particular source contains the
electron’s state, the perturbation of the metric alters the evolution of the state
and collapses it to the corresponding eigenstate.
Is there a realization of the hypothesis? It was already verified that (H2b) can be
realized for any Schrödinger evolution ϕt by a “lensing” effect, i.e., by redirecting ϕt
toward the eigenstate. Also, the white noise process postulated in (H2a) certainly
exists as a mathematical idealization. Moreover, the processes of this kind are
common in physics. Probably the most appropriate example is the thermal noise,
i.e., the random process describing the electric current created by the thermal motion
of electrons inside a conductor.
Could the random fluctuations of the sources in (H2a) be of a similar origin?
During a measurement the measuring device interacts with the measured system. At
the same time, the molecules (atoms, particles) of the device experience a random
thermal motion. In a general (non-stationary) case fluctuations of molecules result
in fluctuations of their states on the space of states. So the main new assumption
made is that the interaction between the measured system and the device also takes
place on the space of states of the system rather than on the classical space alone.
Fluctuations of states of the molecules are then associated with fluctuations of the
field sources along the space of states leading to the postulated stochastic process.
What is the probability dP1 of collapse of a state ϕt to a particular eigenstate
ψ1 at some specific time t = t0 in the hypothesis? Such a probability is equal to the
probability for the U -neighborhood of the corresponding source to contain the state
On the measurement problem 25
at this time. Let (θ0, α0, β0) be coordinates of ϕt0 in the chosen coordinate system
on S3. If U is sufficiently small, the change in the probability density of θt0 across
U can be neglected, and, therefore,
dP1 =
dV. (4.6)
Here dV is the volume of U which in this simplest case is identified with dθdαdβ for
some fixed values of the differentials. 1 According to the hypothesis, the random
variables describing the positions of different sources at t = t0 are independent.
Therefore, the probability dP2 of collapse of the state ϕt0 to another eigenstate ψ2
can be computed in the same way. Finally, since the stochastic processes describing
the sources are uncorrelated in time, the probability for the U -neighborhood of a
source to contain any particular point ϕ0 is not affected by the previous history of
the source. In particular, the probability rule Eq. (4.6) is universally valid.
On the other hand, according to Eq. (2.45), the expression
|c1|2 = cos2
(4.7)
represents the standard probability of transition from the state {ϕt0} to the state
{ψ1}, provided θ0 is the distance between the states in the Fubini-Study metric and
c1 is the coefficient of ψ1 in the decomposition of ϕt0 . Clearly, the distance θ0 in
the formula Eq. (4.7) can be replaced with the the angle θ0 between the states,
explaining the chosen notation. Of course, a similar formula holds true for the
coefficient c2 of ψ2. It follows that the ratio dP1/dP2 coincides with |c1|2/|c2|2. The
conclusion is that the postulated hypothesis yields the Born rule for the probability
of collapse as was claimed.
The “single-push” process of collapse of the state ϕt to an eigenstate can be
replaced with a more elaborate stochastic process. Each encounter with a source in
this process results in a decrease in the distance θ ∈ [0, π] between the state {ϕt}
and the corresponding eigenstate {ψk} by a certain value δ. Between the encounters
the state undergoes the ordinary Schrödinger evolution. Assume for simplicity that
the frequency of encounters is sufficiently high. In this case one can neglect the
Schrödinger evolution of the state during the measurement. The stochastic process
of collapse can be then defined as a finite, time-homogeneous Markov chain with
absorbing boundaries θ = 0 and θ = π and with the number of states equal to
π/δ + 1. The transition matrix for the process can be found via simple formulas
1The volume element for the sphere S3 with the usual metric in the chosen coordinates is
dSV =
sin θ
cos θ
dθdαdβ. It would be more appropriate to associate dSV with the volume of
U . Moreover, the expression sin θ
cos θ
is the derivative of cos2 θ
. This leads one to interesting
models in which at any t = t0 the random variable θt0 is uniformly distributed on (−π, π] and
the coefficient cos2 θ
appears in the (complementary) cumulative distribution function due to the
factor sin θ
cos θ
in the volume element dSV . However, the element dV will be sufficient for the
purpose of this Letter.
26 Alexey A. Kryukov
from the condition that the steady-state transition matrix has the right transition
probabilities from any state θi to the absorbing states {ψ1}, {ψ2} (namely, cos2 θi2
and cos2 π−θi
= sin2 θi
). The resulting process is a generalization of the (biased)
random walk with absorbing boundaries (also known as the gambler’s ruin), in which
the transition probabilities vary with the state. Namely, the transition probabilities
for a step toward an absorbing state {ψi} increase as the electron’s state moves
closer to {ψi}.
Various stochastic processes have been extensively used in modeling collapse
(see reviews Refs. [9] and [10]). In general words, the existing models are based
on adding an external random noise term and a term containing the measured
observable to the Schrödinger equation. The term with the observable provides the
“choices” for observations, while the random noise term is a “chooser” (see Ref.
[9]). The probability density for a particular noise in the models is given by yet
another equation. This equation makes it more probable for the noise to fluctuate
around values associated with the eigenstates of the observable and in such a way
that the probability of the noise also depends on the initial state ϕ0 of the system.
When applied to the process of measurement, the models of this kind explain the
probabilistic results of observations by relating them to the random noise, selected
by the mentioned probability rule. At the same time, the physical reason for a
particular random noise remains unexplained (see Ref. [9]).
Even without analyzing the existing stochastic models of collapse in detail, one
can pinpoint the essential difference of the model considered here. Namely, the
noise in the advocated approach is a process on the space of states which does not
depend on a particular state ϕ of the measured system. In particular, the noise in
the model does not change when the state ϕ changes. This independence of the
noise from the state of the measured system opens a way for associating it with the
measuring device itself. For example, as already discussed, the noise may originate
in the thermal motion of molecules of the device, considered as a process on the
space of states.
Another important observation is that the process of collapse in the model is
a deterministic process on the space of states. In fact, by associating the random
noise with a physical process, one should be able, in principle, to provide a specific
functional form of the noise. In this case it becomes possible to predict the time
and the outcome of collapse for an arbitrary evolution ϕt of the system.
Note that the proposed mechanism of collapse, although particularly simple, is
not the only one satisfying the hypotheses (H1), (H2). Furthermore, as already
mentioned, the proposed mechanism is far from being realistic at this stage. The
ultimate choice of a physically valid scenario of collapse depends crucially on the
field equations on the space of states and cannot be provided at this time. Instead,
let us demonstrate that under the above assumptions even such a simple mechanism
sheds new light on the quantum measurement problem.
On the measurement problem 27
5 THE MEASUREMENT PROBLEM
The observations made so far, combined with the results of Ref. [4] suggest the
following statements about objects and interactions in QM:
(S1) Physical objects in QM are most adequately represented by points of a Hilbert
manifold of states. In this sense, they have a functional nature.
(S2) Physical interactions involving microscopic objects (in particular, the process
of measurement) are most adequately described as processes on the manifold
of states, rather than on the classical space alone. In other words, the manifold
of states represents a new arena for description of physical processes.
(S3) The interactions can be described in terms of the Riemannian metric on the
manifold of states. In particular, the states of microscopic particles move along
geodesics on the sphere of states furnished with a Riemannian metric. In this
sense, the interactions may have a geometric origin.
Let us investigate how these statements together with the hypotheses (H1),(H2)
in the previous section help provide an understanding of the measurement process
in QM. First of all, a particular measuring device can be modeled by a metric field
with sources at the eigenstates of the measured observable. That is, the kind of
measurement performed on the system determines a specific field created by the
device on the manifold of states of the system. Provided the model based on the
hypotheses can be developed into a consistent physical theory, the latter result would
resolve the so-called preferred basis problem in QM. The problem can be formulated
as follows:
(P1) How could the electron’s state ϕ “know”, which basis {ek} to use to associate
the right probabilities to the coefficients ck in decomposition ϕ =
k ckek?
The constructed model suggests the following answer:
(R1) The coefficients of state of the system in the basis of eigenvectors of the ob-
servable describe position of the state relative to the sources of the field created
by the device. As already discussed, this position determines the probability
for the state to be “pushed” by sources to a particular eigenstate point on the
projective space of states. In other words, by creating a surrounding field in
the space of states, the device itself defines the “preferred” basis.
Next, the process of collapse in the model is an objective process driving the state
of the system to an eigenstate of the measured observable. The stochastic nature of
the process is due in the model to random fluctuations of sources associated with
measuring “parts” of the device. These fluctuations could be directly related to the
usual chaotic oscillations of the “parts” extended to the space of states of the system.
28 Alexey A. Kryukov
The “classical world” in the approach is represented by eigenstates of observ-
ables. The set of all eigenstates of an observable Â for a quantum system will be
called the set of Â-classical states (or points) for the system. So the set of Â-classical
states is a subset in the Hilbert space of states (or the corresponding space of phys-
ical states) of the system. Let us point out that there is nothing special about the
classical states as what is “classical” with respect to one observable is “quantum”
with respect to another one.
In the model under investigation, the integral curves of vector fields associated
with observables are geodesics in the Killing metric on the sphere of states. More
generally, the integral curves of vector fields associated with any reasonable set
of physical observables of a quantum system can be shown to be geodesics in an
appropriate Riemannian metric on the sphere of states of the system. The non-
commutativity of observables is then tied to the curvature of the metric.
Let us investigate in this light the “mother of quantum mechanics”, i.e., the
double-slit experiment. There are two main paradoxes associated with the experi-
ment:
(P2) How could the electron pass trough both slits at once?
(P3) How could a measuring device inserted after the screen with the slits instanta-
neously change the way in which the electron has passed through the screen?
The Hilbert space of states in the double-slit experiment is infinite-dimensional.
It would be helpful to consider at the same time a version of the experiment with
a finite dimensional space of states. For this let us return to the motion of electron
in a homogeneous magnetic field. Recall that the spin state of the electron evolves
in accordance with equation Eq. (2.3). If the field is directed along the Y -axis and
the initial state of the electron is
, the solution of Eq. (2.3) is given by
cos µB0
sin µB0
. (5.1)
If t changes, say, between 0 and π
, then the process of passing through the field
results in a “splitting” of the original spin-up eigenstate of the operator σ̂z into a
superposition of spin-up and spin-down states. In this respect the experiment is a
finite dimensional version of the double-slit experiment in which a localized electron
wave packet gets transformed by the screen with the slits into a superposition of
two wave packets.
With this in hand, let us address the above mentioned paradoxes (P2) and
(P3) of the double-slit experiment. Let us call the original double-slit experiment
and the experiment with an electron in a homogeneous magnetic field the E1 and
E2 experiments respectively. The electron in the experiment E2 evolves from the
On the measurement problem 29
original “classical” state
, into superposition
of two eigenstates of
σ̂z. During this evolution the Z-component of the electron’s spin is unknown. The
reason for that is clear: for 0 < t ≤ π
, the trajectory of electron’s state on
S2 = CP 1 does not pass through the classical states, i.e., through the eigenstates
of σ̂z. Classically speaking, one has a paradox here: the electron’s intrinsic angular
momentum is not defined. Instead, the electron is in a superposition of states of
two different angular momenta. In a way, the electron’s spin is up and down at the
same time.
Note however that the state function ϕt is defined for t = 0 as well as for t > 0.
In other words, it describes the classical and the non-classical states equally well.
Moreover, any (physical) state of the electron is just a point on S2. The evolution
of the electron’s state is just a path {ϕt} with values in S2. The classical way of
thinking tells us that the electron somehow splits into two parts that evolve along
different paths. However, the actual evolution of the electron is most adequately
described by a single path ϕt, thereby confirming the statements (S1), (S2).
The situation in experiment E1 is almost identical, although the paradox here
is more dramatic as our classical intuition of position is very strong. Again, the
intuition tells us that the electron splits into two parts which are passing through
different slits. However, the electron’s evolution is most adequately described by a
path in the space of states. Of course, such a path does not “split” and it describes
the evolution of electron before and after the screen with the slits equally well. The
resolution of the paradox (P2) is then as follows:
(R2) The electron in the experiment E2 is not in the spin-up and spin-down states
at once. Rather, it is in the state that is neither a spin-up, nor a spin-down
state. Similarly, the electron in the experiment E1 does not pass through two
slits at once. Rather, it does not pass through the slits at all! Indeed, for
a state to be a spin-up state, for example, it must be at the north pole of
the sphere S2 of states, which is not the case for the electron’s state in the
experiment E2 for t > 0. Similarly, to pass through a slit is to have a state
localized at that slit. But the state of the electron after its interaction with
the screen in the experiment E1 is not localized. In other words, the electron
(i.e., the electron’s state) is located at a point on the space of states that is
different from the point at which an electron passing through the slit would
have been. To put it figuratively, the electron passes over rather than through
the slits.
One can see that the paradox (P2) is resolved by considering the motion of
electron in the experiments E1, E2 as happening in the functional space of states
rather than on the classical space or on the space of angular momenta. Vaguely
speaking, the “functional” (i.e., consistent with (S1) and (S2)) way of thinking
makes the paradox disappear. In light of this, the resolution of the paradox (P3)
30 Alexey A. Kryukov
is now immediate:
(R3) How could a measuring device inserted after the screen change the way the
electron has passed through the screen? The answer is: it does not! If a counter
is inserted behind the screen (and sufficiently far from the screen), the process
of “passing through” the screen is not affected by it. In particular, the counter
can be placed after the electron has already passed “through” the screen and
this will not change the history. Indeed, the evolution of electron is described
by a path ϕt. If only one slit is open, this path passes through a point in the
space of states which is represented by a state function localized at the slit.
If, however, both slits are open, the path does not pass through such a point.
This is true independently of any measurement done behind the screen. What
the counter does is to change the path ϕt for larger values of the parameter t
so as to produce a state localized at one of the slits in a way discussed in the
previous section. As a result, the final state is as if the electron had passed
through only one of the slits. However, no reality should be attached in this
case to the event of passing though the slits. Once again, the electron in the
experiment E1 does not pass through the slits. Likewise, the state function ϕt
does not describe the probability of passing through one of the slits (but only
the probability to be found by one of the slits). Rather, ϕt itself represents a
new “functional” reality of the world which is more adequate in QM than the
familiar classical reality.
To summarize, the paradox (P3) is resolved by accepting the statements (S1)
and (S2), i.e., by recognizing the evolution of electron in the space of states as
physical (i.e., real) and by allowing a “deformation” of such an evolution in the
presence of a measuring device.
Let us finally analyze a measurement performed on a pair of spin-1/2 particles.
This will give a hint as to how to proceed in more general cases. As already discussed
in section 4, the total quantum mechanical state of the pair is a point in the tensor
product of Hilbert spaces of each particle. In particular, the spin state of a pair of
electrons is an element of C2 ⊗ C2. A unit normalized state is a point on the unit
sphere S7 in this four dimensional complex space. Physical spin states of the pair
are then points in the complex projective space CP 3. Note that there may be points
in CP 3 that do not represent a physical state of the pair. In particular, if the total
angular momentum of the pair vanishes, the state of the pair can only be of the
form aϕ+ ⊗ ψ− + bϕ− ⊗ ψ+ with a, b ∈ C. Moreover, if the particles are identical,
one must have a = −b.
The σ̂z-classical points on CP
3 are the points where both particles have a specific
value of the Z-component of spin. These points are represented by the products of
ϕ± and ψ±. In the case when the total angular momentum of the pair vanishes, the
points are represented by ϕ+ ⊗ ψ− and ϕ− ⊗ ψ+. The evolution of the pair is now
a path with values in S7. This path projects down to a path with values on the
underlying space CP 3.
On the measurement problem 31
With these standard ingredients in place, one can analyze now a version of the
famous Einstein-Podolsky-Rosen (EPR) paradox in QM:
(P4) Given a pair of spin-1/2 particles in entangled state atϕ+ ⊗ψ− + btϕ− ⊗ψ+,
how could it be, that by measuring the Z-component of spin of one of them
one fixes the Z-component of spin of the other one, even if the particles are
far apart?
Note that this paradox is similar to the paradox (P3), taking place on the space
of states of the pair. Indeed, the resolution of the paradox is almost identical:
(R4) Physical reality is described by the path ϕt with values in the space S
7 (or
CP 3) of states of the pair. Unless one of the coefficients at, bt in atϕ+ ⊗
ψ− + btϕ− ⊗ ψ+ is zero for some t, the path ϕt does not pass through the
σ̂z-classical points ϕ+ ⊗ ψ− or ϕ− ⊗ ψ+. That is, the particles do not have
any Z-component of spin. To measure the Z-component of spin of a particle
is to make the path ϕt pass through one of the σ̂z-classical points ϕ+ ⊗ψ− or
ϕ− ⊗ ψ+. In this case (and only in this case!) the Z-components of spin of
both particles are defined and take opposite values. Furthermore, as with a
single particle, the interaction with the measuring device is assumed to cause
a “deformation” of the path ϕt. The resulting path ends up then at one of the
classical points via a stochastic process on the space of states.
The full version of the experiment involving a spatial separation of the particles
is even more dramatic. How could the second particle at a point y “know” about
measurement of the Z-component of spin performed on the first particle at a distant
point x?
Again, physical reality of the pair is most adequately described by a path
ϕt = atϕx+ ⊗ ψy− + btϕx− ⊗ ψy+ in the space H of states of the pair. Here
ϕx+ is the spin-up state of the first particle located at x and similarly for the
other state functions in ϕt. The classical points in H have the form ϕx+⊗ψy−
and ϕx−⊗ψy+. If ϕt does not pass through these points the spin of individual
particles is not defined. Intuitively, we think that if a particle is “here” (at
a point x), then it ought to have all attributes of a “real” particle, including
spin. But before the measurement is performed, the particle in the experiment
is not really here! Indeed, it is somewhere else on the sphere of states SH in
H (or on the corresponding projective space CPH). So if reality is associated
with the state function of the pair, the paradox is resolved.
But what about this “spooky action at a distance”? Notice that the new “func-
tional” reality does not use it! Indeed, the equation of geodesics is “local” in the
space of states S3, because it is a differential equation of geodesics on S3. Of course,
this locality in the space of states does not preclude a non-locality in the classical
32 Alexey A. Kryukov
space. Indeed, what is a point in the space of states may represent a pair of well
separated particles in the classical space. Furthermore, what is close in the metric
on the space of functions does not have to be close in the metric on the classical
space (see Ref. [4]). A detailed analysis of this will be, however, a subject for a
different paper.
REFERENCES
1. M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984)
2. J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990)
3. B. Simon, Phys. Rev. Lett. 51, 2167 (1983)
4. A. Kryukov, Found. Phys. 36, 175 (2006)
5. A. Kryukov, Int. J. Math. & Math. Sci. 14, 2241 (2005)
6. A. Kryukov, Found. Phys. 34, 1225 (2004)
7. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous
Media (Springer, 1990)
8. J. Bjorken and S. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964)
9. P. Pearle, “Collapse models,” in Open Systems and Measurement in Relativis-
tic Quantum Theory, F. Petruccione and H.P. Breuer, eds. (Springer, 1999)
10. A. Bassi & G. Ghirardi, Physics Reports 379, 257 (2003)
	GEOMETRY AND QUANTUM MECHANICS
	ELECTRON IN A HOMOGENEOUS MAGNETIC FIELD
	Quantum Mechanics on the Space of States S3
	Quantum Mechanics on the Projective Space of States CP1
	TENSOR PROPERTIES OF EQUATIONS IN THE MODEL
	THE PROCESS OF MEASUREMENT
	THE MEASUREMENT PROBLEM
