arXiv:0704.1084v1  [hep-th]  9 Apr 2007
Vacuum Energy and Renormalization on the Edge
M. Asorey1, D. Garćıa-Álvarez2, J. M. Muñoz-Castañeda1
1 Departamento de F́ısica Teórica. Facultad de Ciencias. Universidad de Zaragoza,
50009 Zaragoza. Spain
2 Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
E-mail: asorey@unizar.es
Abstract. The vacuum dependence on boundary conditions in quantum field
theories is analysed from a very general viewpoint. From this perspective the
renormalization prescriptions not only imply the renormalization of the couplings of the
theory in the bulk but also the appearance of a flow in the space of boundary conditions.
For regular boundaries this flow has a large variety of fixed points and no cyclic orbit.
The family of fixed points includes Neumann and Dirichlet boundary conditions. In
one-dimensional field theories pseudoperiodic and quasiperiodic boundary conditions
are also RG fixed points. Under these conditions massless bosonic free field theories
are conformally invariant. Among all fixed points only Neumann boundary conditions
are infrared stable fixed points. All other conformal invariant boundary conditions
become unstable under some relevant perturbations. In finite volumes we analyse the
dependence of the vacuum energy along the trajectories of the renormalization group
flow providing an interesting framework for dark energy evolution. On the contrary,
the renormalization group flow on the boundary does not affect the leading behaviour
of the entanglement entropy of the vacuum in one-dimensional conformally invariant
bosonic theories.
PACS numbers: 11.10.Hi,11.25.Hf
Keywords: Renormalization group, boundary conditions, vacuum energy.
http://arxiv.org/abs/0704.1084v1
Vacuum Energy and Renormalization on the Edge 2
1. Introduction
The emergence of the dark energy as one of the basic ingredients of the current
standard cosmological scenario, and the absence of an even vague understanding of
its possible origin, opens a window to the analysis of all possible mechanisms that
generate background energy (see e.g. [1] for a review of recent proposals). The main
problem is that the apparent value of the dark energy is very tiny compared with any
physical energy scale. A second problem is that in a generic quantum field theory
there is generation of vacuum energy and any renormalization prescription requires a
fine tuning, which is not very convincing without the quantisation of the gravitational
interaction.
The guess that dark energy might change with the evolution of the Universe can
be understood even if dark energy is just vacuum energy. The finite corrections due to
finite size of the causal Hubble domain decrease as the Universe continues to expand.
The aim of this paper is to analyse the variation of these finite size corrections under
of renormalization group on the space of boundary conditions for scalar field theories in
flat space, although the results are generalisable for more general backgrounds.
The dependence of the vacuum energy on the boundary conditions [2] is well known
since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for
recent revisions). However, boundaries might also be considered as a source of new,
although peculiar, interactions and therefore can undergo renormalization [7, 8]. The
renormalization of boundary conditions modifies the critical behaviour of the theory
[9, 10, 11]. In systems with boundaries or defects, the boundary RG flow induces a
dynamical behaviour on the boundaries. The dynamics of D-branes in string theory
emerges in this way [12].
The renormalization group flow is analysed from a global viewpoint in the most
general framework for boundary conditions of scalar field theories introduced in Ref.
[13]. In particular, we consider the possible existence of topological transitions [14]
induced by the renormalization of boundary conditions or cyclic orbits in the boundary
RG flow [15]. The dependence of the finite size corrections to the vacuum energy and
vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a
very general perspective.
2. Boundary conditions in Field Theory
The action which governs the dynamics of scalar field theory in a bounded domain
Ω of flat space consists of two different of terms, S(φ) = SB(φ) + Sb(φ). The first one
SB(φ) =
g dDx
|φ̇|2 − |∇φ|2 − V (|φ|2)
is defined in terms of the values of the fields in the bulk. The second term
Sb(ϕ) =
dD−1x
|ϕ̇|2+
ϕ∗∂nϕ+
∗)ϕ−|∇ϕ|2
Vacuum Energy and Renormalization on the Edge 3
depends only on the values of the fields at the boundary ∂Ω † . g
denotes the metric
induced on the boundary by the bulk flat metric, and ∂n is the normal derivative at the
boundary
ϕ = φ|
∂nϕ = ∂nφ|∂Ω. (3)
The presence of the boundary term Sb allows the generation of local classical equations
of motion without requiring any specific type of boundary conditions [19, 20]. Indeed,
the gradient term
V = 1
|∇φ|2 (4)
can be rewritten as
φ†∆φ+
φ† ∂nφ (5)
where ∆ is the Laplace-Beltrami operator ∆ = − ∂µ∂µ . In the quantum theory the
Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint
Hamiltonian
H = 1
∆+m2. (6)
for the free field theory (The inclusion of interactions does not changes the picture [21]).
This means that the classical fields must satisfy boundary conditions which make the
operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this
requirement [13] are in one-to-one correspondence with the group of unitary operators
of the boundary Hilbert space L2(∂Ω, C ). For any unitary operator U ∈ L2(∂Ω, C ), the
fields satisfying the boundary condition
ϕ− i ∂nϕ = U (ϕ+ i ∂nϕ) (7)
define a domain where ∆ is a selfadjoint operator.
In the case of open strings, the corresponding conformal 1+1 dimensional scalar
field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety
of admissible boundary conditions described by the unitary group M = U(2). The
unitary matrices
define Dirichlet, Neumann and periodic boundary conditions, which in string theory
correspond to a string attached to a D-brane background, free open and closed string
theories, respectively.
For higher N-dimensional target spaces, or N-component strings, the set of
boundary conditions becomes M = U(2N) which includes matrices which interpolate
between one single closed string or N disconnected strings [13]. The topology change
is described in this picture by a simple change of boundary conditions in L2(∂Ω, C N)
[14].
† We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated
to the presence of irregular boundaries
Vacuum Energy and Renormalization on the Edge 4
If the spectrum of eigenvalues of the unitary operator U does not include the value
±1 (i.e. ±1 /∈ SpU) the boundary condition (7) can be rewritten as
∂nϕ = −i
ϕ (9)
which means that only the boundary values of the fields at the boundary can have an
arbitrary value ϕ whereas its normal derivative is determined by U and ϕ.
The corresponding operator mappings from unitary into selfadjoint operators
A± = −i
are the celebrated Cayley transforms. The inverse Cayley transform
I∓ iA±
I± iA±
recovers the unitary operator U from their selfadjoint Cayley transforms A±.
The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee
the unitarity of the corresponding quantum field theory. Indeed, in the case of free
field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum
of ∆ +m2 must be not only real but also positive which restricts the set of admissible
boundary conditions to a subset M of L2(∂Ω, C ).
Because of the existence of the boundary term in (5) the Hamiltonian H (6) is not
selfadjoint if the spectrum of the unitary operator U intersects the following domain of
phase factors
S1m = {e2αi;−π < α ≤ π, 0 < α <
− arctan m2, or
< −α < π − arctan m2 }.
In any other case, −m2 is a lower bound for the spectrum of the operator ∆ and H is
selfadjoint. One possible source of unitarity loss is the existence of edge estates with
large negative eigenvalues of operator ∆.
The consistency of the quantum field theory imposes, thus, a very stringent
condition on the type of acceptable boundary conditions, even in the case of massive
theories in order to prevent this type of pathological behaviour of vacuum energy.
For real scalar fields there is a further condition. U has to satisfy a CP symmetry
preserving condition
U † = U∗, U = UT . (12)
The usual Neumann and Dirichlet boundary conditions U = ±I satisfy this condition.
In general, for
BT A2
the condition requires that
A1 = A
1 , A2 = A
2 , A1B
∗ +BA
2 = 0 (14)
BB† + A1A
1 = I, A2A
TB∗ = I (15)
Vacuum Energy and Renormalization on the Edge 5
In particular, the quasi-periodic condition ϕ(L) = M−1ϕ(0), ∂nϕ(L) = M∂nϕ(0)
is also compatible if M = M t = M∗.
In the case of one single real massless scalar the set of compatible boundary
conditions has two connected components: M0 given by the operators of the form
Uβ = cos β I+ i sin β σy,
and M1 given by
Uα = cosα σz + sinα σx . (16)
M0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M1 contains the
quasi-periodic boundary conditions
ϕ(L) = tan
ϕ(0); ∂nϕ(L) =
∂nϕ(0) (17)
which include periodic (α = π
) and antiperiodic (α = −π
) boundary conditions.
3. Boundary Conditions and Renormalization Group
Since boundary conditions appear more naturally in the Schrödinger picture of field
theory and the theory is plagued of ultraviolet singularities some doubts were raised
about their relevance for the quantum field theory. The pioneer work of Symanzik [21]
confirmed the consistence of the standard picture even in presence of bulk renormalizable
interactions (see [22] for an explanation of a recent controversy [23]).
Moreover, there is a renormalization of the very boundary conditions because the
boundary terms are the source of new interactions.
The renormalization group can be defined in the continuum approach by
= Λ[φ][φ(x)− ξΛ(x)] (18)
by means of a fluctuating field ξΛ with short range fluctuations of order
. This implies
that the boundary condition
∂nϕ = Aϕ (19)
is renormalised to
∂nϕΛ = AΛϕΛ, (20)
since
= Λ[φ]+1[∂nφ(xb)− ∂nξΛ(xb)] = AΛ[φ]+1φ(xb) = AΛφΛ
with AΛ = ΛA. For more general boundary conditions the continuum renormalization
group is given by
Λ∂ΛUΛ =
Λ − UΛ
t ∂tUt =
t − Ut
Vacuum Energy and Renormalization on the Edge 6
for Λ = Λ0 e
t. Fixed points correspond, therefore, to self-adjoint boundary conditions
U † = U . In particular, Dirichlet and Neumann (U = ∓ I) are renormalization group
fixed points.
For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann
(IR) conditions.
U = e2i arctan e
I. (24)
Critical exponents can be identified with the eigenvalues of the matrix Uc at the
fixed points. Since Uc is also hermitian all critical exponents are either 1 or −1 and
there is no room for cyclic orbits. It is well known, however, that some quantum systems
with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization
group flows. Moreover, some topological field theories (e.g. Russian doll models) present
a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur
for regular boundaries. For the same reasons topological transitions do not occur for
finite scale transformations since the flip of eigenvalues from −1 to +1 requires a change
in the parameter t of the flow from −∞ to ∞ as in (24)).
4. Conformal Invariance and boundary conditions
In 1+1 dimensions the theory of massless scalar fields is formally conformal
invariant. However, boundary conditions might break this symmetry [9, 10, 11].
Conformal invariance is only preserved if the boundary conditions are stable under
the boundary renormalization group flow. The fixed points can easily be identified. For
a complex scalar field, besides the above mentioned fixed points, which correspond
to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are
conformal invariant, there are fixed points corresponding to quasi-periodic boundary
conditions (17). They also preserve the conformal symmetry.
In 1+1 dimensions this exhausts the whole set of conformal invariant boundary
conditions. Any other boundary condition flows toward one of these fixed points.
The most stable fixed point corresponds to Neumann conditions because all its critical
exponents are +1. The most unstable is that of Dirichlet conditions since all critical
exponents −1. This is compatible with the fact that the neighbourhood of Dirichlet
boundary conditions is plagued of singularities
Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and
irrelevant perturbations with critical exponents ±1, respectively. Negative values label
the possible instabilities. Implications of these results for string theory are well known.
Periodic boundary conditions, appear as attractors of systems with quasi-periodic and
pseudo-periodic conditions which stresses the stability of closed string theory vacuum.
For open strings the (stable) attractor points are standard free strings (Neumann). Any
other boundary condition flow toward one of those fixed points.
Notice that the absence of topological transitions in the boundary renormalization
group flow is a consequence of the fact that all relevant perturbations are always
associated with -1 critical exponents.
Vacuum Energy and Renormalization on the Edge 7
In higher dimensions (D > 1) conformal invariance requires, even in the massless
case m = 0, that Neumann boundary conditions have to be modified in order to preserve
conformal invariance with a term
∂nϕ =
D − 1
K ϕ, (25)
proportional to the extrinsic curvature K of the boundary.
In the case of singular boundaries some more interesting boundary renormalization
group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the
boundary renormalization group flow can appear [15, 25, 24] and conformal invariance
can be partially broken to a discrete subgroup Z [24].
5. Vacuum energy and boundary conditions
The infrared properties of quantum field theory are very sensitive to boundary
conditions [26]. In particular, physical properties of the quantum vacuum state like
the vacuum energy may exhibit a very strong dependence on the type of boundary
conditions. This can be explicitly shown in the simple case of a massless field defined
on a finite one-dimensional interval [0, L].
For pseudoperiodic boundary conditions defined by the unitary operator
Uθ = cos θ σx − sin θ σy : ϕ(L) = eiθϕ(0) (26)
the Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by
+n− 1
 (27)
The vacuum energy dependence on θ is in this case relatively smooth. The only
cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely
regular behaviour is obtained for Robin boundary conditions
U = e2αiI : ∂nϕ(0) = tanαϕ(0), ∂nϕ(L) = tanαϕ(L) (28)
which smoothly interpolate between Dirichlet (α = π
) and Neumann (α = π) conditions
when α is restricted to the interval α ∈ [π
, π] [27, 28, 29] .
Finally, the Casimir energy for quasi-periodic boundary conditions [30]
is also dependent on the choice of the parameter α. Two particularly interesting
cases are given by α = 0, UZ = σz; ϕ(L) = 0, ∂nϕ(0) = 0 and α = π,
U ′Z = σz; ϕ(0) = 0, ∂nϕ(L) = 0 which correspond to a Zaremba (mixed) boundary
conditions: one boundary is Dirichlet and the other Neumann.
Vacuum Energy and Renormalization on the Edge 8
6. Vacuum entanglement entropy
The dependence of vacuum energy on boundary conditions seems to suggest that
many other observables may suffer the same effect. In particular, one may wonder
whether or not the entropy of the system is dependent on the type of conditions that
constrain the values of the fields at the boundary. The entropy of the field theory at
finite temperature scales with the volume of physical space. Only in quantum gravity
or string theory the entropy can scale with the area of black hole horizon. However, in
field theory it is possible
Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω
to generate a mixed state from the pure vacuum state Ψ0 by integrating out the
fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1)
Ψ∗0Ψ0. (30)
The entropy of this state Sω = −Tr ρω log ρω, although ultravioletly divergent, provides
a measure of the degree of entanglement of the vacuum state. In the case of a free
massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales
logarithmically with the size lω of ω and the ultraviolet cut-off ǫ introduced to split
apart the domain ω and its complement Ω\ω
, (31)
and in D = 2 dimensions it scales linearly with the perimeter Rω of ω
Sω = c2
− γ (32)
and in D > 2 dimensions as the volume of the boundary of ω
Sω = cDVωǫ
1−D. (33)
In particular in three-dimensional spaces it scales with the area of the boundary of ω
like in the presence of a blackhole [16, 17]. Although the coefficients of the leading terms
c2, cD in (32) and (33) have been explicitly computed, they are not universal because
they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient
Vacuum Energy and Renormalization on the Edge 9
c1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third
of the central charge of the corresponding conformal field theory. Similarly, the finite γ
term in (32) is also universal in D = 2 dimension and is related to a degree of topological
entanglement [31].
It is remarkable that in D = 1 the coefficient c1 = 1/3 is also independent
of the choice of boundary condition in Ω. This in contrast with what happens for
the finite size corrections to vacuum energy. The coefficient of the 1/L term is also
proportional to the central charge but in that case the corresponding factor is very
sensitive to the type of boundary conditions imposed at the boundary of Ω. The above
results indicate that whereas the Casimir energy is closely related with the infrared
properties of the conformal theory which are sensitive to the boundary conditions, the
entanglement entropy is rather associated to the behaviour at the interface between ω
and its complement Ω\ω which do not depend on the choice of boundary conditions at
the edge of the physical space.
7. Conclusions
The description of regular boundary conditions in terms of unitary matrices provides
a very useful framework for the description of the boundary renormalization group flow
and the breaking of conformal invariance due to boundary effects. Neumann conditions
turn out to be the only boundary conditions which are absolutely stable under RG
flow. All other boundary conditions may have some relevant perturbations which are
the source of RG instabilities. However, the global structure of the flow does not permit
topological transitions.
The finite size corrections to vacuum energy are very sensitive to the choice
of boundary conditions which discriminate between the different fixed points of the
renormalization group flow. On the contrary, the leading contribution to entanglement
entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories,
to the change of boundary conditions. In D=2 dimensions the same property holds
for the finite correction to the entanglement entropy of massless scalar theories. This
fact, is very relevant for the implementation of quantum codes with topological stability
[31]. However, these properties do not hold for the leading terms contributing to the
entanglement entropy.
Acknowledgements
We thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting
discussions on closely related subjects. This work is partially supported by CICYT
(grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2).
Vacuum Energy and Renormalization on the Edge 10
References
[1] T. Padmanabhan, Phys. Rept. 380 (2003) 235
[2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063
[3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793
[4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci.,
Singapore (2001)
[5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1
[6] M. Asorey, D. Garćıa-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127
[7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759
[8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445
[9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35
[10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007]
[11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S.
Tsun, Academic Press (2006)
[12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724
[13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001
[14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299
[15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401
[16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373
[17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666
[18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028]
[19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032
[20] A. A. Saharian, Phys. Rev. D69 (2004) 085005
[21] K. Symanzik, Nucl. Phys. B190 (1981) 1
[22] E. Elizalde, J. Phys. A 36 (2003) L567
[23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677
(2004) 379-404
[24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006)
[25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505
[26] M.Asorey, J. Geom. Phys. 11(1993)94
[27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297
[28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039
[29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325
[30] M. Asorey, unpublished (1994)
[31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404

  The vacuum dependence on boundary conditions in quantum field theories is
analysed from a very general viewpoint. From this perspective the
renormalization prescriptions not only imply the renormalization of the
couplings of the theory in the bulk but also the appearance of a flow in the
space of boundary conditions. For regular boundaries this flow has a large
variety of fixed points and no cyclic orbit. The family of fixed points
includes Neumann and Dirichlet boundary conditions. In one-dimensional field
theories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed
points. Under these conditions massless bosonic free field theories are
conformally invariant. Among all fixed points only Neumann boundary conditions
are infrared stable fixed points. All other conformal invariant boundary
conditions become unstable under some relevant perturbations. In finite volumes
we analyse the dependence of the vacuum energy along the trajectories of the
renormalization group flow providing an interesting framework for dark energy
evolution. On the contrary, the renormalization group flow on the boundary does
not affect the leading behaviour of the entanglement entropy of the vacuum in
one-dimensional conformally invariant bosonic theories.

Introduction
The emergence of the dark energy as one of the basic ingredients of the current
standard cosmological scenario, and the absence of an even vague understanding of
its possible origin, opens a window to the analysis of all possible mechanisms that
generate background energy (see e.g. [1] for a review of recent proposals). The main
problem is that the apparent value of the dark energy is very tiny compared with any
physical energy scale. A second problem is that in a generic quantum field theory
there is generation of vacuum energy and any renormalization prescription requires a
fine tuning, which is not very convincing without the quantisation of the gravitational
interaction.
The guess that dark energy might change with the evolution of the Universe can
be understood even if dark energy is just vacuum energy. The finite corrections due to
finite size of the causal Hubble domain decrease as the Universe continues to expand.
The aim of this paper is to analyse the variation of these finite size corrections under
of renormalization group on the space of boundary conditions for scalar field theories in
flat space, although the results are generalisable for more general backgrounds.
The dependence of the vacuum energy on the boundary conditions [2] is well known
since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for
recent revisions). However, boundaries might also be considered as a source of new,
although peculiar, interactions and therefore can undergo renormalization [7, 8]. The
renormalization of boundary conditions modifies the critical behaviour of the theory
[9, 10, 11]. In systems with boundaries or defects, the boundary RG flow induces a
dynamical behaviour on the boundaries. The dynamics of D-branes in string theory
emerges in this way [12].
The renormalization group flow is analysed from a global viewpoint in the most
general framework for boundary conditions of scalar field theories introduced in Ref.
[13]. In particular, we consider the possible existence of topological transitions [14]
induced by the renormalization of boundary conditions or cyclic orbits in the boundary
RG flow [15]. The dependence of the finite size corrections to the vacuum energy and
vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a
very general perspective.
2. Boundary conditions in Field Theory
The action which governs the dynamics of scalar field theory in a bounded domain
Ω of flat space consists of two different of terms, S(φ) = SB(φ) + Sb(φ). The first one
SB(φ) =
g dDx
|φ̇|2 − |∇φ|2 − V (|φ|2)
is defined in terms of the values of the fields in the bulk. The second term
Sb(ϕ) =
dD−1x
|ϕ̇|2+
ϕ∗∂nϕ+
∗)ϕ−|∇ϕ|2
Vacuum Energy and Renormalization on the Edge 3
depends only on the values of the fields at the boundary ∂Ω † . g
denotes the metric
induced on the boundary by the bulk flat metric, and ∂n is the normal derivative at the
boundary
ϕ = φ|
∂nϕ = ∂nφ|∂Ω. (3)
The presence of the boundary term Sb allows the generation of local classical equations
of motion without requiring any specific type of boundary conditions [19, 20]. Indeed,
the gradient term
V = 1
|∇φ|2 (4)
can be rewritten as
φ†∆φ+
φ† ∂nφ (5)
where ∆ is the Laplace-Beltrami operator ∆ = − ∂µ∂µ . In the quantum theory the
Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint
Hamiltonian
H = 1
∆+m2. (6)
for the free field theory (The inclusion of interactions does not changes the picture [21]).
This means that the classical fields must satisfy boundary conditions which make the
operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this
requirement [13] are in one-to-one correspondence with the group of unitary operators
of the boundary Hilbert space L2(∂Ω, C ). For any unitary operator U ∈ L2(∂Ω, C ), the
fields satisfying the boundary condition
ϕ− i ∂nϕ = U (ϕ+ i ∂nϕ) (7)
define a domain where ∆ is a selfadjoint operator.
In the case of open strings, the corresponding conformal 1+1 dimensional scalar
field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety
of admissible boundary conditions described by the unitary group M = U(2). The
unitary matrices
define Dirichlet, Neumann and periodic boundary conditions, which in string theory
correspond to a string attached to a D-brane background, free open and closed string
theories, respectively.
For higher N-dimensional target spaces, or N-component strings, the set of
boundary conditions becomes M = U(2N) which includes matrices which interpolate
between one single closed string or N disconnected strings [13]. The topology change
is described in this picture by a simple change of boundary conditions in L2(∂Ω, C N)
[14].
† We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated
to the presence of irregular boundaries
Vacuum Energy and Renormalization on the Edge 4
If the spectrum of eigenvalues of the unitary operator U does not include the value
±1 (i.e. ±1 /∈ SpU) the boundary condition (7) can be rewritten as
∂nϕ = −i
ϕ (9)
which means that only the boundary values of the fields at the boundary can have an
arbitrary value ϕ whereas its normal derivative is determined by U and ϕ.
The corresponding operator mappings from unitary into selfadjoint operators
A± = −i
are the celebrated Cayley transforms. The inverse Cayley transform
I∓ iA±
I± iA±
recovers the unitary operator U from their selfadjoint Cayley transforms A±.
The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee
the unitarity of the corresponding quantum field theory. Indeed, in the case of free
field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum
of ∆ +m2 must be not only real but also positive which restricts the set of admissible
boundary conditions to a subset M of L2(∂Ω, C ).
Because of the existence of the boundary term in (5) the Hamiltonian H (6) is not
selfadjoint if the spectrum of the unitary operator U intersects the following domain of
phase factors
S1m = {e2αi;−π < α ≤ π, 0 < α <
− arctan m2, or
< −α < π − arctan m2 }.
In any other case, −m2 is a lower bound for the spectrum of the operator ∆ and H is
selfadjoint. One possible source of unitarity loss is the existence of edge estates with
large negative eigenvalues of operator ∆.
The consistency of the quantum field theory imposes, thus, a very stringent
condition on the type of acceptable boundary conditions, even in the case of massive
theories in order to prevent this type of pathological behaviour of vacuum energy.
For real scalar fields there is a further condition. U has to satisfy a CP symmetry
preserving condition
U † = U∗, U = UT . (12)
The usual Neumann and Dirichlet boundary conditions U = ±I satisfy this condition.
In general, for
BT A2
the condition requires that
A1 = A
1 , A2 = A
2 , A1B
∗ +BA
2 = 0 (14)
BB† + A1A
1 = I, A2A
TB∗ = I (15)
Vacuum Energy and Renormalization on the Edge 5
In particular, the quasi-periodic condition ϕ(L) = M−1ϕ(0), ∂nϕ(L) = M∂nϕ(0)
is also compatible if M = M t = M∗.
In the case of one single real massless scalar the set of compatible boundary
conditions has two connected components: M0 given by the operators of the form
Uβ = cos β I+ i sin β σy,
and M1 given by
Uα = cosα σz + sinα σx . (16)
M0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M1 contains the
quasi-periodic boundary conditions
ϕ(L) = tan
ϕ(0); ∂nϕ(L) =
∂nϕ(0) (17)
which include periodic (α = π
) and antiperiodic (α = −π
) boundary conditions.
3. Boundary Conditions and Renormalization Group
Since boundary conditions appear more naturally in the Schrödinger picture of field
theory and the theory is plagued of ultraviolet singularities some doubts were raised
about their relevance for the quantum field theory. The pioneer work of Symanzik [21]
confirmed the consistence of the standard picture even in presence of bulk renormalizable
interactions (see [22] for an explanation of a recent controversy [23]).
Moreover, there is a renormalization of the very boundary conditions because the
boundary terms are the source of new interactions.
The renormalization group can be defined in the continuum approach by
= Λ[φ][φ(x)− ξΛ(x)] (18)
by means of a fluctuating field ξΛ with short range fluctuations of order
. This implies
that the boundary condition
∂nϕ = Aϕ (19)
is renormalised to
∂nϕΛ = AΛϕΛ, (20)
since
= Λ[φ]+1[∂nφ(xb)− ∂nξΛ(xb)] = AΛ[φ]+1φ(xb) = AΛφΛ
with AΛ = ΛA. For more general boundary conditions the continuum renormalization
group is given by
Λ∂ΛUΛ =
Λ − UΛ
t ∂tUt =
t − Ut
Vacuum Energy and Renormalization on the Edge 6
for Λ = Λ0 e
t. Fixed points correspond, therefore, to self-adjoint boundary conditions
U † = U . In particular, Dirichlet and Neumann (U = ∓ I) are renormalization group
fixed points.
For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann
(IR) conditions.
U = e2i arctan e
I. (24)
Critical exponents can be identified with the eigenvalues of the matrix Uc at the
fixed points. Since Uc is also hermitian all critical exponents are either 1 or −1 and
there is no room for cyclic orbits. It is well known, however, that some quantum systems
with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization
group flows. Moreover, some topological field theories (e.g. Russian doll models) present
a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur
for regular boundaries. For the same reasons topological transitions do not occur for
finite scale transformations since the flip of eigenvalues from −1 to +1 requires a change
in the parameter t of the flow from −∞ to ∞ as in (24)).
4. Conformal Invariance and boundary conditions
In 1+1 dimensions the theory of massless scalar fields is formally conformal
invariant. However, boundary conditions might break this symmetry [9, 10, 11].
Conformal invariance is only preserved if the boundary conditions are stable under
the boundary renormalization group flow. The fixed points can easily be identified. For
a complex scalar field, besides the above mentioned fixed points, which correspond
to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are
conformal invariant, there are fixed points corresponding to quasi-periodic boundary
conditions (17). They also preserve the conformal symmetry.
In 1+1 dimensions this exhausts the whole set of conformal invariant boundary
conditions. Any other boundary condition flows toward one of these fixed points.
The most stable fixed point corresponds to Neumann conditions because all its critical
exponents are +1. The most unstable is that of Dirichlet conditions since all critical
exponents −1. This is compatible with the fact that the neighbourhood of Dirichlet
boundary conditions is plagued of singularities
Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and
irrelevant perturbations with critical exponents ±1, respectively. Negative values label
the possible instabilities. Implications of these results for string theory are well known.
Periodic boundary conditions, appear as attractors of systems with quasi-periodic and
pseudo-periodic conditions which stresses the stability of closed string theory vacuum.
For open strings the (stable) attractor points are standard free strings (Neumann). Any
other boundary condition flow toward one of those fixed points.
Notice that the absence of topological transitions in the boundary renormalization
group flow is a consequence of the fact that all relevant perturbations are always
associated with -1 critical exponents.
Vacuum Energy and Renormalization on the Edge 7
In higher dimensions (D > 1) conformal invariance requires, even in the massless
case m = 0, that Neumann boundary conditions have to be modified in order to preserve
conformal invariance with a term
∂nϕ =
D − 1
K ϕ, (25)
proportional to the extrinsic curvature K of the boundary.
In the case of singular boundaries some more interesting boundary renormalization
group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the
boundary renormalization group flow can appear [15, 25, 24] and conformal invariance
can be partially broken to a discrete subgroup Z [24].
5. Vacuum energy and boundary conditions
The infrared properties of quantum field theory are very sensitive to boundary
conditions [26]. In particular, physical properties of the quantum vacuum state like
the vacuum energy may exhibit a very strong dependence on the type of boundary
conditions. This can be explicitly shown in the simple case of a massless field defined
on a finite one-dimensional interval [0, L].
For pseudoperiodic boundary conditions defined by the unitary operator
Uθ = cos θ σx − sin θ σy : ϕ(L) = eiθϕ(0) (26)
the Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by
+n− 1
 (27)
The vacuum energy dependence on θ is in this case relatively smooth. The only
cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely
regular behaviour is obtained for Robin boundary conditions
U = e2αiI : ∂nϕ(0) = tanαϕ(0), ∂nϕ(L) = tanαϕ(L) (28)
which smoothly interpolate between Dirichlet (α = π
) and Neumann (α = π) conditions
when α is restricted to the interval α ∈ [π
, π] [27, 28, 29] .
Finally, the Casimir energy for quasi-periodic boundary conditions [30]
is also dependent on the choice of the parameter α. Two particularly interesting
cases are given by α = 0, UZ = σz; ϕ(L) = 0, ∂nϕ(0) = 0 and α = π,
U ′Z = σz; ϕ(0) = 0, ∂nϕ(L) = 0 which correspond to a Zaremba (mixed) boundary
conditions: one boundary is Dirichlet and the other Neumann.
Vacuum Energy and Renormalization on the Edge 8
6. Vacuum entanglement entropy
The dependence of vacuum energy on boundary conditions seems to suggest that
many other observables may suffer the same effect. In particular, one may wonder
whether or not the entropy of the system is dependent on the type of conditions that
constrain the values of the fields at the boundary. The entropy of the field theory at
finite temperature scales with the volume of physical space. Only in quantum gravity
or string theory the entropy can scale with the area of black hole horizon. However, in
field theory it is possible
Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω
to generate a mixed state from the pure vacuum state Ψ0 by integrating out the
fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1)
Ψ∗0Ψ0. (30)
The entropy of this state Sω = −Tr ρω log ρω, although ultravioletly divergent, provides
a measure of the degree of entanglement of the vacuum state. In the case of a free
massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales
logarithmically with the size lω of ω and the ultraviolet cut-off ǫ introduced to split
apart the domain ω and its complement Ω\ω
, (31)
and in D = 2 dimensions it scales linearly with the perimeter Rω of ω
Sω = c2
− γ (32)
and in D > 2 dimensions as the volume of the boundary of ω
Sω = cDVωǫ
1−D. (33)
In particular in three-dimensional spaces it scales with the area of the boundary of ω
like in the presence of a blackhole [16, 17]. Although the coefficients of the leading terms
c2, cD in (32) and (33) have been explicitly computed, they are not universal because
they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient
Vacuum Energy and Renormalization on the Edge 9
c1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third
of the central charge of the corresponding conformal field theory. Similarly, the finite γ
term in (32) is also universal in D = 2 dimension and is related to a degree of topological
entanglement [31].
It is remarkable that in D = 1 the coefficient c1 = 1/3 is also independent
of the choice of boundary condition in Ω. This in contrast with what happens for
the finite size corrections to vacuum energy. The coefficient of the 1/L term is also
proportional to the central charge but in that case the corresponding factor is very
sensitive to the type of boundary conditions imposed at the boundary of Ω. The above
results indicate that whereas the Casimir energy is closely related with the infrared
properties of the conformal theory which are sensitive to the boundary conditions, the
entanglement entropy is rather associated to the behaviour at the interface between ω
and its complement Ω\ω which do not depend on the choice of boundary conditions at
the edge of the physical space.
7. Conclusions
The description of regular boundary conditions in terms of unitary matrices provides
a very useful framework for the description of the boundary renormalization group flow
and the breaking of conformal invariance due to boundary effects. Neumann conditions
turn out to be the only boundary conditions which are absolutely stable under RG
flow. All other boundary conditions may have some relevant perturbations which are
the source of RG instabilities. However, the global structure of the flow does not permit
topological transitions.
The finite size corrections to vacuum energy are very sensitive to the choice
of boundary conditions which discriminate between the different fixed points of the
renormalization group flow. On the contrary, the leading contribution to entanglement
entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories,
to the change of boundary conditions. In D=2 dimensions the same property holds
for the finite correction to the entanglement entropy of massless scalar theories. This
fact, is very relevant for the implementation of quantum codes with topological stability
[31]. However, these properties do not hold for the leading terms contributing to the
entanglement entropy.
Acknowledgements
We thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting
discussions on closely related subjects. This work is partially supported by CICYT
(grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2).
Vacuum Energy and Renormalization on the Edge 10
References
[1] T. Padmanabhan, Phys. Rept. 380 (2003) 235
[2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063
[3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793
[4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci.,
Singapore (2001)
[5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1
[6] M. Asorey, D. Garćıa-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127
[7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759
[8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445
[9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35
[10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007]
[11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S.
Tsun, Academic Press (2006)
[12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724
[13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001
[14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299
[15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401
[16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373
[17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666
[18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028]
[19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032
[20] A. A. Saharian, Phys. Rev. D69 (2004) 085005
[21] K. Symanzik, Nucl. Phys. B190 (1981) 1
[22] E. Elizalde, J. Phys. A 36 (2003) L567
[23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677
(2004) 379-404
[24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006)
[25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505
[26] M.Asorey, J. Geom. Phys. 11(1993)94
[27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297
[28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039
[29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325
[30] M. Asorey, unpublished (1994)
[31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404
