Composite Structure and Causality
Satish D. Joglekar∗
October 30, 2018
Department of Physics, I.I.T. Kanpur, Kanpur 208016 (INDIA)
Abstract
We study the question of whether a composite structure of elemen-
tary particles, with a length scale 1/Λ, can leave observable effects
of non-locality and causality violation at higher energies (but . Λ).
We formulate a model-independent approach based on Bogoliubov-
Shirkov formulation of causality. We analyze the relation between the
fundamental theory (of finer constituents) and the derived theory (of
composite particles). We assume that the fundamental theory is causal
and formulate a condition which must be fulfilled for the derived the-
ory to be causal. We analyze the condition and exhibit possibilities
which fulfil and which violate the condition. We make comments on
how causality violating amplitudes can arise.
1 Introduction
The standard model (SM), a local quantum field theory, has served so far
as a very good description of elementary particle processes [1]. It is however
widely believed that soon, when higher energies are experimentally accessible,
new phenomena may emerge that require a description that goes beyond the
standard model. Among the various the possibilities, is the possibility that
a composite nature of the standard model constituents may be revealed [2]
and a possible failure of locality [3]. It is possible that the underlying physics
∗e-mail address:sdj@iitk.ac.in
http://arxiv.org/abs/0704.0995v2
is nonlocal at shorter distances which could be a result of composite struc-
ture of particles, or granularity of space-time, or underlying noncommutative
structure of space-time [4]. With a nonlocal interaction, often goes causality
violation that can arise because the interaction region, encloses points sep-
arated by a space-like interval. Causality violation has been studied in the
context of non-local [5] and non-commutative quantum field theories [6, 7]. It
has in fact been suggested [8, 5] that non-local quantum field theories [9, 10]
may indeed serve as effective field theories for a deeper/more fundamental
theory such as a composite model; and the former indeed show causality
violation[10, 5]. An effective tool to study causality has been developed by
Bogoliubov and Shirkov [11] and has been in particular employed for the
causality violation in non-local [5] and non-commutative QFT’s [7]. We wish
to consider the following question: in view of the possible composite nature
of elementary particles, leading to extended structures, will these leave an
observable effect in the form of a violation of causality and locality that can
be detected? A similar question regarding a violation of the Pauli exclusion
principle on account of the compositeness of particles has been earlier ad-
dressed to [12]. This question is particularly interesting since should there
be a signal of CV, it will be detected long before an explicit knowledge of
composite structure is known. In fact, it is has been suggested [5] that the
unknown physics at high energy scales (Λ) from a possible source can effec-
tively be represented in a consistent way (a unitary, gauge-invariant, finite (or
renormalizable) theory) by a nonlocal theory at energies lower than Λ, but
higher than the present ones. In other words, the nonlocal standard model,
with a parameter Λ, can serve as such an effective field theory and will afford
a model-independent way of consistently reparametrizing the effects beyond
standard model . In this model, one finds that there is but a small CV at
low energies, which grows rapidly as energies approach Λ and beyond these,
the fundamental theory is expected to take over and presumably it leads to
no CV again. The aim of the present work is to approach this question in
a model-independent way in connection with a composite structure of SM
constituents.
2 Preliminary
2.1 Definition of the problem
Suppose that the presently known standard model particles are a composite
of a set of finer constituents. Suppose that these underlying constituents
belong to a local causality-preserving fundamental theory. Suppose, further
that at lower energies, one only observes the composite bound states and
their scattering processes. These bound state particles are extended objects.
A priori, their interaction is expected to be non-local. A nonlocal covariant
interaction has, at a given instant, interaction spread over a region in space,
which therefore contains spatially separated points. An obvious question
arises: will the interactions of the composite theory be such that causality
is preserved by this low-energy theory? We need the fundamental theory
for energy scales >> Λ, and for energy scales << Λ, we have the set of
composite particles described by the ”derived” theory. Then the question,
paraphrased differently is, will the phase transition (should there be one)
from the fundamental to the composite be causality preserving or it could
lead to a breakdown of causality at short enough distances?
2.2 Definition of the system
Let, for simplicity, the fundamental theory, denoted by F , be character-
ized by a single coupling constant g. For the purpose of formulation of the
Bogoliubov-Shirkov (BS) criterion of causality, we shall need to formulate
a theory with a variable coupling g(x). Let the low-energy derived theory
be characterized by its own coupling constant λ ≡ λ [g], which for identical
reasons, we shall need to allow to depend on space-time: λ = λ (x). We shall
assume, for simplicity, that the derived theory, denoted by C, is completely
described by its scattering states: i.e. we shall assume that the model admits
no bound states. A scattering state of the derived theory can be looked upon
from two different point of views:
⋆ a scattering state, as t → ±∞, is a state of a certain set of non-
interacting composite particles of the low energy theory with certain
momenta, polarizations etc.
⋆ a scattering state, as t → ±∞, is a (complicated) configuration of fields
of the fundamental theory.
3 Causality formulation for a theory without
a well-defined S-matrix
Bogoliubov and Shirkov have shown [11] that S−operator is causal only if it
satisfies,
δg (x)
δg (y)
= 0, x <∼ y (1)
(Here, x < y⇐⇒ x0 < y0 and x ∼ y ⇐⇒ (x − y)
2 < 0 ). The condition is
obtained from the primary meaning of causality that a disturbance does not
propagate outside the forward light-cone (the disturbance considered is that
in g(x)1), and is independent of any specific field theory formulation. The
BS causality criterion holds for a theory for which an S-operator is defined.
For a theory such as QCD, some of the matrix elements of the S-operator
may not exist on account of the infrared divergences. It is nonetheless true
that an alternate formulation in terms of the U-operator (i.e. U (−T, T ′))
can be given. This is so because, the U operator is unitary as much as the
S-operator and the BS criterion depends on two points x, y with x <∼ y
which can always be chosen to be such that they both lie in (−T, T ′). The
relation would then read
δg (x)
δg (y)
= 0, x <∼ y; −T < x0, y0 < T
′ (2)
It is possible to alternately formulate the causality condition in terms of
the following choices of the couplings. [This way results when we suitably
integrate (2) over x0 < 0 and y0 > 0]. In this approach, we make a comparison
of the following two neighboring theories in the coupling constant space2:
1. Fundamental theory F ′: Coupling constants = g′2 ( a constant value)
for x0 > 0 and g1(a constant value) for x0 < 0. Corresponding derived
theory is C′.
2. Fundamental theory F ′′: Coupling constants = g′′2 ( a constant value)
for x0 > 0 and g1(the same constant value) for x0 < 0. Corresponding
derived theory is C′′.
1One may consider varying g(x) an unphysical operation, but one can look alternately
upon varying g(x) at a point x0 as insertion of a (specific) local operator
∂g(x)
at x0 and
study the propagation of its effects.
2The idea of varying the coupling with time over all space is not an entirely unfamiliar
one: it is also employed in the LSZ formulation.
All the coupling constants are (chosen to be) space-independent. It suffices
for our purpose that g′2 differs infinitesimally from g1 and g
2 . (We can in fact
assume that the infinitesimal change from g1 to g2 is carried out adiabatically
and in an infinitesimal time). Then, we can alternately formulate [13] the
causality condition as,
U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] is independent of g1 (3)
This alternate formulation makes mathematics simpler, though it may lead
to an unusual-looking Physics.
In the following, we shall adopt a ”reductio ad absurdum” approach: We
shall let, if possible, that the theory C be causality-preserving and deduce
the consequences of causality of F for C and analyze these.
4 Relations between the derived theory and
the fundamental theory
4.1 Relations between coupling constants
The coupling constant λ is a function of g. If we allow a space-time dependent
coupling, then λ = λ [g]. A small change3δg (x) in the coupling g (x) about
g(x) = g = constant, will cause a change in λ (x) as given by4 δλ(z) =
δλ(z)
δg(y)
g(y)=g
δg (y). For the BS criterion of causality of Eq. (2), we need
to know the impact of a localized change g(x) → g (x) + δg (x) ≡ g (x) +
εδ4 (x− x̃) on the function λ (x). Now, if causality is valid, λ (x) cannot
be affected for any x0 < x̃0. Assuming that the theory has T-invariance
λ (x) cannot be affected for any x0 > x̃0. Thus, this together with causality
requires that,
λ (x) → λ (x) + Cεδ4 (x− x̃)
3For the argument presented subsequently, we shall go back to a general space-time
dependent coupling and not confine ourselves to the specific couplings presented in the
previous section.
4 We shall assume the existence and non-vanishing of
δλ(z)
δg(y)
g(y)=g
. By translational
invariance, this quantity is a function of (z − y) and is independent of the point z as such.
5We shall need that the theory F with a variable coupling has a T-invariance. This is
possible to formulate a time-reversal transformation for a theory with a variable g (x): we
need to define the action of time-reversal as Tg (x, t)T−1 = g (x,−t).
+ terms having finite order derivatives of delta function
Thus,
δλ (z)
δg (y)
= Cδ4 (z − y)
+ terms having finite order derivatives of delta function.
Then, for a constant small change δg = ε, for all x0 > 0, [i.e. δg(x) = ǫθ(x0)];
we find,
δλ(z) =
δλ(z)
δg(y)
δg (y)
Cδ4 (z − y) + derivatives of delta function
εθ (y0)
= C ′εθ (z0) for z0 > 0.
We shall denote by λ′2 = λ[g
2, g1] and λ
2 = λ[g
2 , g1].
4.2 Relation between states
We shall work in the interaction picture of C. Let the derived theory C′
have as incoming states6 {|c̃m (λ1,−T )〉} which, as −T → −∞, represents
scattering states with a number of free composite particles. We shall keep
T finite and will let T → ∞ only at the end of the argument. Evidently, as
−T → −∞, |c̃m (λ1,−T )〉 depends on λ1 only through the self-interaction
of each individual non-interacting particle in the state. Let H̃ denote the
Hilbert space of states of C′. Then the hypothesis that the scattering states
of C′ forms a complete set implies that the set {|c̃m (λ1,−T )〉} spans H̃:
H̃ ≡ sp {|c̃m (λ1,−T )〉}. We shall denote by H, the Hilbert space of states
of F ′ (and likewise for F ′′ ). Consider a state |c̃m (λ1,−T )〉 ∈ H̃ in the
interaction picture. On physical grounds, we know that there is a corre-
sponding state of F ′ in the interaction picture, denoted by |cm (g1,−T )〉.
We note that H can, in addition, have states linearly independent of the
states {|cm (g1,−T )〉}. We augment this set to complete an orthonormal ba-
sis {|cm (g1,−T )〉} ∪ {|βn (g1,−T )〉} ≡ {|αp (g1,−T )〉} for H. We shall call
6For technical simplicity, we shall assume that the set of states is countably infinite.
the span of {|cm (g1,−T )〉} by Ĥ ⊂ H. A similar discussion holds for F
Let us now consider the time-evolution, from t = −T to t = T ′, of a single
particle state of C′′ denoted by |s̃p (λ1,−T )〉, which belongs to the basis of
H̃. The unitary time evolution operator Ũ [λ1, λ
2;−T, T
′] as applied to the
state leads to
Ũ [λ1, λ
2;−T, T
′] |s̃p (λ1,−T )〉 = |s̃p (λ
′)〉 ∈ H̃ (4)
This state is a single particle state of slightly different mass, on account of a
slightly different self-energy, and interacts with a coupling λ′′2. We shall also
introduce interaction picture states
d̃m (λ
. These states are at t = T ′
and as T ′ → ∞ consist of a set of non-interacting (but self-interacting) parti-
cles of a slightly different mass and coupling constant λ′′2. These are analogues
of the ”out” states. We shall assume that these also span H̃. We shall further
make a convention: Under time reversal, the quantum numbers of particles
in the state |c̃m (λ1,−T )〉 become those of
d̃m (λ1, T
. Now, consider an
exclusive process in C′′. The magnitude of the quantum mechanical ampli-
tude for it, as seen from C′′and F ′′ are identical, as these are, in principle,
experimentally observable:
|ũnm| ≡
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
≡ |〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉| ≡ |unm| (5)
Here, we have introduced states |dn (g
2 , T
′)〉 in H analogous to
d̃n (λ
in H̃. We note that U here is the U−matrix in the interaction picture of
F ′′, as the set of states {|cm (g1,−T )〉} evolve according to the interaction
Hamiltonian H′I (g) (in the interaction picture) of the F
First we note that on account of unitarity of Ũ and Eq. (5),
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
|〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉|
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
|〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉|
So, the unitarity of U implies,
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |βm (g1,−T )〉 = 0
〈βn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉 = 0 (8)
The relations (8) implies that U is a block-diagonal matrix. The unitarity of
U then implies that the block corresponding to the subspace ∧H, viz. Û , is
also unitary. We shall now attempt relate these further. In this connection,
we recall a result for a finite dimensional matrices:
Lemma : Let U and U ′ be two N ×N unitary matrices satisfying: |u′ij| =
|uij|; 1 ≤ i, j ≤ N . Then, there exist phases {θi : i = 1, 2, . . . , N}
and {φi : i = 2, . . . , N} such that u
ij = uij exp [i (θi + φj)] : 1 ≤ i, j ≤
N withφ1 ≡ 0..
Proof : Let the diagonal elements of U ′ and U be related by: u′ii = exp (iΘi)uii.
We define U ′′ by u′′ij = exp (−iΘi)u
ij. Then, u
ii = uii. Now U
′′ is unitary and
thus, a priori, has N2 independent parameters. The information on moduli
of elements constitutes (N − 1)2 independent conditions, corresponding to
an (N − 1)× (N − 1) dimensional submatrix; the rest of the (2N − 1) mod-
uli being determined by relations implying that the norm of each row and
column is unity. The relations u′′ii = uii imply additional N relations on the
phases on uii.This leaves N
2 − (N − 1)2 −N = N − 1 free parameters. The
phases of u′′1j , 2 ≤ j ≤ N are unconstrained by |u
ij| = |uij| : 1 ≤ i, j ≤ N
and u′′ii = uii and we define u
1j = u1j exp (iφj), 2 ≤ j ≤ N . Then, there
are no free parameters and must lead to a unique U ′′. Now, U ′′ specified by
u′′ij = uij exp (iφj − iφi), 1 ≤ i, j ≤ N (φ1 ≡ 0) is such a solution. This
together with u′′ij = exp (−iΘi)u
ij leads to the result; with the definition
Θi − φi = θi.
Thus, in view of the unitarity of Ũ , and Û and (5), we write,
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
≡ 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉 × exp (iθ
n + iφm) (9)
We shall assume that F and C are have time-reversal invariance and derive
the consequences. Under time reversal, we know then that,
〈β|S |α〉 = 〈T α|S |T β〉 (10)
where |T β〉 is the state obtained by time-reversing the quantum numbers of
the state |β〉. In this case, it would imply, keeping in mind our choice of
definitions,
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
d̃m (λ1, T )
Ũ [λ1, λ
2;−T, T
′] |c̃n (λ
′)〉 (11)
We write a similar relation for F . Putting T ′ = T , (or equivalently, noting
that the matrix elements are insensitive to T ′ and T ), we find,
φp(λ2, λ1) = θp(λ1, λ2) (12)
5 Consequence of Causality of F for C
We shall assume that the fundamental theory F is causal and deduce the
consequences for the derived theory C(C′, C′′). The causality of F implies
U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
is independent of g1. Hence,
Mnm ≡ 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] |dm (g
is also independent of g1 since the state vectors 〈dn (g
2 , T
′)| and |dm (g
are independent of g1 with g
2 and g
2 fixed. We shall re-express Mnm in
terms of the matrix elements of the derived theory C(C′, C′′) and deduce the
consequences. We note,
Mnm = 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] |dm (g
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |αp (g1,−T )〉
× 〈αp (g1,−T )|U
† [g1, g
2;−T, T
′] |dm (g
′)〉 (13)
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cp (g1,−T )〉
× 〈cp (g1,−T )|U
† [g1, g
2;−T, T
′] |dm (g
′)〉 (14)
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃p (λ1,−T )〉 exp [−i(θ̃
p − θ̃
× 〈c̃p (λ1,−T )| Ũ
† [λ1, λ
2;−T, T
d̃m (λ
exp−[i(θ′′n − θ
m)](15)
≡ M̃nm(λ
2, λ1) (16)
In the 3rd step, we have employed the equations (8) and in the second step,
we have employed the closure relation for F .
In the above, θ′′n ≡ θn(λ
2, λ1), θ
m ≡ θm(λ
2, λ1), and θ̃
p ≡ θp(λ1, λ
2) etc.
Thus, the expression (15) is independent of λ1:
∂M̃nm(λ
2, λ1)
= 0 (17)
6 Analysis of Causality Condition
We shall now analyze the condition (17) obtained as an implication of causal-
ity of F . For this purpose, we shall find it useful to Taylor-expand θn as
follows7:
2, λ1) = θn(λ
2, λ1(0)) + βn∆1 + γn∆2 + δn∆1∆2 + · · ·
≡ αn + βn∆1 + γn∆2 + δn∆1∆2 + · · ·
2, λ1) = αm + βm∆1 + · · · (18)
Here, ∆1 ≡ λ1 − λ1(0); ∆2 ≡ λ
2 − λ
2; β, γ, δ refer to appropriate partial
derivatives at (λ′2, λ1(0)) and λ1(0) is some value near λ1.
We note that if
θn(λ2, λ1) is a function only of its first argument (I)
then, (θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
2) is zero and (θ
n − θ
m) ≡ θn(λ
2, λ1)−
2, λ1) is independent of λ1. Also, we can then carry out the sum over p
using the completeness relation and find that the independence from λ1 of
M̃nm(λ
2, λ1)
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] Ũ † [λ1, λ
2;−T, T
d̃m (λ
× exp−[i(θ′′n − θ
m)] (19)
7Throughout, we have employed only the infinitesimal variations in the couplings.
These are sufficient to determine the first order partial derivatives with respect to each λ1
and λ2. Hence, we shall content ourselves with expansion only upto O(∆1∆2)
for all m,n implies Ũ [λ1, λ
2;−T, T
′] Ũ † [λ1, λ
2;−T, T
′] is independent of λ1.
This condition is indeed necessary for causality of C. In fact, in this case, we
can rewrite8
d̃n (λ2, T
Ũ [λ1, λ2;−T, T
′] |c̃m (λ1,−T )〉
≡ 〈dn (g2, T
′)|U [g1, g2;−T, T
′] |cm (g1,−T )〉
× exp (iθn(λ2, λ1) + iθm(λ1, λ2)) (20)
d̃∗n (λ2, T
Ũ [λ1, λ2;−T, T
∣c̃∗m (λ1,−T )
≡ 〈dn (g2, T
′)|U [g1, g2;−T, T
′] |cm (g1,−T )〉 (21)
by redefining states by absorbing phases:
∣c̃∗m (λ1,−T )
= e−iθm(λ1) |c̃m (λ1,−T )〉)
etc. We note that this redefinition of the states is meaningful and compatible
with causality when θn is independent of its second argument. If on the
other hand, θn is dependent on its second argument (excepting a possibility
below), we cannot absorb a phase in a manner compatible with causality : a
state
∣c̃∗m
at t = −T cannot be made to depend on the value of coupling
λ2 it would have at a later time t > 0.
We can, in fact, liberalize somewhat the above condition by requiring
that,
βn = β and δn = 0 ∀ n (II)
In this case,
(θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
= β∆2 + · · · (22)
is independent of λ1 and does not depend also on p and thus comes out of
the summation in (15). The summation in (15) can be carried out using
the completeness relation. Also, (θ′′n − θ
m) ≡ θn(λ
2, λ1) − θm(λ
2, λ1) is still
independent of λ1. Thus, the entire discussion proceeds as before: in par-
ticular, as a little analysis shows, the phases can again be absorbed into the
definition of states in a manner compatible with causality.
8We have dropped primes on λ2.
While we shall not provide the general analysis of (17), we shall establish
examples of a few specific sufficient conditions for causality violation. (These
are simple conditions that, in fact, contradict I or II above) We can easily
verify the following results:
1. There is causality violation if (i) for some n, δn 6= 0 and (ii) βn =
βm ∀ n,m
2. There is causality violation if there be m 6= n such that
M̃nm(λ
2, λ1) 6= 0, when evaluated to O(∆), and βm 6= βn.
Proof : We shall let, if possible, C be causal. We can then write,
Ũ [λ1, λ
2;−T, T
′] = Ũ [λ′′2; 0, T
′] Ũ [λ1;−T, 0] (23)
Then, we can write the expression (15) as,
M̃nm(λ
2, λ1) =
d̃n (λ
2, 0)
c̃p (λ1, 0)〉 exp [−i(θ̃
p − θ̃
× 〈c̃p (λ1, 0)
d̃m (λ
2, 0)
exp−[i(θ′′n − θ
d̃n (λ
2, 0)
d̃m (λ
2, 0)
exp [−i(θ′′n − θ
m)] (24)
where X ≡
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃
p − θ̃
p)]. We shall now expand
the quantities involved to the first order in the infinitesimals as in (18). In
addition, we note that to the zeroth order in ∆2, (i.e. λ”2 − λ
2 = 0), we
have, (θ̃′′p − θ̃
p) = 0 and the completeness relation leads to X = 1. We further
define,
d̃n (λ
2, 0)
d̃m (λ
2, 0)
= δnm + iηnm∆2 + · · · (25)
Proof of (i): We define δ0 ≡ max{|δn|}; and let ±δ0 = δq for some q. We
now have,
(θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
= β∆2 + δp∆1∆2 + · · · (26)
and thus,
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃
p − θ̃
= exp (−iβ∆2)
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| × exp [−i(δp∆1∆2)]
= exp (−iβ∆2)
I − i∆1∆2
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| δp
Thus,
exp (iβ∆2)
d̃q (λ
2, 0)
d̃q (λ
2, 0)
= 1 + iηqq∆2 − i∆1∆2
δp|upq|
2 + · · · (28)
where upq ≡
d̃q (λ2, 0)
c̃p (λ1, 0)〉 (We can ignore primes on λ2 in this term).
The multiplicative exponential factor in (24) becomes:
exp (−iγq∆2 − iδq∆1∆2 + · · ·) ≈ 1− iγq∆2 − iδq∆1∆2 + · · ·
. Thus,
M̃qq = 1 + iηqq∆2 − iγq∆2 − iδq∆1∆2 − i∆1∆2
δp|upq|
2 + · · ·
= 1 + iηqq∆2 − iγq∆2 − i∆1∆2
[δq + δp]|upq|
2 + · · · (29)
In view of the fact that either δp + δq ≥ 0 ∀ p or δp+ δq ≤ 0 ∀ p the last term
is necessarily non-vanishing and dependent on ∆1
Proof of (ii): Consider the matrix element
M̃nm(λ
2, λ1) ≡
d̃n (λ
2, 0)
d̃m (λ
2, 0)
exp−[i(θ′′n − θ
m)] 6= 0(30)
for n 6= m. To the first order in the infinitesimals, the nonzero matrix element
d̃n (λ
2, 0)
d̃m (λ
2, 0)
is independent of ∆1. The multiplicative phase factor,
exp−[i(θ′′n − θ
m)] = exp{−i(αn − αm)− i(βn − βm)∆1 − iγn∆2}
is necessarily dependent on ∆1, thus implying causality violation.
9There is the obvious exception that δp = −δq for every such p such thatupq 6= 0; and
this has to be valid for each such q for which δq = ±δ0.
7 Additional comments
We comment in a qualitative way upon how a phase factor depending on
both values of the coupling can arise. Suppose that the derived theory C
is actually correctly described by a nonlocal covariant theory with a finite
non-zero non-locality scale ∆ ∼ 1/Λ. Since the theory is covariant, it is also
non-local in time. We write,
Ũ(λ1, λ2;−T, T
′) = Ũ(λ2; ∆, T
′)Ũ(λ1, λ2;−∆,∆)Ũ(λ1;−T,−∆) (31)
where the first and the third factors on the right hand side depends only on
one value of the coupling due to finite size of non-locality in time. The sec-
ond factor however depends on both couplings because in this time-interval
(−∆,∆), time evolution depends on both values of the coupling λ. On the
other hand, the fundamental theory, being local and causal, however has no
such analogue . The matrix Ũ(λ1, λ2;−∆,∆) can then give rise to phases
depending on both couplings in relation (9).
Naively, one may expect that if the fundamental theory is causal, the de-
rived theory should be so. Examples are however known where the diagrams
of the fundamental theory are associated with a different weight in the actual
phenomenology. For example, OZI rule in hadronic phenomenology gives a
suppression of a subset of the QCD diagrams. While such a possibility is
distinct from what is discussed in this work, generally such a modification
of the amplitudes within the fundamental theory may alter the underlying
properties of the fundamental theory such as causality.
References
[1] See e.g. Reviews of Particle Properties in W.-M. Yao et al., Journal of
Physics G 33, 1 (2006).
[2] See e.g. H. Harari, Phys.Rept.104:159,1984
[3] C. Bourrely , N.N. Khuri , Andre Martin , J. Soffer , Tai Tsun Wu
hep-ph/0511135; N.N. Khuri hep-ph/9512386
[4] See e.g. R.J. Szabo, Phys.Rep. 278,207 (2003).
http://arxiv.org/abs/hep-ph/0511135
http://arxiv.org/abs/hep-ph/9512386
[5] S. D. Joglekar, and A. Jain, Int. J.Mod. Phys. 19, (2004), S.D. Joglekar,
hep-th/0601006;
[6] See e.g. M. Chaichian, K. Nishijima, A. Tureanu, Phys.Lett. B568:146-
152,2003; O.W. Greenberg, Phys.Rev.D73:045014,2006
[7] A. Haque and S.D. Joglekar, hep-th/0701171
[8] S.D.Joglekar, J. Phys. A 34, 2765 (2001); S.D.Joglekar,Int.J.Mod.
Phys.A 16, (2001).
[9] E. D. Evens et al, Phys Rev D43, 499 (1991)
[10] G. Kleppe, and R. P. Woodard, Nucl. Phys. B 388, 81(1992).
[11] N. N. Bogoliubov, and D. V. Shirkov, Introduction to the theory of
quantized fields (John Wiley, New York, 1980) pg. 200-220.
[12] K. Akama et al Phys. Rev. Lett. 68, 1826 (1991); O.W. Greenberg, R.N.
Mohapatra Phys.Rev.Lett.59:2507,1987, Erratum-ibid.61:1432,1988;
Phys.Rev.Lett.62:712,1989, Erratum-ibid.62:1927,1989
[13] For a simpler and intuitive understanding of the causality condition in
either form, see e.g. S.D. Joglekar, hep-th/0601006
http://arxiv.org/abs/hep-th/0601006
http://arxiv.org/abs/hep-th/0701171
http://arxiv.org/abs/hep-th/0601006
	 Introduction
	 Preliminary
	 Definition of the problem
	 Definition of the system
	 Causality formulation for a theory without a well-defined S-matrix
	 Relations between the derived theory and the fundamental theory
	 Relations between coupling constants
	 Relation between states
	Consequence of Causality of F for C
	Analysis of Causality Condition
	Additional comments

  We study the question of whether a composite structure of elementary
particles, with a length scale $1/\Lambda$, can leave observable effects of
non-locality and causality violation at higher energies (but $\lesssim
\Lambda$). We formulate a model-independent approach based on
Bogoliubov-Shirkov formulation of causality. We analyze the relation between
the fundamental theory (of finer constituents) and the derived theory (of
composite particles). We assume that the fundamental theory is causal and
formulate a condition which must be fulfilled for the derived theory to be
causal. We analyze the condition and exhibit possibilities which fulfil and
which violate the condition. We make comments on how causality violating
amplitudes can arise.

Introduction
The standard model (SM), a local quantum field theory, has served so far
as a very good description of elementary particle processes [1]. It is however
widely believed that soon, when higher energies are experimentally accessible,
new phenomena may emerge that require a description that goes beyond the
standard model. Among the various the possibilities, is the possibility that
a composite nature of the standard model constituents may be revealed [2]
and a possible failure of locality [3]. It is possible that the underlying physics
∗e-mail address:sdj@iitk.ac.in
http://arxiv.org/abs/0704.0995v2
is nonlocal at shorter distances which could be a result of composite struc-
ture of particles, or granularity of space-time, or underlying noncommutative
structure of space-time [4]. With a nonlocal interaction, often goes causality
violation that can arise because the interaction region, encloses points sep-
arated by a space-like interval. Causality violation has been studied in the
context of non-local [5] and non-commutative quantum field theories [6, 7]. It
has in fact been suggested [8, 5] that non-local quantum field theories [9, 10]
may indeed serve as effective field theories for a deeper/more fundamental
theory such as a composite model; and the former indeed show causality
violation[10, 5]. An effective tool to study causality has been developed by
Bogoliubov and Shirkov [11] and has been in particular employed for the
causality violation in non-local [5] and non-commutative QFT’s [7]. We wish
to consider the following question: in view of the possible composite nature
of elementary particles, leading to extended structures, will these leave an
observable effect in the form of a violation of causality and locality that can
be detected? A similar question regarding a violation of the Pauli exclusion
principle on account of the compositeness of particles has been earlier ad-
dressed to [12]. This question is particularly interesting since should there
be a signal of CV, it will be detected long before an explicit knowledge of
composite structure is known. In fact, it is has been suggested [5] that the
unknown physics at high energy scales (Λ) from a possible source can effec-
tively be represented in a consistent way (a unitary, gauge-invariant, finite (or
renormalizable) theory) by a nonlocal theory at energies lower than Λ, but
higher than the present ones. In other words, the nonlocal standard model,
with a parameter Λ, can serve as such an effective field theory and will afford
a model-independent way of consistently reparametrizing the effects beyond
standard model . In this model, one finds that there is but a small CV at
low energies, which grows rapidly as energies approach Λ and beyond these,
the fundamental theory is expected to take over and presumably it leads to
no CV again. The aim of the present work is to approach this question in
a model-independent way in connection with a composite structure of SM
constituents.
2 Preliminary
2.1 Definition of the problem
Suppose that the presently known standard model particles are a composite
of a set of finer constituents. Suppose that these underlying constituents
belong to a local causality-preserving fundamental theory. Suppose, further
that at lower energies, one only observes the composite bound states and
their scattering processes. These bound state particles are extended objects.
A priori, their interaction is expected to be non-local. A nonlocal covariant
interaction has, at a given instant, interaction spread over a region in space,
which therefore contains spatially separated points. An obvious question
arises: will the interactions of the composite theory be such that causality
is preserved by this low-energy theory? We need the fundamental theory
for energy scales >> Λ, and for energy scales << Λ, we have the set of
composite particles described by the ”derived” theory. Then the question,
paraphrased differently is, will the phase transition (should there be one)
from the fundamental to the composite be causality preserving or it could
lead to a breakdown of causality at short enough distances?
2.2 Definition of the system
Let, for simplicity, the fundamental theory, denoted by F , be character-
ized by a single coupling constant g. For the purpose of formulation of the
Bogoliubov-Shirkov (BS) criterion of causality, we shall need to formulate
a theory with a variable coupling g(x). Let the low-energy derived theory
be characterized by its own coupling constant λ ≡ λ [g], which for identical
reasons, we shall need to allow to depend on space-time: λ = λ (x). We shall
assume, for simplicity, that the derived theory, denoted by C, is completely
described by its scattering states: i.e. we shall assume that the model admits
no bound states. A scattering state of the derived theory can be looked upon
from two different point of views:
⋆ a scattering state, as t → ±∞, is a state of a certain set of non-
interacting composite particles of the low energy theory with certain
momenta, polarizations etc.
⋆ a scattering state, as t → ±∞, is a (complicated) configuration of fields
of the fundamental theory.
3 Causality formulation for a theory without
a well-defined S-matrix
Bogoliubov and Shirkov have shown [11] that S−operator is causal only if it
satisfies,
δg (x)
δg (y)
= 0, x <∼ y (1)
(Here, x < y⇐⇒ x0 < y0 and x ∼ y ⇐⇒ (x − y)
2 < 0 ). The condition is
obtained from the primary meaning of causality that a disturbance does not
propagate outside the forward light-cone (the disturbance considered is that
in g(x)1), and is independent of any specific field theory formulation. The
BS causality criterion holds for a theory for which an S-operator is defined.
For a theory such as QCD, some of the matrix elements of the S-operator
may not exist on account of the infrared divergences. It is nonetheless true
that an alternate formulation in terms of the U-operator (i.e. U (−T, T ′))
can be given. This is so because, the U operator is unitary as much as the
S-operator and the BS criterion depends on two points x, y with x <∼ y
which can always be chosen to be such that they both lie in (−T, T ′). The
relation would then read
δg (x)
δg (y)
= 0, x <∼ y; −T < x0, y0 < T
′ (2)
It is possible to alternately formulate the causality condition in terms of
the following choices of the couplings. [This way results when we suitably
integrate (2) over x0 < 0 and y0 > 0]. In this approach, we make a comparison
of the following two neighboring theories in the coupling constant space2:
1. Fundamental theory F ′: Coupling constants = g′2 ( a constant value)
for x0 > 0 and g1(a constant value) for x0 < 0. Corresponding derived
theory is C′.
2. Fundamental theory F ′′: Coupling constants = g′′2 ( a constant value)
for x0 > 0 and g1(the same constant value) for x0 < 0. Corresponding
derived theory is C′′.
1One may consider varying g(x) an unphysical operation, but one can look alternately
upon varying g(x) at a point x0 as insertion of a (specific) local operator
∂g(x)
at x0 and
study the propagation of its effects.
2The idea of varying the coupling with time over all space is not an entirely unfamiliar
one: it is also employed in the LSZ formulation.
All the coupling constants are (chosen to be) space-independent. It suffices
for our purpose that g′2 differs infinitesimally from g1 and g
2 . (We can in fact
assume that the infinitesimal change from g1 to g2 is carried out adiabatically
and in an infinitesimal time). Then, we can alternately formulate [13] the
causality condition as,
U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] is independent of g1 (3)
This alternate formulation makes mathematics simpler, though it may lead
to an unusual-looking Physics.
In the following, we shall adopt a ”reductio ad absurdum” approach: We
shall let, if possible, that the theory C be causality-preserving and deduce
the consequences of causality of F for C and analyze these.
4 Relations between the derived theory and
the fundamental theory
4.1 Relations between coupling constants
The coupling constant λ is a function of g. If we allow a space-time dependent
coupling, then λ = λ [g]. A small change3δg (x) in the coupling g (x) about
g(x) = g = constant, will cause a change in λ (x) as given by4 δλ(z) =
δλ(z)
δg(y)
g(y)=g
δg (y). For the BS criterion of causality of Eq. (2), we need
to know the impact of a localized change g(x) → g (x) + δg (x) ≡ g (x) +
εδ4 (x− x̃) on the function λ (x). Now, if causality is valid, λ (x) cannot
be affected for any x0 < x̃0. Assuming that the theory has T-invariance
λ (x) cannot be affected for any x0 > x̃0. Thus, this together with causality
requires that,
λ (x) → λ (x) + Cεδ4 (x− x̃)
3For the argument presented subsequently, we shall go back to a general space-time
dependent coupling and not confine ourselves to the specific couplings presented in the
previous section.
4 We shall assume the existence and non-vanishing of
δλ(z)
δg(y)
g(y)=g
. By translational
invariance, this quantity is a function of (z − y) and is independent of the point z as such.
5We shall need that the theory F with a variable coupling has a T-invariance. This is
possible to formulate a time-reversal transformation for a theory with a variable g (x): we
need to define the action of time-reversal as Tg (x, t)T−1 = g (x,−t).
+ terms having finite order derivatives of delta function
Thus,
δλ (z)
δg (y)
= Cδ4 (z − y)
+ terms having finite order derivatives of delta function.
Then, for a constant small change δg = ε, for all x0 > 0, [i.e. δg(x) = ǫθ(x0)];
we find,
δλ(z) =
δλ(z)
δg(y)
δg (y)
Cδ4 (z − y) + derivatives of delta function
εθ (y0)
= C ′εθ (z0) for z0 > 0.
We shall denote by λ′2 = λ[g
2, g1] and λ
2 = λ[g
2 , g1].
4.2 Relation between states
We shall work in the interaction picture of C. Let the derived theory C′
have as incoming states6 {|c̃m (λ1,−T )〉} which, as −T → −∞, represents
scattering states with a number of free composite particles. We shall keep
T finite and will let T → ∞ only at the end of the argument. Evidently, as
−T → −∞, |c̃m (λ1,−T )〉 depends on λ1 only through the self-interaction
of each individual non-interacting particle in the state. Let H̃ denote the
Hilbert space of states of C′. Then the hypothesis that the scattering states
of C′ forms a complete set implies that the set {|c̃m (λ1,−T )〉} spans H̃:
H̃ ≡ sp {|c̃m (λ1,−T )〉}. We shall denote by H, the Hilbert space of states
of F ′ (and likewise for F ′′ ). Consider a state |c̃m (λ1,−T )〉 ∈ H̃ in the
interaction picture. On physical grounds, we know that there is a corre-
sponding state of F ′ in the interaction picture, denoted by |cm (g1,−T )〉.
We note that H can, in addition, have states linearly independent of the
states {|cm (g1,−T )〉}. We augment this set to complete an orthonormal ba-
sis {|cm (g1,−T )〉} ∪ {|βn (g1,−T )〉} ≡ {|αp (g1,−T )〉} for H. We shall call
6For technical simplicity, we shall assume that the set of states is countably infinite.
the span of {|cm (g1,−T )〉} by Ĥ ⊂ H. A similar discussion holds for F
Let us now consider the time-evolution, from t = −T to t = T ′, of a single
particle state of C′′ denoted by |s̃p (λ1,−T )〉, which belongs to the basis of
H̃. The unitary time evolution operator Ũ [λ1, λ
2;−T, T
′] as applied to the
state leads to
Ũ [λ1, λ
2;−T, T
′] |s̃p (λ1,−T )〉 = |s̃p (λ
′)〉 ∈ H̃ (4)
This state is a single particle state of slightly different mass, on account of a
slightly different self-energy, and interacts with a coupling λ′′2. We shall also
introduce interaction picture states
d̃m (λ
. These states are at t = T ′
and as T ′ → ∞ consist of a set of non-interacting (but self-interacting) parti-
cles of a slightly different mass and coupling constant λ′′2. These are analogues
of the ”out” states. We shall assume that these also span H̃. We shall further
make a convention: Under time reversal, the quantum numbers of particles
in the state |c̃m (λ1,−T )〉 become those of
d̃m (λ1, T
. Now, consider an
exclusive process in C′′. The magnitude of the quantum mechanical ampli-
tude for it, as seen from C′′and F ′′ are identical, as these are, in principle,
experimentally observable:
|ũnm| ≡
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
≡ |〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉| ≡ |unm| (5)
Here, we have introduced states |dn (g
2 , T
′)〉 in H analogous to
d̃n (λ
in H̃. We note that U here is the U−matrix in the interaction picture of
F ′′, as the set of states {|cm (g1,−T )〉} evolve according to the interaction
Hamiltonian H′I (g) (in the interaction picture) of the F
First we note that on account of unitarity of Ũ and Eq. (5),
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
|〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉|
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
|〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉|
So, the unitarity of U implies,
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |βm (g1,−T )〉 = 0
〈βn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉 = 0 (8)
The relations (8) implies that U is a block-diagonal matrix. The unitarity of
U then implies that the block corresponding to the subspace ∧H, viz. Û , is
also unitary. We shall now attempt relate these further. In this connection,
we recall a result for a finite dimensional matrices:
Lemma : Let U and U ′ be two N ×N unitary matrices satisfying: |u′ij| =
|uij|; 1 ≤ i, j ≤ N . Then, there exist phases {θi : i = 1, 2, . . . , N}
and {φi : i = 2, . . . , N} such that u
ij = uij exp [i (θi + φj)] : 1 ≤ i, j ≤
N withφ1 ≡ 0..
Proof : Let the diagonal elements of U ′ and U be related by: u′ii = exp (iΘi)uii.
We define U ′′ by u′′ij = exp (−iΘi)u
ij. Then, u
ii = uii. Now U
′′ is unitary and
thus, a priori, has N2 independent parameters. The information on moduli
of elements constitutes (N − 1)2 independent conditions, corresponding to
an (N − 1)× (N − 1) dimensional submatrix; the rest of the (2N − 1) mod-
uli being determined by relations implying that the norm of each row and
column is unity. The relations u′′ii = uii imply additional N relations on the
phases on uii.This leaves N
2 − (N − 1)2 −N = N − 1 free parameters. The
phases of u′′1j , 2 ≤ j ≤ N are unconstrained by |u
ij| = |uij| : 1 ≤ i, j ≤ N
and u′′ii = uii and we define u
1j = u1j exp (iφj), 2 ≤ j ≤ N . Then, there
are no free parameters and must lead to a unique U ′′. Now, U ′′ specified by
u′′ij = uij exp (iφj − iφi), 1 ≤ i, j ≤ N (φ1 ≡ 0) is such a solution. This
together with u′′ij = exp (−iΘi)u
ij leads to the result; with the definition
Θi − φi = θi.
Thus, in view of the unitarity of Ũ , and Û and (5), we write,
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
≡ 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cm (g1,−T )〉 × exp (iθ
n + iφm) (9)
We shall assume that F and C are have time-reversal invariance and derive
the consequences. Under time reversal, we know then that,
〈β|S |α〉 = 〈T α|S |T β〉 (10)
where |T β〉 is the state obtained by time-reversing the quantum numbers of
the state |β〉. In this case, it would imply, keeping in mind our choice of
definitions,
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃m (λ1,−T )〉
d̃m (λ1, T )
Ũ [λ1, λ
2;−T, T
′] |c̃n (λ
′)〉 (11)
We write a similar relation for F . Putting T ′ = T , (or equivalently, noting
that the matrix elements are insensitive to T ′ and T ), we find,
φp(λ2, λ1) = θp(λ1, λ2) (12)
5 Consequence of Causality of F for C
We shall assume that the fundamental theory F is causal and deduce the
consequences for the derived theory C(C′, C′′). The causality of F implies
U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
is independent of g1. Hence,
Mnm ≡ 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] |dm (g
is also independent of g1 since the state vectors 〈dn (g
2 , T
′)| and |dm (g
are independent of g1 with g
2 and g
2 fixed. We shall re-express Mnm in
terms of the matrix elements of the derived theory C(C′, C′′) and deduce the
consequences. We note,
Mnm = 〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′]U † [g1, g
2;−T, T
′] |dm (g
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |αp (g1,−T )〉
× 〈αp (g1,−T )|U
† [g1, g
2;−T, T
′] |dm (g
′)〉 (13)
〈dn (g
2 , T
′)|U [g1, g
2 ;−T, T
′] |cp (g1,−T )〉
× 〈cp (g1,−T )|U
† [g1, g
2;−T, T
′] |dm (g
′)〉 (14)
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] |c̃p (λ1,−T )〉 exp [−i(θ̃
p − θ̃
× 〈c̃p (λ1,−T )| Ũ
† [λ1, λ
2;−T, T
d̃m (λ
exp−[i(θ′′n − θ
m)](15)
≡ M̃nm(λ
2, λ1) (16)
In the 3rd step, we have employed the equations (8) and in the second step,
we have employed the closure relation for F .
In the above, θ′′n ≡ θn(λ
2, λ1), θ
m ≡ θm(λ
2, λ1), and θ̃
p ≡ θp(λ1, λ
2) etc.
Thus, the expression (15) is independent of λ1:
∂M̃nm(λ
2, λ1)
= 0 (17)
6 Analysis of Causality Condition
We shall now analyze the condition (17) obtained as an implication of causal-
ity of F . For this purpose, we shall find it useful to Taylor-expand θn as
follows7:
2, λ1) = θn(λ
2, λ1(0)) + βn∆1 + γn∆2 + δn∆1∆2 + · · ·
≡ αn + βn∆1 + γn∆2 + δn∆1∆2 + · · ·
2, λ1) = αm + βm∆1 + · · · (18)
Here, ∆1 ≡ λ1 − λ1(0); ∆2 ≡ λ
2 − λ
2; β, γ, δ refer to appropriate partial
derivatives at (λ′2, λ1(0)) and λ1(0) is some value near λ1.
We note that if
θn(λ2, λ1) is a function only of its first argument (I)
then, (θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
2) is zero and (θ
n − θ
m) ≡ θn(λ
2, λ1)−
2, λ1) is independent of λ1. Also, we can then carry out the sum over p
using the completeness relation and find that the independence from λ1 of
M̃nm(λ
2, λ1)
d̃n (λ
Ũ [λ1, λ
2;−T, T
′] Ũ † [λ1, λ
2;−T, T
d̃m (λ
× exp−[i(θ′′n − θ
m)] (19)
7Throughout, we have employed only the infinitesimal variations in the couplings.
These are sufficient to determine the first order partial derivatives with respect to each λ1
and λ2. Hence, we shall content ourselves with expansion only upto O(∆1∆2)
for all m,n implies Ũ [λ1, λ
2;−T, T
′] Ũ † [λ1, λ
2;−T, T
′] is independent of λ1.
This condition is indeed necessary for causality of C. In fact, in this case, we
can rewrite8
d̃n (λ2, T
Ũ [λ1, λ2;−T, T
′] |c̃m (λ1,−T )〉
≡ 〈dn (g2, T
′)|U [g1, g2;−T, T
′] |cm (g1,−T )〉
× exp (iθn(λ2, λ1) + iθm(λ1, λ2)) (20)
d̃∗n (λ2, T
Ũ [λ1, λ2;−T, T
∣c̃∗m (λ1,−T )
≡ 〈dn (g2, T
′)|U [g1, g2;−T, T
′] |cm (g1,−T )〉 (21)
by redefining states by absorbing phases:
∣c̃∗m (λ1,−T )
= e−iθm(λ1) |c̃m (λ1,−T )〉)
etc. We note that this redefinition of the states is meaningful and compatible
with causality when θn is independent of its second argument. If on the
other hand, θn is dependent on its second argument (excepting a possibility
below), we cannot absorb a phase in a manner compatible with causality : a
state
∣c̃∗m
at t = −T cannot be made to depend on the value of coupling
λ2 it would have at a later time t > 0.
We can, in fact, liberalize somewhat the above condition by requiring
that,
βn = β and δn = 0 ∀ n (II)
In this case,
(θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
= β∆2 + · · · (22)
is independent of λ1 and does not depend also on p and thus comes out of
the summation in (15). The summation in (15) can be carried out using
the completeness relation. Also, (θ′′n − θ
m) ≡ θn(λ
2, λ1) − θm(λ
2, λ1) is still
independent of λ1. Thus, the entire discussion proceeds as before: in par-
ticular, as a little analysis shows, the phases can again be absorbed into the
definition of states in a manner compatible with causality.
8We have dropped primes on λ2.
While we shall not provide the general analysis of (17), we shall establish
examples of a few specific sufficient conditions for causality violation. (These
are simple conditions that, in fact, contradict I or II above) We can easily
verify the following results:
1. There is causality violation if (i) for some n, δn 6= 0 and (ii) βn =
βm ∀ n,m
2. There is causality violation if there be m 6= n such that
M̃nm(λ
2, λ1) 6= 0, when evaluated to O(∆), and βm 6= βn.
Proof : We shall let, if possible, C be causal. We can then write,
Ũ [λ1, λ
2;−T, T
′] = Ũ [λ′′2; 0, T
′] Ũ [λ1;−T, 0] (23)
Then, we can write the expression (15) as,
M̃nm(λ
2, λ1) =
d̃n (λ
2, 0)
c̃p (λ1, 0)〉 exp [−i(θ̃
p − θ̃
× 〈c̃p (λ1, 0)
d̃m (λ
2, 0)
exp−[i(θ′′n − θ
d̃n (λ
2, 0)
d̃m (λ
2, 0)
exp [−i(θ′′n − θ
m)] (24)
where X ≡
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃
p − θ̃
p)]. We shall now expand
the quantities involved to the first order in the infinitesimals as in (18). In
addition, we note that to the zeroth order in ∆2, (i.e. λ”2 − λ
2 = 0), we
have, (θ̃′′p − θ̃
p) = 0 and the completeness relation leads to X = 1. We further
define,
d̃n (λ
2, 0)
d̃m (λ
2, 0)
= δnm + iηnm∆2 + · · · (25)
Proof of (i): We define δ0 ≡ max{|δn|}; and let ±δ0 = δq for some q. We
now have,
(θ̃′′p − θ̃
p) ≡ θp(λ1, λ
2)− θp(λ1, λ
= β∆2 + δp∆1∆2 + · · · (26)
and thus,
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃
p − θ̃
= exp (−iβ∆2)
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| × exp [−i(δp∆1∆2)]
= exp (−iβ∆2)
I − i∆1∆2
|c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| δp
Thus,
exp (iβ∆2)
d̃q (λ
2, 0)
d̃q (λ
2, 0)
= 1 + iηqq∆2 − i∆1∆2
δp|upq|
2 + · · · (28)
where upq ≡
d̃q (λ2, 0)
c̃p (λ1, 0)〉 (We can ignore primes on λ2 in this term).
The multiplicative exponential factor in (24) becomes:
exp (−iγq∆2 − iδq∆1∆2 + · · ·) ≈ 1− iγq∆2 − iδq∆1∆2 + · · ·
. Thus,
M̃qq = 1 + iηqq∆2 − iγq∆2 − iδq∆1∆2 − i∆1∆2
δp|upq|
2 + · · ·
= 1 + iηqq∆2 − iγq∆2 − i∆1∆2
[δq + δp]|upq|
2 + · · · (29)
In view of the fact that either δp + δq ≥ 0 ∀ p or δp+ δq ≤ 0 ∀ p the last term
is necessarily non-vanishing and dependent on ∆1
Proof of (ii): Consider the matrix element
M̃nm(λ
2, λ1) ≡
d̃n (λ
2, 0)
d̃m (λ
2, 0)
exp−[i(θ′′n − θ
m)] 6= 0(30)
for n 6= m. To the first order in the infinitesimals, the nonzero matrix element
d̃n (λ
2, 0)
d̃m (λ
2, 0)
is independent of ∆1. The multiplicative phase factor,
exp−[i(θ′′n − θ
m)] = exp{−i(αn − αm)− i(βn − βm)∆1 − iγn∆2}
is necessarily dependent on ∆1, thus implying causality violation.
9There is the obvious exception that δp = −δq for every such p such thatupq 6= 0; and
this has to be valid for each such q for which δq = ±δ0.
7 Additional comments
We comment in a qualitative way upon how a phase factor depending on
both values of the coupling can arise. Suppose that the derived theory C
is actually correctly described by a nonlocal covariant theory with a finite
non-zero non-locality scale ∆ ∼ 1/Λ. Since the theory is covariant, it is also
non-local in time. We write,
Ũ(λ1, λ2;−T, T
′) = Ũ(λ2; ∆, T
′)Ũ(λ1, λ2;−∆,∆)Ũ(λ1;−T,−∆) (31)
where the first and the third factors on the right hand side depends only on
one value of the coupling due to finite size of non-locality in time. The sec-
ond factor however depends on both couplings because in this time-interval
(−∆,∆), time evolution depends on both values of the coupling λ. On the
other hand, the fundamental theory, being local and causal, however has no
such analogue . The matrix Ũ(λ1, λ2;−∆,∆) can then give rise to phases
depending on both couplings in relation (9).
Naively, one may expect that if the fundamental theory is causal, the de-
rived theory should be so. Examples are however known where the diagrams
of the fundamental theory are associated with a different weight in the actual
phenomenology. For example, OZI rule in hadronic phenomenology gives a
suppression of a subset of the QCD diagrams. While such a possibility is
distinct from what is discussed in this work, generally such a modification
of the amplitudes within the fundamental theory may alter the underlying
properties of the fundamental theory such as causality.
References
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hep-ph/0511135; N.N. Khuri hep-ph/9512386
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http://arxiv.org/abs/hep-ph/0511135
http://arxiv.org/abs/hep-ph/9512386
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[7] A. Haque and S.D. Joglekar, hep-th/0701171
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[13] For a simpler and intuitive understanding of the causality condition in
either form, see e.g. S.D. Joglekar, hep-th/0601006
http://arxiv.org/abs/hep-th/0601006
http://arxiv.org/abs/hep-th/0701171
http://arxiv.org/abs/hep-th/0601006
	 Introduction
	 Preliminary
	 Definition of the problem
	 Definition of the system
	 Causality formulation for a theory without a well-defined S-matrix
	 Relations between the derived theory and the fundamental theory
	 Relations between coupling constants
	 Relation between states
	Consequence of Causality of F for C
	Analysis of Causality Condition
	Additional comments
