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Dispersion Relations Alone Cannot Guarantee Causality
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Dispersion Relations Alone Cannot Guarantee Causality
L. Gavassino,1M. Disconzi,1and J. Noronha2
1Department of Mathematics, Vanderbilt University, 1326 Stevenson Center Lane, Nashville, Tennessee 37240, USA
2Illinois Center for Advanced Studies of the Universe & Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, Illinois 61801-3003, USA
(Received 2 August 2023; accepted 29 March 2024; published 17 April 2024)
We show that linear superpositions of plane waves involving a single-valued, covariantly stable
dispersion relation ω?k?always propagate outside the light cone unless ω?k??a?bk. This implies that
there is no notion of causality for individual dispersion relations since no mathematical condition on the
function ω?k?(such as the front velocity or the asymptotic group velocity conditions) can serve as a
sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emergefrom a careful cancellation that occurs when one superimposes all the excitation branches of a physical
model. This happens automatically in local theories of matter that are covariantly stable. Hence, we find
that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need forantiparticles in relativistic quantum mechanics.
DOI: 10.1103/PhysRevLett.132.162301
Introduction. —The“practical ”definition of relativistic
causality is universally accepted: it is impossible to transmit
information faster than the vacuum speed of light [1–3].
The question is how to translate this principle into
mathematical constraints that our physical theories mustobey. In some cases, this question has an unambiguous
answer. In classical field theory, causality demands that the
characteristics of the field equations lay inside or upon thelight cone [4–8]. In quantum field theory, the commutator
of spacelike-separated observables must vanish [9–11].I n
other contexts, the mathematical nature of causality is lessunderstood.
Consider a homogeneous system in thermodynamic
equilibrium, and let ω?k?be the eigenfrequency of one
of its (linear) excitations, which is a function of the wave
number k[12]. Under which conditions is such dispersion
relation compatible with causality? Most attemptedanswers revolve around imposing inequalities on the phase
velocity ?Reω?=k, or on the group velocity d?Reω?=dk
[13–15]. However, no fully consistent and universally
reliable criterion has been found. The most widely accepted
constraint is that the “front velocity ”[16,17] , or the
“asymptotic group velocity ”[18], both of which usually
coincide by L ’Hopital ’s rule, should not exceed the speed of
light c(?1, in our units), namelyv
f? lim
k→∞;k∈RReω
k?L’Hlim
k→∞;k∈RdReω
dk∈??1;1/C138:?1?
Unfortunately, this condition is far from satisfactory as
many famous acausal equations in physics fulfill (1)even if
the theory of partial differential equations tells us that theypropagate information at infinite speeds [4,8]. Three not-
able examples are the diffusion equation, the Euclidean
wave equation, and the linearized Benjamin-Bona-Mahony(BBM) [19] equation, respectively:
??
t??2x?φ?0?ω??ik2;
??2t??2x?φ?0?ω?/C6ik;
??t??x??t?2x?φ?0?ω?k
1?k2: ?2?
All these equations have vf?0. Furthermore, their phase
and group velocities are (sub)luminal for all k. Neverthe-
less, these three models are strongly acausal. The BBM
equation is particularly striking because one cannot attri-bute the causality violation to the imaginary part of ω,
given that ωis real for real k. Yet, it is acausal as the lines
t?const are spacelike characteristics [20].
This Letter shows that the limitations of (1)are mani-
festations of a fundamental impossibility. Namely, unlessω?k??a?bkfor all k(with a,bconstant), a single
dispersion relation ω?k?cannot be causal. Rather,
“causality ”is a collective property of the system, which
describes how all the excitation branches ω
n?k?combine
when the full initial value problem is set up. Therefore,
apart from ω?k??a?bk, it is impossible to formulate a
sufficient condition for causality in the form of anPublished by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI. Funded by SCOAP3.PHYSICAL REVIEW LETTERS 132, 162301 (2024)
0031-9007 =24=132(16) =162301(6) 162301-1 Published by the American Physical Societyinequality that ω?k?should obey. This is why, given a
causality criterion like (1), one can always find models that
fulfill it and are acausal, such as (2).
Nevertheless, we also show that one can overcome these
difficulties by appealing to specific structures present inmany (but not all) physical theories, which guarantee thatthe dispersion branches combine “correctly ”to ensure
causality. In particular, if the operator governing the
dynamics is local and the system is covariantly stable
(in precise senses defined below), all superluminal tails
cancel out; see Theorem 1 for a precise statement.
A key inequality. —Our analysis relies on the following
inequality, which must hold for all dispersion branches
describing disturbances around the equilibrium state of a
stable system in relativity [22,23] :
Imω?k?≤jImkj: ?3?
This covariant bound can be derived from the study of
retarded causal correlators of stable phases of matter, and it
is textbook material [24], whose importance in constraining
transport properties of matter was demonstrated inRef.[22]. Independently from the principle of causality, (3)
constitutes the physical requirement that a stable system
should be simultaneously stable in every inertial frame ofRef. [25]. In fact, if (3)were violated, namely if there were
some k∈Cfor which Imω>jImkj, then a boost with
velocity v?Imk=Imωwould lead us to a new reference
frame where Imω
0>0andImk0?0[23]. This would
imply that there is an observer who can detect a growing
Fourier mode, signaling, an instability [26–28]. For this
reason, we assume (3)holds as a basic stability property of
the system.
Single dispersion branches are superluminal. —Fix some
level of description of matter, which may be, e.g., quantum
field theory, kinetic theory, or hydrodynamics. Using esta-blished techniques [29–32], one can compute all the
(possibly infinite) dispersion branches predicted by such
theory. Choose one of interest, ω?k?. According to conven-
tional wisdom [13–18], the relation ω?k?determines how
the corresponding excitation “propagates ”,a n dt h e r e
should be some causality criterion for ω?k?, e.g., (1),
which guarantees that the excitation propagates sublumi-
nally. Now we prove that this intuitive interpretation
can be consistently maintained only in the trivial caseω?k??a?bk. In dispersive media, causality can never be
argued from ω?k?alone.
First, let us make the above (incorrect) intuition about the
causality of ω?k?more precise. Let φ?x
μ?∈Cbe the linear
perturbation to a local observable of interest. For example,
φ?xμ?may be the local energy density fluctuation. Then,
consider a 1?1dimensional profile φ?t; x?that is con-
structed by superimposing plane waves all belonging solely
to the selected excitation branch ω?k?, i.e.,φ?t; x??Z?∞
?∞φ?k?ei?kx?ω?k?t/C138dk
2π: ?4?
By setting t?0, we find that φ?k?is the Fourier transform
of the initial data, φ?0;x?. The straightforward definition of
“causal dispersion relation ”is the following: If φ?0;x?has
support in a set R, then the support of φ?t; x?at later times
should be contained inside the future light cone of R, see
Fig. 1. As a consequence, if φ?0;x?has compact spatial
support, one should find that φ?t; x?has compact spatial
support for each fixed t>0. Now we will show that, in
practice, this is never the case for φgiven by (4). On the
contrary, single-branched excitations of the form (4)always
“travel ”at infinite speed unless ω?a?bk(i.e., when the
medium is not dispersive).
A simple argument. —Ifφ?t; x?vanishes outside the
future light cone of a compact set R, then also ?tφ?t; x?
must vanish there. Hence, to prove that φ?t; x?exits the
light cone, it suffices to show that, for some t0>0, the
spatial profiles of φ?t0;x?and?tφ?t0;x?cannot both have
compact support simultaneously. We assume that φ?t; x?is
smooth, but the argument can be generalized.
Fixt0>0. From (4)and the uniqueness of the Fourier
transform, we have that the spatial Fourier transform of
φ?t0;x?is given by φ?t0;k??φ?k?e?iω?k?t0. Now, suppose
thatφ?t0;x?is compactly supported. Then, φ?t0;k?extends
to an entire function of k∈C[33]. Under our assumptions,
we can bring time derivatives under the integral to concludethat ?
tφ?t0;x?has spatial Fourier transform ˙φ?t0;k??
?iω?k?φ?t0;k?. Corollary 1.1 of [22] tells us that, if
ω?k?obeys (3), then it cannot be an entire function (unless
ω?a?bk). Therefore, ˙φ?t0;k?is the product of an entire
function with a function that is not entire. Such a function
can be entire only in the remote eventuality in which the
discrete zeroes of φ?k?happen to cancel the singularities of
ω?k?. This requires a perfect fine-tuning of the initial data,
and it does not happen in general [34]. Furthermore,
according to Theorem 2 of [22],i fω?k?satisfies (3), then
FIG. 1. The principle of causality. If a perturbation has initial
support inside a region of space R(blue segment), then it cannot
propagate outside the set J??R?called the “causal future of
R”[6],o r“future light cone of R”[5](red region).PHYSICAL REVIEW LETTERS 132, 162301 (2024)
162301-2its singularities are never poles or essential singularities.
Instead, they are expected to be branch points, which
cannot be erased by multiplying ω?k?with an entire
nonzero function. Thus, ˙φ?t0;k?is not an entire function
and, therefore, ?tφ?t0;x?cannot have compact support,
as desired.
Application 1: Hegerfeldt paradox .—The above argu-
ment is a generalization of the well-known result (due to
Hegerfeldt [35]) that relativistic single-particle wave func-
tions of the form
φ?t; x??Z?∞
?∞φ?k?ei?kx???????????
m2?k2p
t?dk
2π?5?
must propagate outside the light cone [9]. Indeed, it can
be easily verified that the dispersion relation of the free
particle, ω?????????????????
m2?k2p
, obeys (3), and this forces it to be
nonanalytical, as testified by the square root. Hence, the
support of (5)expands at infinite speed [36], even if the
group velocity, vg?k??k=????????????????
m2?k2p
, is subluminal.
Application 2: Necessity of nonhydrodynamic modes .—
An immediate corollary of our analysis is that the retarded
Green ’s function of any theory for diffusion having only
one dispersion relation, ω?k???iDk2?O?k3?, always
exits the light cone. Thus, to build a subluminal Green ’s
function, we need at least two dispersion relations (see [37],
Sec. 7.4). This explains why an additional (usually gapped)
mode is needed for causality [22].
Explanation. —The superluminal behavior of (4)in
causal matter seems absurd, but there is a simple explan-
ation: excitations of the form (4)cannot be truly localized,
unless ω?a?bk. They may seem to have compact
support, if φ?0;x?is supported in R, but, in principle,
an observer can detect the excitation from outside R
already at t?0by measuring some other observable. In
fact, we recall that the dispersion relation ω?k?is derived
from some underlying physical theory (e.g., quantum field
theory, kinetic theory, or hydrodynamics), which may
possess several other local observables besides φ. The fact
that the initial profile φ?0;x?has support inside Rdoes
not imply that all the measurable fields affected by the
excitation are unperturbed outside R. Instead, it may be the
case that, due to this excitation, the perturbation to a second
observable ψ?xμ?of the theory has already unbounded
support at t?0. This is how, in principle, φcan propagate
outside the light cone without necessarily violating the
principle of causality in the full theory: there is no super-
luminal propagation of information if such information was
already accessible through the measurement of ψ?0;x?
outside R. Indeed, below we prove that if the perturbations
toallthe observables are initially supported inside a
compact region R(i.e., the excitation is truly localized),
and the dynamics is governed by a local operator, then
φ?t; x?cannot be expressed in the form (4), and it must
always combine at least two dispersion branches, unless
ω?a?bk.Compactly supported excitations. —We assume that the
state of the system at a given time can be characterized, in
the linear regime, by a collection of smooth perturbation
fields Ψ?xμ?∈CD, which all vanish at equilibrium. We
assume that Dis finite, although it can be as large as the
number of particles in a material volume element. In mostphysical theories currently available (e.g., electrodynamics,
elasticity theory, or hydrodynamics), the 1?1dimensional
equation of motion of the system takes the form
?
tΨ?L??x?Ψ; ?6?
where is L??x?a polynomial of finite degree in ?x, i.e.,
L??x??A0?A1?x?/C1/C1/C1? AM?Mx, where Ajare constant
D×Dmatrices, and M∈N[38]. This is what we mean by
a“local operator. ”In fact, operators involving an infinite
series of derivatives can produce nonlocalities and causality
violations. For example, if we set L??x??ea?x, Eq. (6)
becomes ?tΨ?t; x??Ψ?t; x?a?, which is clearly a non-
local theory. Indeed, the main reason why the BBM
equation in (2)is acausal is that its dynamical operator,
L??x??? ?2x?1??1?x, is nonlocal [39,40] .
The general formal solution to (6)reads
Ψ?t; x??Z?∞
?∞eL?ik?tΨ?k?eikxdk
2π; ?7?
where Ψ?k?is the Fourier transform of the initial data
Ψ?0;x?. Now, the field φ?t; x?, being a linearized local
observable, is a local linear functional of the degreesof freedom, namely φ?V??
x?Ψ, where Vis also a
polynomial of finite degree in ?x, i.e., V??x??B0?
B1?x?/C1/C1/C1? BN?Nx. Here, Bjare constant row vectors of
length D, and N∈N. Therefore, we have the following
formula:
φ?t; x??Z?∞
?∞V?ik?eL?ik?tΨ?k?eikxdk
2π: ?8?
Assume that the excitation is initially supported inside R.
Then, all the components of Ψ?0;x?are compactly sup-
ported, and all the components of Ψ?k?are entire functions.
Furthermore, L?ik?andV?ik?are entire in k∈C, being
polynomials. Also, the matrix exponential is an analyticfunction of the components of the matrix in the exponent,
so that e
L?ik?tis entire in k. Combining these results, we can
conclude that the integrand of (8)is an entire function of k
for all t. For this reason, it cannot coincide with (4), unless
ω?k??a?bk(i.e., in dispersion-free systems). This
shows that, when we construct the state (4)in a dispersive
medium, we implicitly allow some component of Ψto have
unbounded support already at t?0, which is what we
wanted to prove. In the Supplemental Material [21],w e
analyze the explicit example of the Klein-Gordon equation.PHYSICAL REVIEW LETTERS 132, 162301 (2024)
162301-3Causality criterion for stable matter. —The above analy-
sis suggests that if the dynamics of the system is governed
by a local operator L, then all the dispersion branches will
automatically combine in a way to cancel the infinite tails
of the individual excitations (4). This intuition can be made
rigorous through the following theorem, according towhich, schematically,
/C18local
equations/C19
?/C18stability in
all frames/C19
?/C18relativistic
causality/C19
:
More precisely, note the following:
Theorem 1 .—IfL?ik?andV?ik?are polynomials of
finite degree in ik, and the eigenvalues ω
n?k?ofiL?ik?
obey the stability requirement (3)for all k∈C, then all
smooth linear excitations propagate subluminally, in the
sense that the support of φ?t; x?, as given by (8),i s
contained within the future light cone of the support
ofΨ?0;x?.
Proof .—We will focus on the case where Ψ?0;x?has
support inside the interval ??1;0/C138. We will verify that
φ?t; x?vanishes for x>t≥0. More general cases can be
recovered from here by invoking linearity, translation
invariance, and closure of the solution space.
Consider the complex integral
I?Z
ΓV?ik?eL?ik?tΨ?k?eikxdk
2π; ?9?
where Γis the closed loop in complex kspace in Fig. 2, in the
limit of large R. Since the integrand is entire, I?0. Let us
now show that, if x>t≥0, then the contribution coming
from the upper semicircle decays to zero as R→?∞, so that
0?I?φ?t; x?; see Eq. (8). To this end, we first note that,
according to the Jordan-Chevalley decomposition theorem,the matrix L?ik?can be expressed as
L?ik???iX
nωn?k?Pn?k??N?k?; ?10?
where Pnare complementary eigenprojectors (so that
PmPn?δmnPn,P
nPn?I), andNis a nilpotent matrix
(Na?0for some a∈N) which commutes with all Pn.
Thus, the integrand in (9)can be rewritten as
X
nXa?1
j?0V?Nt?j
j!Pnei?kx?ωnt?Ψ: ?11?
The matrix elements of NjandPngrow at most like powers
ofjkj. This follows from [41] [Chap. 2, Eqs. (1.21) and
(1.26)], applied to the matrix ?ik??ML?ik?regarded as a
polynomial in ?ik??1→0, combined with the fact that
?ik??ML?ik?andL?ik?have the same invariant subspaces.
On the other hand, if Imk≥0, and x>t≥0, we have the
following estimates:jei?kx?ωnt?j?e?xImk?tImωn≤e??x?t?Imk≤1;
jΨ?k?j ?/C12/C12/C12/C12Z
0
?1e?xImkΨ?0;?x?e?i?xRekd?x/C12/C12/C12/C12≤L
1?Ψ?0;x?/C138:
?12?
In the first line, we have invoked the inequality (3). In the
second line, we have used the fact that e?xImk≤1inside
the interval ??1;0/C138. Note that Ψ?0;x?, being continuous
and compactly supported, has finite L1norm. From the
estimates (12), we can conclude that (11)decays exponen-
tially to zero when Imk→?∞. Furthermore, since
Ψ?Rek?iImk?, regarded as a function of Rek, is the
Fourier transform of the Schwartz function exImkΨ?0;x?,i t
is itself a Schwartz function [33], meaning that (11)decays
to zero faster than any power also when Rek→∞.I t
follows that, as R2??Rek?2??Imk?2→?∞, the inte-
gral over the semicircle converges to zero since the integrand
decays faster than any power of R. ?
Most derivations of (1)rely on the assumption that ω≈
vfkfor large k∈C, so that (1)is a direct consequence
of(3). However, (3)is a much more stringent condition,
as it automatically rules out the acausal equations (2).
Indeed, the apparent success of (1)in many situations can
be traced back to (3)through the following theorem
proven below:
Theorem 2 .—If(6)is a hyperbolic first-order system,
withL??x???Ξ?M?x, then (3)implies (1), and the
characteristic velocities coincide with the front velocities.
Proof .—The ratios ωn=kare eigenvalues of the matrix
iL?ik?
k?M??ik??1Ξ: ?13?
If we regard the right-hand side as a polynomial in ?ik??1,
we can take the limit as ?ik??1→0and apply the continuity
property of eigenvalues [41] to conclude that ωn=kmust
converge to eigenvalues of Mfor large k∈C. But the
eigenvalues of Mare the characteristic velocities of the
system, and they are real (by hyperbolicity), so that
lim
k→∞;k∈Cωn
k?vch;n∈R: ?14?
Restricting the above limit to real k, and using the
continuity of Re, we find that the characteristic velocities
FIG. 2. Path of integration for the proof of Theorem 1.PHYSICAL REVIEW LETTERS 132, 162301 (2024)
162301-4coincide with the front velocities. Restricting the limit to
imaginary k, we find that (3)implies causality.
vch;n?Re/C20
lim
k→∞;k∈Rωn
k/C21
? lim
k→∞;k∈RReωn
k?vf;n;
vch;n?Re/C20
lim
k→∞;k∈iRωn
iImk/C21
? lim
k→∞;k∈iRImωn
Imk∈??1;1/C138:
?15?
This completes our proof. ?
While the above analysis was restricted to classical initial
value problems, its broad implications may also be extrapo-lated to quantum systems. For example, some conformalfield theories are known to be acausal [42]. Given that such
theories are local, we can “apply ”our Theorem 1 to conclude
that such theories are not covariantly stable and violate thebound (3), in agreement with Sec. III.A of [25].
Correlators in QFT. —Causality requires multiple
dispersion relations also in QFT. Given a local observable
operator ?φ?x
μ?, the correlator G?xμ??h ? ?φ?xμ?;?φ?0?/C138ihas
support inside the light cone [9]. But since the slices of the
light cone at constant time are compact spheres, the spatialFourier transform G?t;k?must be entire in kfor all t[33].
This is why introducing momentum cutoffs or “patching ”
correlators in momentum space leads to causality violations[43]: it breaks analyticity. Furthermore, if G?t;k?can be
expressed as a superposition of modes of the form e
?iωn?k?t
(see e.g., [44]), then we know that all the nonanalyticities of
the individual frequencies ωn?k?must cancel out.
Final remarks. —Consider the following puzzle: All
solutions of the relativistic Schr?dinger equation i?tφ???????????????? ?
m2??2xp
φare also solutions of the Klein-Gordon equa-
tion??2tφ??m2??2x?φ. Nevertheless, the former is noto-
riously acausal [9], while the latter is causal. This defies the
intuition of causality as a statement about the propagation
speed of φ. How can the same function φ?t; x?be super-
luminal when viewed as a solution of one equation andsubluminal when viewed as a solution of another equation?
Here, we solved this puzzle by showing that causality is
not an intrinsic property of the fields themselves. Rather, itis a property of how we “attach information ”to the fields by
defining the physical state. The existence of faster-than-
light motion does not result in causality violation if themotion carries no new information about the state. Indeed,relativistic Schr?dinger and Klein-Gordon differ by the way
they define the physical state at a given time: fφ?x?gin
the former, and fφ?x?;?
tφ?x?gin the latter. The puzzle
arises because compactly supported field states within
relativistic Schr?dinger (i.e., localized φprofiles) must
have unbounded support within Klein-Gordon (i.e., cannotbe localized in ?
tφ); see Supplemental Material [21].
Starting from this intuition, we showed that nonhydro-
dynamic modes become necessary for relativistic viscoushydrodynamics for the same reason that antiparticles arenecessary for relativistic quantum mechanics: defining a
notion of locality in dispersive systems requires at least twodispersion relations.
Note added. —Recently, other formulations of Theorems 1
and 2 were proposed [45,46] .
L. G. and J. N. would like to thank P. Kovtun and R.
Hoult for a stimulating discussion. We also thank P.Lowdon for bringing the issue of correlators in QFT toour attention. L. G. is partially supported by a VanderbiltSeeding Success Grant. M. M. D. is partially supportedby NSF Grant No. DMS-2107701, a Chancellor ’s Faculty
Fellowship, DOE Grant No. DE-SC0024711, and a
Vanderbilt Seeding Success Grant. J. N. is partially sup-ported by the U.S. Department of Energy, Office ofScience, Office for Nuclear Physics under AwardNo. DE-SC0023861. L. G. and J. N. would like to thankP. Kovtun and R. Hoult for a stimulating discussion. Wealso thank P. Lowdon for bringing the issue of correlatorsin QFT to our attention. The authors thank KITP SantaBarbara for its hospitality during “The Many Faces of
Relativistic Fluid Dynamics ”Program. This research was
supported in part by the National Science Foundation underGrant No. NSF PHY-1748958.
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