RAPID COMMUNICATIONS
PHYSICAL REVIEW E 79, 010101共R兲 共2009兲
Time parameters and Lorentz transformations of relativistic stochastic processes
Jörn Dunkel,1,2,* Peter Hänggi,2 and Stefan Weber3
1
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
2
Institut für Physik, Universität Augsburg, Universitätsstraße 1, Augsburg, Germany
3
School of Operations Research and Information Engineering, 279 Rhodes Hall, Cornell University, Ithaca, New York 14853, USA
共Received 28 August 2008; published 21 January 2009兲
Rules for the transformation of time parameters in relativistic Langevin equations are derived and discussed.
In particular, it is shown that, if a coordinate-time-parametrized process approaches the relativistic JüttnerMaxwell distribution, the associated proper-time-parametrized process converges to a modified momentum
distribution, differing by a factor proportional to the inverse energy.
DOI: 10.1103/PhysRevE.79.010101
PACS number共s兲: 05.40.Jc, 02.50.Ey, 47.75.⫹f
Stochastic processes 共SPs兲 present an ubiquitous tool for
modeling complex phenomena in physics 关1–3兴, biology
关4,5兴, or economics and finance 关6–9兴. Stochastic concepts
provide a promising alternative to deterministic models
whenever the underlying microscopic dynamics of a relevant
observable is not known exactly but plausible assumptions
about the underlying statistics can be made. A specific area
where the formulation of consistent microscopic interaction
models becomes difficult 关10–12兴 concerns classical relativistic many-particle systems. Accordingly, SPs provide a useful phenomenological approach to describing, e.g., the interaction of a relativistic particle with a fluctuating environment
关13–17兴. Applications of stochastic concepts to relativistic
problems include thermalization processes in quark-gluon
plasmas, as produced in relativistic heavy-ion colliders
关18–21兴, or complex high-energy processes in astrophysics
关22–25兴.
While these applications illustrate the practical relevance
of relativistic SPs, there still exist severe conceptual issues
which need clarification from a theoretical point of view.
Among these is the choice of the time parameter that quantifies the evolution of a relativistic SP 关26兴. This problem
does not arise within a nonrelativistic framework, since
Newtonian physics postulates the existence of a universal
time which is the same for any inertial observer; thus, it is
natural to formulate nonrelativistic SPs by making reference
to this universal time. By contrast, in special relativity
关27,28兴 the notion of time becomes frame dependent, and it
is necessary to carefully distinguish between different time
parameters when constructing relativistic SPs. For example,
if the random motion of a relativistic particle is described in
a t-parametrized form, where t is the time coordinate of some
fixed inertial system ⌺, then one may wonder if and how this
process can be reexpressed in terms of the particle’s proper
time ␶ and vice versa. Another closely related question 关17兴
concerns the problem of how a certain SP appears to a moving observer: i.e., how does a SP behave under a Lorentz
transformation?
The present paper aims at clarifying the above questions
for a broad class of relativistic SPs governed by relativistic
Langevin equations 关13–17兴. First, we will discuss a heuristic approach that suffices for most practical calculations and
*jorn.dunkel@physics.ox.ac.uk
1539-3755/2009/79共1兲/010101共4兲
clarifies the basic ideas. Subsequently, these heuristic arguments will be substantiated with a mathematically rigorous
foundation by applying theorems for the time change of 共local兲 martingale processes 关29兴. The main results can be summarized as follows: If a relativistic Langevin-Itô process has
been specified in the inertial frame ⌺ and is parametrized by
the associated ⌺-coordinate time t, then this process can be
reparametrized by its proper time ␶ and the resulting process
is again of the Langevin-Itô type. Furthermore, the process
can be Lorentz transformed to a moving frame ⌺⬘, yielding a
Langevin-Itô process that is parametrized by the ⌺⬘
coordinate-time t⬘. In other words, similar to the case of
purely deterministic relativistic equations of motions, one
can choose freely between different time parametrizations in
order to characterize these relativistic SPs—but the noise
part needs to be transformed differently than the deterministic part.
We
adopt
the
metric
convention
共␩␣␤兲
= diag共−1 , 1 , . . . , 1兲 and units such that the speed of light c
= 1. Contravariant space-time and energy-momentum fourvectors are denoted by 共x␣兲 = 共x0 , xi兲 = 共x0 , x兲 = 共t , x兲 and 共p␣兲
= 共p0 , pi兲 = 共p0 , p兲, respectively, with Greek indices ␣
= 0 , 1 , . . . , d and Latin indices i = 1 , . . . , d, where d is the
number of space dimensions. Einstein’s summation convention is applied throughout.
As a starting point, we consider the t-parametrized random motion of a relativistic particle 共rest mass M兲 in the
inertial laboratory frame ⌺. The laboratory frame is defined
by the property that the thermalized background medium
共heat bath兲 causing the stochastic motion of the particle is at
rest in ⌺ 共on average兲. We assume that the particle’s trajectory (X共t兲 , P共t兲) = (Xi共t兲 , Pi共t兲) in ⌺ is governed by a stochastic differential equation 共SDE兲 of the form 关13–17兴
dX␣共t兲 = 共P␣/P0兲dt,
共1a兲
dPi共t兲 = Aidt + CijdB j共t兲.
共1b兲
Here, dX0共t兲 = dt and dXi共t兲 ª Xi共t + dt兲 − Xi共t兲 denote the time
and position increments and dPi共t兲 ª Pi共t + dt兲 − Pi共t兲 the momentum change. P0共t兲 ª 共M 2 + P2兲1/2 is the relativistic energy, and Vi共t兲 ª dXi / dt = Pi / P0 are the velocity components
in ⌺. In general, the functions Ai and Cij may depend on the
time, position, and momentum coordinates of the particle.
The random driving process B共t兲 = (B j共t兲) is taken to be a
010101-1
©2009 The American Physical Society
RAPID COMMUNICATIONS
PHYSICAL REVIEW E 79, 010101共R兲 共2009兲
DUNKEL, HÄNGGI, AND WEBER
冉
d-dimensional t-parametrized standard Wiener process
关29–31兴; i.e., B共t兲 has continuous paths. For s ⬎ t the increments are normally distributed,
2
e−兩u兩 /关2共s−t兲兴 d
d u,
P兵B共s兲 − B共t兲 苸 关u,u + du兴其 =
关2␲共s − t兲兴d/2
共2兲
and independent for nonoverlapping time intervals.1
Upon naively dividing Eq. 共1b兲 by dt, we see that Ai can
be interpreted as a deterministic force component, while
CijdB j共t兲 / dt represents random “noise.” However, for the
Wiener process the derivatives dB j共t兲 / dt are not well defined
mathematically, so the differential representation 共1兲 is in
fact shorthand for a stochastic integral equation 关29,31兴 with
CijdB j signifying an infinitesimal increment of the Itô integral
关32,33兴. Like a deterministic integral, stochastic integrals can
be approximated by Riemann-Stieltjes sums, but the coefficient functions need to be evaluated at the left end point t of
any time interval 关t , t + dt兴 in the Itô discretization.2 In contrast to other discretization rules 关1,29,31,34,35兴, the Itô discretization implies that the mean value of the noise vanishes;
i.e., 具CijdB j共t兲典 = 0 with 具·典 indicating an average over all realizations of the Wiener process B共t兲. In other words, Itô
integrals with respect to B共t兲 are 共local兲 martingales 关29兴.
Upon applying Itô’s formula 关29,31兴 to the mass-shell condition P0共t兲 = 共M 2 + P2兲1/2, one can derive from Eq. 共1b兲 the
following equation for the relativistic energy:
冊 冋
册
1 ⳵
⳵ pi ⳵
⳵
+ 0 i f = i − Ai f +
共Dik f兲 ,
2 ⳵ pk
⳵t p ⳵x
⳵p
where f is a Lorentz scalar 关37兴 and p0 = 共M 2 + p2兲1/2.4 Deterministic initial data X共0兲 = x0 and P共0兲 = p0 translate into the
localized initial condition f共0 , x , p兲 = ␦共x − x0兲␦共p − p0兲.
Physical constraints on the coefficients Ai共t , x , p兲 and
Cri共t , x , p兲 may arise from symmetries and/or thermostatistical considerations. For example, neglecting additional external force fields and considering a heat bath that is stationary,
isotropic, and position independent in ⌺, one is led to the
ansatz
Ai = − ␣共p0兲pi,
Cij = 关2D共p0兲兴1/2␦ij ,
dP 共t兲 = A dt +
A0 ª
0
冋
Cr0dBr共t兲,
册
Ai Pi Dij ␦ij Pi P j
+
−
,
P0
2 P0 共P0兲3
C0j ª
PiCij
,
P0
共3兲
where Ai ª Ai, Dij ª Dij = 兺rCriCrj, and Cir ª Cir.
Equations 共1兲 define a straightforward relativistic generalization 关13–15兴 of the classical Ornstein-Uhlenbeck process 关36兴, representing a standard model of Brownian motion
theory.3 The structure of Eq. 共1a兲 ensures that the velocity
remains bounded, 兩V兩 ⬍ 1, even if the momentum P were to
become infinitely large. When studying SDEs of the type 共1兲,
one is typically interested in the probability f共t , x , p兲ddx dd p
of finding the particle at time t in the infinitesimal phasespace interval 关x , x + dx兴 ⫻ 关p , p + dp兴. Given Eqs. 共1兲, the
non-negative, normalized probability density f共t , x , p兲 is
governed by the Fokker-Planck equation 共FPE兲
1
For simplicity, we have assumed that B共t兲 is d dimensional, implying that Cij is a square matrix. However, all results still hold if
B共t兲 has a different dimension.
2
One could also consider other discretization rules
关1,29,31,34,35兴, but then the rules of stochastic differential calculus
must be adapted.
3
In the nonrelativistic limit c → ⬁, P0 → M in Eq. 共1a兲.
共5a兲
where the friction and noise coefficients ␣ and D depend on
the energy p0 only. Moreover, if the stationary momentum
distribution is expected to be a thermal Jüttner function
关38,39兴—i.e., if f ⬁ ª limt→⬁ f ⬀ exp共−␤ p0兲 in ⌺—then ␣ and
D must satisfy the fluctuation-dissipation condition 关13,14兴
0 ⬅ ␣共p0兲p0 + dD共p0兲/dp0 − ␤D共p0兲.
共5b兲
In this case, one still has the freedom to adapt one of the two
functions ␣ or D.
In the remainder of this paper, we shall discuss how the
process 共1兲 can be reparametrized in terms of its proper time
␶ and how it transforms under the proper Lorentz group 关28兴.
The
stochastic
proper-time
differential
d␶共t兲
= 共1 − V2兲1/2dt may be expressed as
d␶共t兲 = 共M/P0兲dt.
0
共4兲
共6a兲
The inverse of the function ␶ is denoted by X̂0共␶兲 = t共␶兲 and
represents the time coordinate of the particle in the inertial
frame ⌺, parametrized by the proper time ␶. Our goal is to
find SDEs for the reparametrized processes X̂␣共␶兲
ª X␣(t共␶兲) and P̂␣共␶兲 = P␣(t共␶兲) in ⌺. The heuristic derivation
is based on the relation
dB 共t兲 ⯝ 冑dt =
j
冉冊
P̂0
M
1/2
冑d␶ ⯝
冉冊
P̂0
M
1/2
dB̂ j共␶兲,
共6b兲
where B̂ j共␶兲 is a standard Wiener process with time parameter ␶. The rigorous justification of Eq. 共6b兲 is given below.
Inserting Eqs. 共6a兲 and 共6b兲 into Eqs. 共1兲, one finds
dX̂␣共␶兲 = 共P̂␣/M兲d␶ ,
共7a兲
dP̂i共␶兲 = Âid␶ + ĈijdB̂ j共␶兲,
共7b兲
and
Ĉij
where
Âi ª 共P̂0 / M兲Ai共X̂0 , X̂ , P̂兲
i
ª 共P̂0 / M兲1/2C j共X̂0 , X̂ , P̂兲. The FPE for the associated probability density f̂共␶ , x0 , x , p兲 reads
4
Equation 共4兲 is not covariant, because we are considering here
the “true” phase-space density f共t , x , p兲 rather than the “extended”
phase-space density f̃共t , x , p0 , p兲.
010101-2
RAPID COMMUNICATIONS
PHYSICAL REVIEW E 79, 010101共R兲 共2009兲
TIME PARAMETERS AND LORENTZ TRANSFORMATIONS…
PDF / (M c)−1
1
dB j共t兲 ⯝ 冑dt =
t = 15
φJ
τ = 15
φMJ
0.75
0.5
冑dt⬘ ⯝
冋
共⌳−1兲␮0 P⬘␮
P ⬘0
册
1/2
dB⬘ j共t⬘兲,
where B⬘ j共t⬘兲 is a Wiener process with time t⬘. Furthermore,
defining primed coefficient functions in ⌺⬘ by
0.25
0
1
2
3
4
A⬘i共x⬘0,x⬘,p⬘兲 ª 关共⌳−1兲␮0 p⬘␮/p⬘0兴
5
|P | / (M c)
FIG. 1. “Stationary” probability density function 共PDF兲 of the
absolute momentum 兩P兩 measured at time t = 15 共⫻兲 and ␶ = 15 共䊊兲
from 10 000 sample trajectories of the one-dimensional 共d = 1兲 relativistic Ornstein-Uhlenbeck process 关13兴, corresponding to coefficients D共p0兲 = const and ␣共p0兲 = ␤D / p0 in Eqs. 共1兲, 共5a兲, and 共5b兲.
Simulation parameters: dt = 0.001, M = c = ␤ = D = 1.
冊 冋
册
1 ⳵
⳵ p␣ ⳵
⳵
i
+
共D̂ik f̂兲 ,
␣ f̂ =
i − Â f̂ +
2 ⳵ pk
⳵␶ M ⳵x
⳵p
Y ⬘␯共t兲 ª ⌳␮␯X␮共t兲,
G⬘␯共t兲 ª ⌳␮␯ P␮共t兲.
C⬘j i共x⬘0,x⬘,p⬘兲 ª 关共⌳−1兲␮0 p⬘␮/p⬘0兴1/2
⫻⌳␯i C␯j „共⌳−1兲␮0 x⬘␮,共⌳−1兲␮i x⬘␮,共⌳−1兲␮i p⬘␮…,
the particle’s trajectory (X⬘共t⬘兲 , P⬘共t⬘兲) in ⌺⬘ is again governed by a SDE of the standard form
共9兲
where ⌳−1 is the inverse Lorentz transformation. Thus, a
similar heuristics as in Eq. 共6b兲 gives
dX⬘␣共t⬘兲 = 共P⬘␣/P⬘0兲dt⬘ ,
共11a兲
dP⬘i共t⬘兲 = A⬘idt⬘ + C⬘j idB⬘ j共t⬘兲.
共11b兲
We will now rigorously derive the transformations of
SDEs under time changes and thereby show that the heuristic
transformations leading to Eqs. 共7兲 and 共11兲 are justified; i.e.,
we are interested in a time change t 哫 t̆ of a generic SDE
dY共t兲 = E dt + F j dB j共t兲,
共12a兲
where E and F j will typically be smooth functions of the
state variables 共Y , . . . 兲,5 and B共t兲 = B j共t兲 is a d-dimensional
standard Wiener process.6 We consider a time change t 哫 t̆
specified in the form 关cf. Eqs. 共6a兲 and 共9兲兴
dt̆ = H dt,
t̆共0兲 = 0,
共12b兲
with H being a strictly positive smooth function7 of 共Y , . . . 兲.
The inverse of t̆共t兲 is denoted by t共t̆兲. We would like to show
that Eq. 共12a兲 can be rewritten as
dY̆共t̆兲 = Ĕ dt̆ + F̆ j dB̆ j共t̆兲,
共12c兲
where Y̆共t̆兲 ª Y(t共t̆兲), Ĕ共t̆兲 ª E(t共t̆兲) / H(t共t̆兲), F̆ j共t̆兲 ª F j(t共t̆兲) /
冑H(t共t̆兲), and
5
Then we replace t by the coordinate time t⬘ of ⌺⬘ to obtain
processes X⬘␣共t⬘兲 = Y ⬘␣(t共t⬘兲) and P⬘␣共t⬘兲 = G⬘␣(t共t⬘兲). Note
that dt⬘共t兲 = dY ⬘0共t兲 = ⌳␮0 dX␮共t兲, and, hence,
⌳␮0 P␮
G ⬘0
P⬘0„t⬘共t兲…
dt,
dt
=
dt
=
P0
P0
共⌳−1兲␮0 P⬘␮„t⬘共t兲…
⫻⌳␯i A␯„共⌳−1兲␮0 x⬘␮,共⌳−1兲␮i x⬘␮,共⌳−1兲␮i p⬘␮…,
共8兲
We
note
that
where
now
D̂ik ª 兺rĈirĈkr.
0
0 d
d
f̂共␶ , x , x , p兲dx d x d p gives the probability of finding the
particle at proper time ␶ in the interval 关t , t + dt兴 ⫻ 关x , x
+ dx兴 ⫻ 关p , p + dp兴 in the inertial frame ⌺.
Remarkably, if the coefficient functions satisfy the constraints 共5a兲 and 共5b兲—so that the stationary solution f ⬁ of
Eq. 共4兲 is a Jüttner function ␾J共p兲 = Z−1 exp共−␤ p0兲—then the
stationary solution f̂ ⬁ of the corresponding proper-time FPE
共8兲 is given by a modified Jüttner function ␾MJ共p兲
= Ẑ−1 exp共−␤ p0兲 / p0. The latter can be derived from a relative entropy principle using a Lorentz invariant reference
measure in momentum space 关40兴. Physically, the difference
between f ⬁ and f̂ ⬁ is due to the fact that measurements at t
= const and ␶ = const are nonequivalent even if ␶ , t → ⬁. This
can also be confirmed by direct numerical simulation of Eqs.
共1兲; see Fig. 1.
Having discussed the proper-time reparametrization, we
next show that a similar reasoning can be applied to transform the SDEs 共1兲 to a moving frame ⌺⬘ 关17兴.
Neglecting time reversals, we consider a proper Lorentz
transformation 关28兴 from the laboratory frame ⌺ to ⌺⬘, mediated by a constant matrix ⌳␮␯ with ⌳00 ⬎ 0, that leaves the
metric tensor ␩␣␤ invariant. We proceed in two steps: First
we define
dt⬘共t兲 =
1/2
共10兲
0
冉
冉 冊
P0
P ⬘0
The state variables of the system are assumed to have continuous
paths and need to satisfy suitable integrability conditions. More
generally, E = E共t兲 and F j = F j共t兲 can be assumed to be continuous
adapted processes.
6
The Wiener process is defined on a complete filtered probability
space 共⍀ , F , F , P兲 that satisfies the usual hypotheses 关29兴. The increasing family F = 共Ft兲 is called a filtration. Ft denotes the information that will be available to an observer at time t who follows
the particle.
7
More precisely, in general H = H共t兲 is a strictly positive, continuous adapted process such that P关兰t0H共s兲ds ⬍ ⬁ ∀ t兴 = 1 and
P关兰⬁0 H共s兲ds = ⬁兴 = 1.
010101-3
RAPID COMMUNICATIONS
PHYSICAL REVIEW E 79, 010101共R兲 共2009兲
DUNKEL, HÄNGGI, AND WEBER
dB̆ j共t̆兲 ª 冑H dB j共t兲
共12d兲
Y̆共t̆兲 =
兺
n→⬁ k=0
关L j,L j兴共t兲 ª lim
再 冉 冊 冉 冊冎
Lj
共k + 1兲t
kt
− Lj n
2n
2
2
is given by 关L j , L j兴共t兲 = 兰t0H共s兲ds.9 For the quadratic variation
of B̆ j共t̆兲 = L j(t共t̆兲) we then obtain 关B̆ j , B̆ j兴共t̆兲 = 关L j , L j兴(t共t̆兲)
For
i ⫽ j,
we
have
关B̆ j , B̆i兴共t̆兲
= 兰t共t̆兲
0 H共s兲ds = t̆.
10
t共t̆兲
j
i
= 兰0 H共s兲d关B , B 兴共s兲 = 0. Thus, Lévy’s theorem implies
that B̆共t̆兲 = B̆ j共t̆兲 is a d-dimensional standard Wiener process.
Finally, using the definitions of Y̆, Ĕ, and F̆ j, we find11
8
The information available to an observer of the particle at time t̆
is denoted by Gt̆. The corresponding filtration is denoted by G
= 共Gt̆兲; cf. Chap. I.1 in 关29兴. The mathematically precise statement
regarding the time change is that B̆共t̆兲 is a standard Wiener process
with respect to G.
9
Convergence is uniform on compacts in probability; see 关29兴 for
a definition of the quadratic covariation 关Li , L j兴t.
10
See Theorem II.8.40, p. 87, in 关29兴.
11
The second equality follows from Eqs. 共12b兲 and 共12d兲 by approximating the processes E and F j by simple predictable processes; see p. 51 and Theorem II.5.21 in 关29兴.
关1兴
关2兴
关3兴
关4兴
关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
P. Hänggi and H. Thomas, Phys. Rep. 88, 207 共1982兲.
P. Hänggi et al., Rev. Mod. Phys. 62, 251 共1990兲.
E. Frey and K. Kroy, Ann. Phys. 14, 20 共2005兲.
L. J. S. Allen, Stochastic Processes with Applications to Biology 共Prentice-Hall, Upper Saddle River, NJ, 2002兲.
D. Wilkinson, Stochastic Modelling for Systems Biology
共Chapman & Hall/CRC, London, 2006兲.
L. Bachelier, Ann. Sci. Ec. Normale Super. 17, 21 共1900兲.
F. Black and M. Scholes, J. Polit. Econ. 81, 81 共1973兲.
R. C. Merton, J. Financ. 29, 449 共1974兲.
S. E. Shreve, Stochastic Calculus for Finance I & II 共Springer,
Berlin, 2004兲.
A. D. Fokker, Z. Phys. 58, 386 共1929兲.
J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157
共1945兲.
H. Van Dam and E. P. Wigner, Phys. Rev. 138, B1576 共1965兲.
F. Debbasch et al., J. Stat. Phys. 88, 945 共1997兲.
J. Dunkel and P. Hänggi, Phys. Rev. E 71, 016124 共2005兲.
R. Zygadło, Phys. Lett. A 345, 323 共2005兲.
J. Angst and J. Franchi, J. Math. Phys. 48, 083101 共2007兲.
C. Chevalier and F. Debbasch, J. Math. Phys. 49, 043303
共2008兲.
B. Svetitsky, Phys. Rev. D 37, 2484 共1988兲.
P. Roy et al., Nucl. Phys. A 624, 687 共1997兲.
H. van Hees and R. Rapp, Phys. Rev. C 71, 034907 共2005兲.
t共t̆兲
=
t̆
0
=
冕
冕冑
冕
t共t̆兲
E共s兲ds +
0
is a d-dimensional Wiener process with respect to the new
time parameter t̆.8
First, we need to prove that Eq. 共12d兲 or, equivalently,
冑H共s兲dB j共s兲 does indeed define a Wiener process.
B̆ j共t̆兲 ª 兰t共t̆兲
0
To this end, we note that for fixed j 苸 兵1 , . . . , d其 the process
L j共t兲 ª 兰t0冑H共s兲dB j共s兲 is a continuous local martingale,
whose quadratic variation
2n−1
冕
冕
冕
0
E„t共s̆兲…
H„t共s̆兲…
t̆
ds̆ +
t̆
0
F j共s兲dB j共s兲
0
t̆
Ĕ共s̆兲ds̆ +
F j„t共s̆兲…
H„t共s̆兲…
F̆ j共s̆兲dB̆ j共s̆兲,
dB̆ j共s̆兲
共13兲
0
which is just the SDE 共12c兲 written in integral notation.
The above discussion shows how relativistic Langevin
equations can be Lorentz transformed and reparametrized
within a common framework. Thus, mathematically, the special relativistic Langevin theory 关13–17兴 is now as complete
as the classical theories of nonrelativistic Brownian motions
and deterministic relativistic motions, respectively, both of
which are included as special limit cases. From a physics
point of view, the most remarkable observation consists in
the fact that the ␶-parametrized Brownian motion converges
to a modified Jüttner function 关40兴 if the corresponding
t-parametrized process converges to a Jüttner function 关38兴.
This illustrates that it is necessary to distinguish different
notions of “stationarity” in special relativity. While the t parametrization appears more natural when describing diffusion processes from the viewpoint of an external observer
关18–25兴, the ␶ parametrization is more convenient when extending the above theory to include particle creation and annihilation processes, because a particle’s lifetime is typically
quantified in terms of its proper time ␶. Last but not least, the
proper-time parametrization paves the way toward generalizing the above concepts to general relativity.
关21兴 R. Rapp et al., Nucl. Phys. A 774, 685 共2006兲.
关22兴 P. J. D. Mauas and D. O. Gomez, Astrophys. J. 398, 682
共1997兲.
关23兴 N. Itoh et al., Astrophys. J. 502, 7 共1998兲.
关24兴 B. Wolfe and F. Melia, Astrophys. J. 638, 125共2006兲.
关25兴 P. A. Becker et al., Astrophys. J. 647, 539 共2006兲.
关26兴 R. Hakim, J. Math. Phys. 9, 1805 共1968兲.
关27兴 A. Einstein, Ann. Phys. 322, 891 共1905兲.
关28兴 R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles
共Springer, Wien, 2001兲.
关29兴 P. E. Protter, Stochastic Integration and Differential Equations
共Springer-Verlag, New York, 2004兲.
关30兴 N. Wiener, J. Math. Phys. 共Cambridge, Mass.兲 2, 131 共1923兲.
关31兴 I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic
Calculus 共Springer, New York, 1991兲.
关32兴 K. Ito, Proc. Imp. Acad. 共Tokyo兲 20, 519 共1944兲.
关33兴 K. Ito, Mem. Am. Math. Soc. 4, 51 共1951兲.
关34兴 D. Fisk, Trans. Am. Math. Soc. 120, 369 共1965兲.
关35兴 R. L. Stratonovich, SIAM J. Control 4, 362 共1966兲.
关36兴 G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823
共1930兲.
关37兴 N. G. Van Kampen, Physica 共Amsterdam兲 43, 244 共1969兲.
关38兴 F. Jüttner, Ann. Phys. 339, 856 共1911兲.
关39兴 D. Cubero et al., Phys. Rev. Lett. 99, 170601 共2007兲.
关40兴 J. Dunkel et al., New J. Phys. 9, 144 共2007兲.
010101-4