statistical origin of special and doubly special relativity

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Proceedings of the 3rd Galileo–Xu Guangqi MeetingInternational Journal of Modern Physics: Conference SeriesVol. 23 (2013) 373–378c© World Scientific Publishing CompanyDOI: 10.1142/S201019451301163X

STATISTICAL ORIGIN OF SPECIAL AND DOUBLYSPECIAL RELATIVITY

PETR JIZBA

Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in Prague, Brehova 7, 115 19 Praha 1, Czech Republic

[email protected]

FABIO SCARDIGLI

Institute of Physics, Academia Sinica, Taipei 115, [email protected]

We show how a Brownian motion on a short scale can originate a relativistic motion onscales larger than particle’s Compton wavelength. Special relativity appears to be not aprimitive concept, but rather it statistically emerges when a coarse graining average overdistances of order, or longer than the Compton wavelength is taken. Our scheme accom-modates easily also the doubly special relativistic dynamics. A previously unsuspected,common statistical origin of the two frameworks is brought to light for the first time.

Keywords: Relativistic dynamics; path integrals; doubly special relativity.

PACS numbers: 03.65.Ca, 03.30.+p, 05.40.-a, 04.60.-m

1. Introduction

Without doubts the discovery of Lorentz symmetry (LS) irrevocably changed thetheoretical landscape in physics. Up to now LS has been confirmed to unprecedentprecision, and during the last century it has powerfully constrained theories in away that has proved instrumental in discovering new laws of physics. It thus seemsnatural to assume that Lorentz invariance is an exact symmetry of nature which isvalid for an arbitrary boost. There are, however, pressing reasons to suspect that LSmay fail at some critical energy or boost. For instance, in quantum field theory boththe ultraviolet divergences and Landau poles are direct artifacts of the assumptionthat the spectrum of field degrees of freedom is boost invariant. Another soundreason comes from quantum gravity where profound difficulties associated with theproblem of time1,2 indicate that an underlying preferred time may be necessary inorder to reconcile gravitational and quantum physics. In particular, general argu-ments imply that a radical departure from standard space-time symmetries at thePlanck scale is necessary.3–11 However, the idea of LS violation is not new and ithas been considered by a number of authors over the last forty years or so [see, e.g.Refs. 12, 13, 14 and citations therein].

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http://dx.doi.org/10.1142/S201019451301163X


374 P. Jizba & F. Scardigli

Building on our previous work,15,16 in this paper we show that a relativisticquantum mechanics, as formulated through path integrals (PI), bears in itself aseed of understanding how LS can be broken at short spatio-temporal scales andyet emerge as an apparently exact symmetry at large scales. The short spatial scale,smaller than the Compton length, describes a Wiener (i.e., non-relativistic) processwith a fluctuating Newtonian mass. This might be visualized as if the particle wouldbe randomly propagating (in the sense of Brownian motion) through a granular or“polycrystalline” medium. The large spatial scale corresponds, on the other hand,to distances that are larger than particle’s Compton length. At such a scale theparticle evolves according to a genuine relativistic motion, with a sharp value of themass coinciding with the Einstein rest mass.

The outlined scenario can be also conveniently applied in various doubly specialrelativistic (DSR) models. In those models a further invariant scale �, besides thespeed of light c, is introduced, and � is assumed typically to be of the order of thePlanck length. In the present framework, the scale � can be naturally identifiedwith the minimal grain size of the polycrystalline medium. By following the samestrategy as in the special relativistic context, the doubly special relativistic motioncan also be described as emergent from an underlying non relativistic Brownianmotion.

2. Emergent Special Relativity

As an application of the fundamental relations studied in Refs. 15, 16, 17, therelativistic path integral for a scalar particle can be rewritten in the form∫ x(t) = x

x(0) = x′Dx

Dp

(2π)Dexp

{∫ t

0

dτ[ip · x − c

√p2 + m2c2

]}

=

Z ∞

0dm f 1

2

`m, tc2, tc2m2

´ Z x(t) = x

x(0) = x′Dx

Dp(2π)D

eR t0 dτ (ip ·x−p2/2m − mc2), (1)

where tb − ta ≡ t − 0 = t, and

fp(z, a, b) =(a/b)p/2

2Kp(√

ab)zp−1 e−(az+b/z)/2, (2)

is the generalized inverse Gaussian distribution18 (Kp is the modified Bessel func-tion of the second kind with index p). The structure of (1) implies that m canbe interpreted as a Newtonian mass which takes on continuous values distributedaccording to f 1

2

(m, tc2, tc2m2

)with 〈m〉 = m+1/tc2 and var(m) = m/tc2 +2/t2c4.

As a result one may view a single-particle relativistic theory as a single-particlenon-relativistic theory where the particle’s Newtonian mass m represents a fluc-tuating parameter which approaches on average the Einstein rest mass m in thelarge t limit. We stress that the time t in question should be understood as a timeafter which the observation (i.e., the position measurement) is made. In particu-lar, during the period t the system remains unperturbed. In the long run all mass

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Statistical Origin of Special and Doubly Special Relativity 375

fluctuations will be washed out and only a sharp time-independent effective masswill be perceived. The expression for 〈m〉 suggests however also another interest-ing physical consideration. Note that for t large enough we surely have m > 1/tc2

which we can read as mc2t > 1. The latter means that, for large t, the relativis-tic Heisenberg inequality for the energy/time variables is satisfied, ∆E∆t ≥ 1. Onthe other hand, for t � 1/mc2, the fluctuations of the Newtonian mass m aroundthe average m are huge. The motion takes place inside each specific space-grain,as a classical, i.e. non relativistic, Brownian motion, controlled by the Hamiltonianp2/2m. There, the relativistic Heisenberg uncertainty relation is clearly violated, infact mc2t < 1 (remind that m is the Einstein rest mass). However, if we computethe non-relativistic Heisenberg relation, using the Newtonian mass m and the non-relativistic kinetic energy Ekin ∼ mv2, we find that the non-relativistic Heisenbergrelation is not violated, confirming that the motion on short scales is genuinelyclassical (in the sense of non-relativistic, see also Ref. 17).

Fluctuations of the Newtonian mass can be depicted as originating from parti-cle’s evolution in an “inhomogeneous” or a “polycrystalline” medium. Granularity,as we know, for example, from solid-state systems, typically leads to correctionsin the local dispersion relation19 and hence to alterations in the local effectivemass. The following picture thus emerges: on the short-distance scale, a non-relativistic particle can be envisaged as propagating through a single grain witha local mass m, in a classical Brownian motion. This fast-time process has a timescale ∼ 1/mc2. An averaged value of the time scale can be computed with the help ofthe smearing distribution f 1

2

(m, tc2, tc2m2

), which gives a transient temporal scale

〈1/mc2〉 = 1/mc2. The latter coincides with particle’s Compton time tC . At timescales much longer than tC (large-distance scale), the probability that the particleencounters a grain which endows it with a mass m is f 1

2

(m, tc2, tc2m2

). Then the

particle acquires a sharp mass equal to Einstein’s (i.e., Lorentz invariant) mass, andthe motion becomes effectively relativistic (see Fig. 1).

3. Emergent Doubly Special Relativity

Surely Doubly (or Deformed) Special Relativity stands among the prominent ideasintroduced in physics during the last decade, and also among the most controversialones. Many foundational issues about this theory are still being debated, in partic-ular for example the multi-particle sector of the theory (the so called Soccer BallProblem).20 In a nutshell, DSR is a theory which coherently tries to implement asecond invariant, besides the speed of light, into the transformations among inertialreference frames. This new invariant comes directly from the research in quantumgravity, and it is usually assumed to be an observer-independent length-scale — thePlanck length �p, or its inverse, i.e., the Planck energy Ep = c�−1

p . Thus, it is notso surprising that an important area of investigation has been that of the relationsbetween DSR and various quantum gravity models.21 In a particularly suggestiveapproach,22 DSR has been presented as the low energy limit of Quantum Gravity.

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A

t

2m

m

m

tC

m1

m3

m2 B

d = 2H

d = 1H

m

Fig. 1. The roughness of the representative trajectories in the relativistic path integral (1)depends on a spatial/temporal scale. On a fine scale (A), where t � tC (or � � λC) a parti-cle can be considered as propagating with a sharp Newtonian mass m into a single spatial grainwith a Brownian motion controlled by the Hamiltonian p2/2m. On the intermediate scale of orderλC the particle propagates with an average velocity equal to the speed of light c. On a coarserscale (B) the particle appears to follow a Brownian process with a sharp Lorentz invariant massm, and the particle’s net velocity is then less than c — as it should be for a massive relativisticparticle governed by the relativistic Hamiltonian c(p2 + m2c2)1/2.

Connections between DSR and other theories (non commutative geometry, AdSspace-time, etc.) have also been recently investigated.23 Clearly, the most impor-tant connection is the one between DSR and Special Relativity (SR) itself. Howeverin literature does not exist any conceptually deeper elaboration of this connection,apart form the obvious statement that for energy scales much smaller than Ep, DSRshould reduce to conventional SR, with leading corrections of first or higher orderin the ratio of energy scales to Ep. In this respect, the findings of the present paperseem to open up new vistas. In fact, Special Relativity or DSR seem to emerge fromparticular choices of the microstructure of space-time, and from a non-relativisticHamiltonian mechanics. To extend our reasonings to DSR, we start by consideringthe modified invariant, or deformed dispersion relation,

ηabpapb

(1 − �pp0)2= m2c2, (3)

proposed by Magueijo and Smolin,24 in which m plays the role of the DSR invari-ant mass. Assuming a metric signature (+,−,−,−), we can solve (3) in respect to

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Statistical Origin of Special and Doubly Special Relativity 377

p0, which essentially coincides with the physical Hamiltonian H = cp0, the gener-ator of the temporal translations in respect to the coordinate time t. Our startingHamiltonian is therefore

H = c−m2c2� ∓ √

p2(1 − m2c2�2) + m2c2

1 − m2c2�2. (4)

Now, in close analogy with (1), we derive the superstatistics identity17

∫ x(t)=x

x(0)=x′Dx

Dp

(2π)Dexp

∫ t

0

dτ

ip · x + c

(m2c2� − √

p2(1 − m2c2�2) + m2c2)

(1 − m2c2�2)

=∫ ∞

0

dm f 12

(m, tc2λ, tc2m2λ

)∫ x(t) = x

x(0) = x′Dx

Dp

(2π)Dexp

{∫ t

0

dτ

[ip · x− p2

2m− E0

]},

(5)

where E0 = mc2/(1 + mc�) is the particle’s rest energy [see, e.g., Ref.24] and λ =1/(1−m2c2�2) is the deformation parameter. From (2), it is easy to see that 〈m〉 =m + 1/(tc2λ) and var(m) = m/tc2λ + 2/t2c4λ2. From the structure of 〈m〉 we canobtain further useful insights. Similarly as in the SR framework, the fluctuatingNewtonian mass m converges rapidly, at long times t, to the SR rest mass m. Butthis time the rate of convergence is controlled also by the parameter λ. Remindingthat Ep = c/�p, we see that λ = 1/(1 − E2/E2

p). So, 〈m〉 can converge rapidly tothe Einstein value m, even at short times, provided that the particle’s energy E beclose to the Planck energy Ep. The correlation distance is now given by ∼ 1/(mcλ),and since λ > 1, then 1/(mcλ) < 1/(mc) always.

4. Conclusions

In this paper we have shown that both SR and DSR systems can arise by statisti-cally coarse-graining underlying non-relativistic (Wiener) process, making the lattermore fundamental and the former in some sense emergent. The coarse-graining canbe viewed as arising from superposition of two stochastic processes. On a shortspatial scale (much shorter than particle’s Compton wavelength) the particle movesaccording to a Brownian, non-relativistic, motion. Its Newtonian mass fluctuatesaccording to an inverse Gaussian distribution. On a time scale much larger than theCompton time, the particle then behaves as a relativistic particle with a sharp massequal to Einstein’s (i.e., Lorentz invariant) mass. In this case the particle moves witha net velocity which is less than c. The presented concept of statistical emergence,which is shared both by SR and DSR, can offer a new valid insight into the Planck-scale structure of space-time. The existence of a discrete polycrystalline substratemight be welcomed in various quantum gravity constructions. In fact, it has beenspeculated for long time that quantum gravity may lead to a discrete structure ofspace and time which can cure classical singularities.25 As for future work, a deeperunderstanding of a dynamical origin of our smearing would be highly desirable. In

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fact, the exact LS is due to a very special form of the Newtonian-mass distribution.The exact LS has no fundamental significance in our model, but it is controlled bya specific form of the mass distribution. We have seen that a small departure fromits shape brings to a departure from LS and leads, e.g., to DSR. Finally, it is alsohoped that the essence of our results will continue to hold in curved spacetimes.This could be an important step in addressing the issue of quantum gravity.

Acknowledgments

For their useful comments, we are grateful to S. Mignemi, H. Kleinert, C. Schubert,F. Bastianelli, and L.S. Schulman. The figure is due to the art of M. Nespoli.

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statistical origin of special and doubly special relativity

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