On lorentz invariance of random processes

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On Lorentz invariance of random processes

Soutchilin V. PhD, Chief Technology

Filderstadt (Germany)

Annotation

The condition of Lorentz invariance of random processes is formulated, which consists in the fact that these processes should be self-similar. It is shown that white and pink noise, as well as the Brownian process, correspond to this condition. Thus, not limited to these examples, the list of relativistic invariants can be supplemented any random process with the property of self-similarity

Keywords: Brownian process, Lorentz invariance, pink noise, random processes, self-similar process, self-similarity, white noise

Сучилин В.А. К вопросу о Лоренц-инвариантности случайных процессов

Аннотация. Сформулировано условие Лоренц-инвариантности случайных процессов, которое состоит в том, что эти процессы должны быть самоподобными. Показано, что этому условию соответствуют белый и розовый шум, а также броуновский процесс. Таким образом, не ограничиваясь этими примерами, перечень релятивистских инвариантов может быть дополнен любым случайным процессом, обладающим свойством самоподобия.

Ключевые слова: белый шум, броуновский процесс, Лоренц-инвариантность, розовый шум, самоподобие, самоподобный процесс, случайные процессы

Introduction

The Lorentz invariance refers to the property of mathematical relations for physical processes or functions to retain their form by transition from one inertial reference frame (IRF) to another [1]. Without loss of generality, it can be assumed that all IRF are collinear. Therefore, in this article will be used Lorentz transformation of the kind [2]:

ф` = лф (1)

л= where:

ф, ф` - the time intervals in the fixed and moving IRF`s, respectively

v - IRF relative speed

c - light speed in vacuum

The Lorentz invariance property referring to a random process was already considered in [3]. In this paper, it will be generalized to the class of self-similar random processes (hereinafter, SSR processes) [4].

Self-Similar Random Processes

Consider a nontrivial random process defined by the S (ф). This process is self-similar under condition, that for a given numbers g >1 (scale factor) and h (characteristic index) the relation

S (ф) = g^h·(gф) (2)

is satisfied [4].

Relation (2) expresses the property of the SSR process, according to which this process reproduces itself at any selected time interval to be extended with scale factor g.

It will be shown that for any specified SSR process the corresponding value of h can be determined.

Lorentz Invariance and Self-Similar Random Processes

In the framework of the mathematical representation, the condition of Lorentz invariance for the random process (ф) is of the kind:

S (ф) = µ·(ф`) (3)

where µ is a unique real number.

Consider relation (3) in terms of autocorrelation. Then equality in (3) formally expresses the fact that autocorrelation function of the random process retains its form by transition to the moving IRF. In this regard, such a random process will be interpreted as Lorentz-invariant [1].

At the same time, if (ф) has been represented as SSR process, taking into account (1) relation (3) can be rewritten in the form:

S (ф) = µ·(лф) (4)

Comparing (2) and (4) one can obtain:

S (ф) = л^h·(лф) (5)

The relation above is identical with (2) up to substitution л = g. Thus, in order the random process (ф) to be Lorentz-invariant, it must be an SSR process.

Fig. 1. Lorentz invariance of SSR process with ф and ф` set by mean of Lorentz transformation

Instances of Lorentz-Invariant SSR processes

lorentz invariance self-similarity random process

By the instances of the random processes considered below, these are considered by use of autocorrelation functions. Moreover, in regard to the Lorentz relation (1) below will be always assumed g = л.

1. White noise process.

This process has the autocorrelation function of the kind [5]:

S_w(ф) = д(ф) (6)

where д(ф) is Dirac function.

Then, taking into account (1), one can obtain:

S_w(ф`) = д(лф) = д(ф ) (7)

Thus, the autocorrelation function of the white noise retains obviously its form, and it could be considered:

w(ф) х л w(лф) = w(лф) (8)

Therefore, according to (3) the white noise as SSR process is Lorentz-invariant with characteristic index h = 0.

2. Pink noise process.

This process is characterized by the constant spectral density (without loss of generality, it will be taken equal to 1) at -d ? щ ? d , and zero like spectral density in the rest of the frequency range. The autocorrelation function of the process (using Fourier integral) is of the kind [5]:

S_p (ф) = (9)

Then in the moving IRF taking (1) into account it could be considered:

S_p (ф`) = S (10)

where: ? = лщ and ? = d/л.

Thus, the autocorrelation function of the pink noise retains obviously its form, and it could be considered:

S_p(ф) = л^-1·S_p(лф) (11)

Therefore, according to (3) the pink noise as SSR process is Lorentz-invariant with characteristic index h = -1.

3. Brownian process.

Brownian process is well-known random Wiener process with self-similarity (thus, SSR process) which in terms of relation (2) is of the kind [6]:

S_b(ф) = g^-Ѕ·S_b(gф) (12)

Here, in regard to (1) we assume g = л and then obtain:

S_b(ф`) = л^-Ѕ·S_b(лф) (13)

Therefore, in regard to (2):

S_b(ф) = µ·S_b(ф`) (14)

where µ = л^-Ѕ.

Thus, in view of (3) one can consider the Brownian process as Lorentz-invariant with characteristic index h = -Ѕ.

Conclusion

Above, the Lorentz invariance condition of random processes to be self-similar is formulated. It is shown that the random processes, such as white and pink noise as well as Brownian process, correspond to this condition. Thus, the physical invariants of the special theory of relativity can be supplemented by the considered representatives of the random processes. At the same time, not limited to these, any process which corresponds to the Lorentz invariance condition considered here may be the subject of further reviews.

References

1. Dermisek R. Lorentz invariance // [Электронный ресурс].

2. Forshaw J., Smith G. Dynamics and relativity. John Wiley & Sons: 2014, 344 p.

3. Сучилин В.А. Автокорреляция сигналов в инерциальных системах отсчета // Современные научные исследования и инновации. 2018. №5 [Электронный ресурс]

4. Pardo J.C. A brief introduction to self-similar processes. [Электронный ресурс].

5. Miller S., Childers D. Probability and Random Processes. Epub eBook: 2012, 522 p.

6. Karatzas I., Shreve S.E. Brownian Motion and Random Calculus (Graduate Texts in Mathematics). Springer, New York: 1997, 496 p.

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