Summary
To overcome difficulties in the conventional relativistic generalization of statistical mechanics, we formulate four-dimensional statistical mechanics on the basis of common relativity discussed in previous papers. The new fourth dimension, «lightime»w=bt, in common relativity consists of a function «ligh»b and a scalar common timet, which allow for the concept of canonical evolution, the invariant 6N-dimensional phase space and the invariant volumeV I of a system of relativistic particles. Based on a Poincaré-invariant Hamiltonian dynamics with constraints and common time, we derive an invariant Liouville equation. Common relativity also gives a new invariant «genergy»G, closely related to the energy of a particle. This quantityG enables us to have a generalized Maxwell-Boltzmann distribution, entropy and temperature, which are all invariant under the four-dimensional lightime-space transformation of common relativity.
Riassunto
Per superare le difficoltà nella generalizzazione relativistica convenzionale, si formula una meccanica statistica quadridimensionale sulla base della relatività comune discussa in lavori precedenti. La nuova quarta dimensiona «lightime»w=bt, nella relatività comune, consiste di una funzione «ligh»b e di un tempo scalare comunet, che tengono conto del concetto di evoluzione canonica, dello spazio delle fasi invariante a 6N dimensioni e del volume invarianteV I di un sistema di particelle relativistiche. Basandoci su una dinamica hamiltoniana invariante secondo Poincaré con vincoli e tempo comune, si deriva un’equazione di Liouville invariante. La relatività comune dà anche una nuova «genergia» invarianteG, strettamente legata all’energia di una particella. Questa quantitàG ci permette di avere una distribuzione di Maxwell-Boltzmann generalizata, entropia e temperatura, che sono tutte invarianti secondo la trasformazione quadridimensionale lightime-spazio della relatività comune.
Резюме
С целью преодолеть трудности общепринятого релятивистского обобщения статистической механики, мы формулируем четырех-мерную статистическую механику на основе общей теории относительности, которая обсуждалась в предыдущих работах. Новое четвертое измерениеw=bt в общей теории относительности состоит из функцииb и скалярного общего времениt, которые учитывают концепцию канонической эволюции, инвариантное 6N-мерное фазовое пространство и инвариантный объемV I системы релятивистских частиц. На основе Пуанкаре-инвариантной Гамильтоновой динамики с ограничениями и общим временем, мы выводим инвариантное уравнение Лиувилля. Общая теория относительности также дает новую инвариантную функциюG, тесно связанную с энергией частицы. Эта величина дает обобщенное распределение Максвелла-Больцмана для энтропии и температуры, которые являются инвариантными относительно четырех-мерных преобразований пространства иw=bt в общей теории относительности.
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References
For comprehensive discussions of these problems, seeR. Hakim:J. Math. Phys. (N. Y.),8, 1315 (1967);J. L. Synge:The Relativistic Gas (New York, N. Y., 1957).
J. P. Hsu:Nuovo Cimento B,74, 67 (1983);Phys. Lett. A,97, 137 (1983);J. P. Hsu andT. N. Sherry:Found. Phys.,10, 57 (1980). See alsoNature editorial (Nature (London),303, 129 (1983)) for a discussion of common time, etc. So far common relativity and special relativity cannot be distinguished by known experiments such as the time dilatation of unstable particles, etc. Nevertheless, special relativity appears to be too restrictive conceptually and inherently suffered from profound difficulties of divergence in quantum field theory, and yet common relativity provides a more general conceptual framework which allows a new principle of universal probability for quantized field oscillators to overcome divergence difficulties in all local field theories. SeeJ. P. Hsu:Nuovo Cimento B,78, 85 (1983).
Since the choice of the frameF is arbitrary and no privileged frame is assumed, the common time is not unique and hence not absolute in the sense of Newton. See ref. (2).J. P. Hsu:Nuovo Cimento B,74, 67 (1983).
In fact, one may not bother to set up theF′-clocks. One can simply require that all observers in different frames use the same clock system (set up in the particular frameF) to record time. This is what actually happens in daily life: the observers in a train use the nearby clocks on the ground to record time, so that the observers on the ground and in the train have a common time.
Equation (5) is equivalent to the law of motion of a free particlep μ2 −m 2=0,i.e. (dx μ/ds)2=1.
This resembles Newtonian physics and is in sharp contrast with special relativity.
L. D. Landau andE. M. Lifshitz:Classical Theory of Fields (New York, N. Y., 1951), p. 31.
C. Møller:The Theory of Relativity (London, 1962), p. 117.
See, for example,J. P. Hsu:Nuovo Cimento B,75, 185 (1983). The «action» function ∝m ds does not contain the nonuniversal speed of light explicitly. Such an action can be easily generalized to more complicated situations and gives a simple way to find out truly universal constants. For a discussion of a four-dimensional Hamiltonian dynamics with a scalar evolution variable, seeJ. P. Hsu andT. Y. Shi:Phys. Rev. D,26, 2745 (1982).
See, for example,D. C. Montgomery andD. A. Tidman:Plasma Kinetic Theory (New York, N. Y., 1964), p. 85.
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The work is supported in part by southeastern Massachusetts University.
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Hsu, J.P. Four-dimensional symmetry from a broad viewpoint. Nuov Cim B 80, 201–216 (1984). https://doi.org/10.1007/BF02722259
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DOI: https://doi.org/10.1007/BF02722259