Symmetry of Physical Laws Part I. Symmetry in Space-Time and Balance Theorems

Abstract

In default of the theorem of "detailed balance": ${P}_{\mathrm{ij}}={P}_{\mathrm{ji}}$, with regard to elementary transition probabilities, several "balance" theorems are introduced and proved on the basis of symmetry of physical laws in space-time. (1) First theorem of "averaged balance" (Sec. 5): We can establish ${\overline{P}}_{\mathrm{ij}}={\overline{P}}_{\mathrm{ji}}$ by averaging over quantities of "minus class." Table V (Sec. 3) gives a list of "minus" quantities. (2) The so-called "detailed balance of collisions" in classical physics is a special case of Theorem I. (3) Heitler-Coester's theorem of "semidetailed balance" is also a special case of Theorem I. (4) Second theorem of "averaged balance" (Sec. 5): We can establish ${\overline{P}}_{\mathrm{ij}}={\overline{P}}_{\mathrm{ji}}$ by averaging over quantities with ${\ensuremath{\rho}}_{R}=\ensuremath{-}1$. The quantities with ${\ensuremath{\rho}}_{R}=\ensuremath{-}1$ are listed in Table II (Sec. 2). (5) Theorem of "cyclic balance" (Sec. 7): In classical physics, a chain of transitions $i\ensuremath{\rightarrow}j\ensuremath{\rightarrow}k\ensuremath{\rightarrow}\ensuremath{\cdots}\ensuremath{\rightarrow}i$ repeats itself cyclically. (6) Theorem of "long-range balance" (Sec. 7): The time average of transition probability from $i$ to $j$ is equal to the time average of transition probability from $j$ to $i$. Theorems I, II, and III, are direct consequences of inversibility (covariance for space-and-time inversion). Theorem IV is a consequence of reversibility (covariance for time reversal). Theorems V and VI are connected with ergodicity of Markoff's chains. This ergodicity is proved by the condition of bilateral normalization of transition probabilities: $\ensuremath{\Sigma}{j}^{}{P}_{\mathrm{ij}}=1$, $\ensuremath{\Sigma}{i}^{}{P}_{\mathrm{ij}}=1$. This bilateral normalization in turn can be derived from either reversibility or inversibility. The limits of validity of all these balance theorems in actual applications are carefully examined in the text.