Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Identical Particles, Crossing, and CPT

Paul L. Csonka, Michael J. Moravcsik

Published May 15, 1972• 4 citations
Publisher

Abstract

It is shown that if crossing is a valid operation, and if identical quantum-mechanical particles are really indistinguishable, then the reaction $A+\overline{B}\ensuremath{\rightarrow}\overline{A}+B$ is $\mathrm{CPT}$-invariant. Consequently, (a) there is no model-independent test of $\mathrm{TCP}$ invariance which can be carried out on one single reaction; (b) there is no model-independent way at all to distinguish between $\mathrm{TCP}$ conservation and total $\mathrm{TCP}$ violation (except possibly certain interference experiments); (c) no "violation of statistics" can be expected in the reaction $A+A\ensuremath{\rightarrow}B+B$ if the reaction $A+\overline{B}\ensuremath{\rightarrow}\overline{A}+B$ is $\mathrm{CPT}$-invariant.

DOI
10.1103/physrevd.5.2546

Related papers

Peccate fortiter! or probing CPT invariance & novel tests of quantum mechanics in φ →

I. I. Bigi

A review of the experimental status of CPT invariance leads to two conclusions: (i) There is no hint of a CPT violation in the data. (ii) CPT invariance has not been probed with a quantitative sensitivity that greatly exceeds that achieved in studies of CP symmetry. I sketch some models where CPT violations could be implemented. Yet it is hard to see how those types of ansatz could generate observable effects in the transitions of neutral K mesons. On the other hand the quite special quantum aspects of the reaction φ → KL KS can be harnessed to probe quantum mechanics in novel ways by analyzing EPR like correlations of a kind that have not been studied before. Some comparisons are made with Weinberg's recent formulation of non-linear (though homogenous) quantum mechanics and with generalizations that can be obtained from Bohm's interpretation of quantum mechanics.

Nuclear Physics B - Proceedings Supplements•Aug 1, 1991•5 citations

Quantum Statistics of Identical Particles

J. C. Garrison

Abstract The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group.

arXiv:2207.13231•Foundations of Physics•Jul 14, 2022•1 citation

Precision Tests Find No Violation of Bose Statistics

Bertram M. Schwarzschild

The symmetrization postulate of quantum mechanics asserts that the multiparticle wavefunction for any collection of identical particles must be either wholly symmetric or wholly antisymmetric under the exchange of labels between any two particles. In nonrelativistic quantum mechanics, this postulate is a somewhat ad hoc assumption tacked onto the theory; wholly symmetric or antisymmetric wave functions are not the only ways of preserving the indistinguishability of identical particles.

Physics Today•Nov 1, 1996•2 citations

Identical particles

Gennaro Auletta, Mauro Fortunato, Giorgio Parisi A

In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.

Cambridge University Press eBooks•Apr 16, 2009•0 citations

Quantum Mechanics From Inertia Transforming Into Independent E and p Values? Part 2

Francesco R. Ruggeri

In Part 1, we argued that the Lorentz invariant -Et+px contains E and p which are physically different types of interaction variables. We argued that nature must somehow keep E and p independent in -Et+px (if nature also respects this Lorentz invariant). As a result, we suggested a math function exp(-iEt+ipx) which actually acts like a probability in terms of products and ensures conservation of E and p. Here, we wish to consider the situation from a different viewpoint. Like in Part 1, we do not postulate a priori any probability. Rather, we argue that there exist interactions in nature which conserve both E and p (elastic collisions) and that these two interaction variables are associated with different physical processes. For example, the notion of action-reaction is enough to see conservation of momentum, but conservation of energy is an independent process. In other words, E is like an enhanced inertia (mo and v), while p is like a moving inertia. An interaction which conserves both is complicated, yet we argue that the notion of the Lorentz invariant -Et+px should still hold for two interacting particles. For a single particle, the center-of-mass moves as x=vt and so one would expect that t and x in -Et+px are correlated. If one has two particles coming together to interact, then by conservation of E,p one has E1+E2 and p1+p2 (one dimension), but x and t for the two particles should be the same and it seems that one is not concerned about x=vt. Furthermore, a given outcome of the interaction E3+E4 and p3+p4 must also be linked to -Et+px and there must be conservation of total E and p. Thus: -(-E3-E4+E1+E2) t + (-p3-p4+p1+p2) x = 0. This suggests mathematically that t and x are independent variables in direct contrast to x=vt. Furthermore, the system must somehow ensure that x and t are independent for any Ei,Ej and pi,pk such that Ei+Ej=E1+E2 and pi+pj = p1+p2 as this is all the information that the Lorentz invariant -Et+px displays. Given that E and p are linked with different interactions, we suggest that there must exist an a priori property of a particle which allows for the above considerations. This would then explain how any Ei+Ej=E1+E2 and pi+pj = p1+p2 holds. In Part 1, we argued that this condition may occur if a particle is associated with fluctuating x and t values in terms of interactions given by exp(-iEt+ipx). The center-of-mass still moves as x=vt, but in an interaction it is the interaction process which is key, not the particle center-of-mass. One requires a description of such an interaction which can handle both E and p and is consistent with whatever information special relativity allows. We suggest that the exp(-iEt+ipx) is such a description, but that it carries the notion of periodicity in space and time and that this periodicity is then linked with E and p. This then becomes associated with Planck’s constant to link periodicity to E,p.

•Apr 8, 2026•0 citations

Symmetrization Postulate and Its Experimental Foundation

Albert Messiah, Oscar Wallace Greenberg

The symmetrization postulate (SP) that states of more than one identical particle are either symmetric or antisymmetric under permulations is studied from the theoretical and experimental points of view. The theoretical analysis is carried out within the framework of particle quantum mechanics; the field-theory approach to identical particles using Bose, Fermi, para-Bose and para-Fermi quantization is not considered in this article. Particles not obeying SP can be accommodated in quantum mechanics, provided some modifications are made in the usual quantum-mechanical formalism. The main modification is to replace the usual ray by a many-dimensional "generalized ray" as the representative of a physical state. The properties of one-body measurements of systems having several identical particles are discussed, and the unobservability in such measurements of interferences between states having different irreducible permutation symmetries is pointed out. The condition of indistinguishability of identical particles is formulated precisely, and is analyzed both for interactions which conserve particle number and for general interactions which do not. For such general interactions, with the additional assumptions of time-reversal invariance and of coherence of the states having given values of charge, baryon number, and lepton number, it is shown that there is an absolute selection rule forbidding transitions between states ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ which contain any number of particles of species which obey SP but at most one particle of a species not obeying SP, and states which violate SP. Since only states in ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ are now avaiable as initial states of experiments, this selection rule forbids production of SP-violating states in any experiment which is feasible at present. Because of this, presently proposed experimental tests of SP are in fact tests of the quantum-mechanical description of identical particles together with time-reversal invariance and coherence of states in a given superselecting sector. The inclusion of internal variables, such as isospin, for particles violating SP is discussed. A comprehensive discussion is given of direct experimental tests of the SP selection rules. Such tests are more difficult to perform than appears at first sight, because in many cases the indistinguishability of identical particles or the conservation laws already imply the consequences of SP. Criteria for valid tests of SP are given. Several different types of tests are described, with illustrative examples of each. A survey is given of the direct experimental evidence for SP for the various particles. The Fermi character of electrons and positrons and of nucleons is accepted, as is the Bose character of photons. There is good evidence for the Bose nature of pions, especially from the absence of $2\ensuremath{\pi}$ decay of ${{K}_{2}}^{0}$. There is no direct evidence for the statistics of $K$, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, $\ensuremath{\Xi}$, or $\ensuremath{\mu}$. Feasible tests are proposed for the statistics of $K$ and of those hyperons which have an asymmetric decay; but no such tests were found for the other hyperons or for $\ensuremath{\mu}$.

Physical Review•Oct 12, 1964•367 citations
It is shown that if crossing is a valid operation, and if identical quantum-mechanical particles are really indistinguishable, then the reaction A+B‾\ensuremath→A‾+BA+\overline{B}\ensuremath{\rightarrow}\overline{A}+BA+B\ensuremath→A+B is CPT\mathrm{CPT}CPT-invariant. Consequently, (a) there is no model-independent test of TCP\mathrm{TCP}TCP invariance which can be carried out on one single reaction; (b) there is no model-independent way at all to distinguish between TCP\mathrm{TCP}TCP conservation and total TCP\mathrm{TCP}TCP violation (except possibly certain interference experiments); (c) no "violation of statistics" can be expected in the reaction A+A\ensuremath→B+BA+A\ensuremath{\rightarrow}B+BA+A\ensuremath→B+B if the reaction A+B‾\ensuremath→A‾+BA+\overline{B}\ensuremath{\rightarrow}\overline{A}+BA+B\ensuremath→A+B is CPT\mathrm{CPT}CPT-invariant.
A review of the experimental status of CPT invariance leads to two conclusions: (i) There is no hint of a CPT violation in the data. (ii) CPT invariance has not been probed with a quantitative sensitivity that greatly exceeds that achieved in studies of CP symmetry. I sketch some models where CPT violations could be implemented. Yet it is hard to see how those types of ansatz could generate observable effects in the transitions of neutral K mesons. On the other hand the quite special quantum aspects of the reaction φ → KL KS can be harnessed to probe quantum mechanics in novel ways by analyzing EPR like correlations of a kind that have not been studied before. Some comparisons are made with Weinberg's recent formulation of non-linear (though homogenous) quantum mechanics and with generalizations that can be obtained from Bohm's interpretation of quantum mechanics.
Abstract The empirical rule that systems of identical particles always obey either Bose or Fermi statistics is customarily imposed on the theory by adding it to the axioms of nonrelativistic quantum mechanics, with the result that other statistical behaviors are excluded a priori. A more general approach is to ask what other many-particle statistics are consistent with the indistinguishability of identical particles. This strategy offers a way to discuss possible violations of the Pauli Exclusion Principle, and it leads to some interesting issues related to preparation of states and a superselection rule arising from invariance under the permutation group.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
In Part 1, we argued that the Lorentz invariant -Et+px contains E and p which are physically different types of interaction variables. We argued that nature must somehow keep E and p independent in -Et+px (if nature also respects this Lorentz invariant). As a result, we suggested a math function exp(-iEt+ipx) which actually acts like a probability in terms of products and ensures conservation of E and p. Here, we wish to consider the situation from a different viewpoint. Like in Part 1, we do not postulate a priori any probability. Rather, we argue that there exist interactions in nature which conserve both E and p (elastic collisions) and that these two interaction variables are associated with different physical processes. For example, the notion of action-reaction is enough to see conservation of momentum, but conservation of energy is an independent process. In other words, E is like an enhanced inertia (mo and v), while p is like a moving inertia. An interaction which conserves both is complicated, yet we argue that the notion of the Lorentz invariant -Et+px should still hold for two interacting particles. For a single particle, the center-of-mass moves as x=vt and so one would expect that t and x in -Et+px are correlated. If one has two particles coming together to interact, then by conservation of E,p one has E1+E2 and p1+p2 (one dimension), but x and t for the two particles should be the same and it seems that one is not concerned about x=vt. Furthermore, a given outcome of the interaction E3+E4 and p3+p4 must also be linked to -Et+px and there must be conservation of total E and p. Thus: -(-E3-E4+E1+E2) t + (-p3-p4+p1+p2) x = 0. This suggests mathematically that t and x are independent variables in direct contrast to x=vt. Furthermore, the system must somehow ensure that x and t are independent for any Ei,Ej and pi,pk such that Ei+Ej=E1+E2 and pi+pj = p1+p2 as this is all the information that the Lorentz invariant -Et+px displays. Given that E and p are linked with different interactions, we suggest that there must exist an a priori property of a particle which allows for the above considerations. This would then explain how any Ei+Ej=E1+E2 and pi+pj = p1+p2 holds. In Part 1, we argued that this condition may occur if a particle is associated with fluctuating x and t values in terms of interactions given by exp(-iEt+ipx). The center-of-mass still moves as x=vt, but in an interaction it is the interaction process which is key, not the particle center-of-mass. One requires a description of such an interaction which can handle both E and p and is consistent with whatever information special relativity allows. We suggest that the exp(-iEt+ipx) is such a description, but that it carries the notion of periodicity in space and time and that this periodicity is then linked with E and p. This then becomes associated with Planck’s constant to link periodicity to E,p.
The symmetrization postulate (SP) that states of more than one identical particle are either symmetric or antisymmetric under permulations is studied from the theoretical and experimental points of view. The theoretical analysis is carried out within the framework of particle quantum mechanics; the field-theory approach to identical particles using Bose, Fermi, para-Bose and para-Fermi quantization is not considered in this article. Particles not obeying SP can be accommodated in quantum mechanics, provided some modifications are made in the usual quantum-mechanical formalism. The main modification is to replace the usual ray by a many-dimensional "generalized ray" as the representative of a physical state. The properties of one-body measurements of systems having several identical particles are discussed, and the unobservability in such measurements of interferences between states having different irreducible permutation symmetries is pointed out. The condition of indistinguishability of identical particles is formulated precisely, and is analyzed both for interactions which conserve particle number and for general interactions which do not. For such general interactions, with the additional assumptions of time-reversal invariance and of coherence of the states having given values of charge, baryon number, and lepton number, it is shown that there is an absolute selection rule forbidding transitions between states ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ which contain any number of particles of species which obey SP but at most one particle of a species not obeying SP, and states which violate SP. Since only states in ${\mathcal{F}}^{\ifmmode\times\else\texttimes\fi{}}$ are now avaiable as initial states of experiments, this selection rule forbids production of SP-violating states in any experiment which is feasible at present. Because of this, presently proposed experimental tests of SP are in fact tests of the quantum-mechanical description of identical particles together with time-reversal invariance and coherence of states in a given superselecting sector. The inclusion of internal variables, such as isospin, for particles violating SP is discussed. A comprehensive discussion is given of direct experimental tests of the SP selection rules. Such tests are more difficult to perform than appears at first sight, because in many cases the indistinguishability of identical particles or the conservation laws already imply the consequences of SP. Criteria for valid tests of SP are given. Several different types of tests are described, with illustrative examples of each. A survey is given of the direct experimental evidence for SP for the various particles. The Fermi character of electrons and positrons and of nucleons is accepted, as is the Bose character of photons. There is good evidence for the Bose nature of pions, especially from the absence of $2\ensuremath{\pi}$ decay of K20{{K}_{2}}^{0}K2​0. There is no direct evidence for the statistics of KKK, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, $\ensuremath{\Xi}$, or $\ensuremath{\mu}$. Feasible tests are proposed for the statistics of KKK and of those hyperons which have an asymmetric decay; but no such tests were found for the other hyperons or for $\ensuremath{\mu}$.