By Alexander Bach

During this e-book the concept that of indistinguishability is outlined for exact debris by way of the symmetry of the nation. It applies, for this reason, to either the classical and the quantum framework. the writer describes symmetric statistical operators and classifies those via severe issues. For the outline of infinitely extendible interchangeable random variables de Finetti's theorem is derived and generalizations overlaying the Poisson restrict and the relevant restrict are offered. A characterization and interpretation of the crucial representations of classical photon states in quantum optics are derived in abelian subalgebras. Unextendible indistinguishable debris are analyzed within the context of nonclassical photon states. The booklet addresses mathematical physicists and philosophers of technological know-how.

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Proof. It is sufficient to show that all n marginal probabilities P (XI < Xl, ... , X m - l < Xm-l, X m +l < Xm+l, ... , X n < x n ), 1 ::; m ::; n, agree and are symmetric. Then we can proceed by induction. Since P(X I < Xl,· .. , X m - l < Xm-l, X m +l < Xm+l,· .. s. is invariant under permutations. Using the same equation and permutation invariance shows that all marginal probabilities agree. 4. (a) Interchangeable random variables are identically distri" -buted. The converse statement does not hold.

X n ) E M~(Rn) is symmetric. 38 2. 1. Assume that W E S( Q9 1£) and suppose that the probabilities of the random variables J; : (Q, F, P) --+ {I, ... 85) then the random variables J;, 1 ::; i ::; n, are interchangeable iff W is symmetric. Proof. It is obvious from the definitions that the induced measure is symmetric iff P is symmetric. 4. Ey an occupation number we understand a vector k E {a, 1, ... , n}d that assigns to any one-particLe-state the number of particles in that state. This implies that k is subject to the constraint L: k; = n.

1. Assume that W E S( Q9 1£) and suppose that the probabilities of the random variables J; : (Q, F, P) --+ {I, ... 85) then the random variables J;, 1 ::; i ::; n, are interchangeable iff W is symmetric. Proof. It is obvious from the definitions that the induced measure is symmetric iff P is symmetric. 4. Ey an occupation number we understand a vector k E {a, 1, ... , n}d that assigns to any one-particLe-state the number of particles in that state. This implies that k is subject to the constraint L: k; = n.

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