Trace Formalism for Quantum Mechanical Expectation Values

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Abstract
The trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis‐invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A. An expectation‐value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bounded operators, these include positivity, normalization, additivity, and a condition which may be considered as a requirement of regularity for the probability function induced by M. Modifications of these requirements are imposed for unbounded operators, and von Neumann's statistical formula is proved: there is a unique bounded positive operator W with unit trace for which M(A) = Tr(WA). The requirements placed on M are weaker than those of von Neumann, and are in fact satisfied by Tr(WA) as a function of A.

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