A Why are we allowed to use the trace cyclicity here?

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The discussion centers on the use of trace cyclicity for operators in quantum mechanics, particularly in relation to finite and infinite-dimensional Hilbert spaces. The original query questions whether trace cyclicity applies when dealing with products of operators, specifically in cases where the operators are bounded and the Hilbert space is finite-dimensional. It is noted that while trace cyclicity is valid for trace-class operators, the conditions for its application in infinite-dimensional spaces require that both products AB and BA be trace-class. The conversation highlights the relevance of this concept in quantum field theory, where operators like exp(-H) are commonly encountered. Understanding the conditions under which trace cyclicity holds is crucial for accurate calculations in these contexts.
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trace cyclicity with operators?
Hi Pf
i am reading this article: pillet.univ-tln.fr/data/pdf/KMS-states.pdf
I know that the trace cyclicity can be used when there is a product of matrices. But here we have operators (an hamiltonian , an operator which can be the position operators) . the author take the trace of a product. is this product trace class? are we allowed to use the cyclicity formula of the trace here?
thanks.
 
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The operators are bounded because the Hilbert space is finite-dimensional. So we're talking here about finite matrices.
 
yes but my question is more general. Consider 2 operators A and B on an infinite dimensional Hilbert space. If the products AB and BA are class trace it is meaningful to consider Tr(AB) and Tr(BA)
Are there conditions on these operators so that trace cyclicity is true.
Qft is on infinite dimensional hilbert spaces and exp(-H) often appears as one of the operators in the trace.
 

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