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

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The discussion centers on the application of trace cyclicity in the context of operators on Hilbert spaces, specifically addressing the conditions under which the cyclicity formula for the trace can be applied. The participants clarify that while trace cyclicity is valid for finite-dimensional Hilbert spaces with bounded operators, the generalization to infinite-dimensional spaces requires that the products of operators, such as AB and BA, are trace class. The mention of quantum field theory (QFT) highlights the relevance of these concepts in advanced physics applications.

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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.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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