Why are we allowed to use the trace cyclicity here?

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In summary, the conversation is discussing the use of trace cyclicity in calculating the trace of a product of operators. The article being read explores this concept in the context of finite matrices, but the question is raised about its applicability to infinite dimensional Hilbert spaces. The presence of exponential operators in the trace is also mentioned. The main question is whether there are any conditions on the operators for trace cyclicity to hold.
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Heidi
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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.
 
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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.
 

1. Why is trace cyclicity important in scientific research?

Trace cyclicity is important in scientific research because it allows us to understand the cyclical patterns and processes that occur in nature. By analyzing the trace elements in different samples, we can track changes over time and make predictions about future events.

2. How does trace cyclicity help us understand climate change?

Trace cyclicity is a key tool in understanding climate change because it provides evidence of past climate variations. By studying the trace elements in sediment or ice cores, scientists can reconstruct past climate conditions and compare them to current conditions to better understand the causes and impacts of climate change.

3. What techniques are used to analyze trace cyclicity?

There are several techniques used to analyze trace cyclicity, including X-ray fluorescence, mass spectrometry, and electron microscopy. These techniques allow scientists to identify and quantify trace elements present in samples, which can then be used to track changes over time.

4. How does trace cyclicity relate to the study of Earth's history?

Trace cyclicity is closely related to the study of Earth's history because it provides a record of past events and processes. By analyzing trace elements in sedimentary rocks, for example, scientists can reconstruct the history of Earth's climate, ocean chemistry, and even the evolution of life on our planet.

5. Can trace cyclicity be used to make predictions about future events?

Yes, trace cyclicity can be used to make predictions about future events. By understanding the cyclical patterns and processes in nature, scientists can make educated guesses about how these patterns may continue or change in the future. However, it is important to note that these predictions are not guaranteed and are subject to change as new data and research become available.

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