Calculating Determinants of Operators

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    Determinant Operator
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Discussion Overview

The discussion revolves around the calculation of the determinant of the operator \(\frac{\partial}{\partial t} \delta (t-\tau)\). It explores theoretical aspects related to path integrals and their implications for normalization in various contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the method to calculate the determinant of the operator \(\frac{\partial}{\partial t} \delta (t-\tau)\).
  • Another participant suggests that the determinant may originate from a path integral and proposes using a discretized version of the operator.
  • A follow-up response agrees that the source of the determinant is indeed a path integral and questions whether the determinant could be a constant that might be absorbed by normalization.
  • Further, a participant elaborates that the possibility of absorption into normalization depends on the specific calculations being performed, particularly in relation to integrating out degrees of freedom for effective generating functionals or evaluating fluctuations around classical paths.
  • This participant also notes that the value of the determinant is influenced by boundary conditions, specifically mentioning periodic boundary conditions in the context of thermal field theory.
  • Lastly, a recommendation is made for H. Kleinert's work on path integrals as a useful resource for detailed calculations related to this topic.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the determinant's value and its treatment in normalization, indicating that multiple competing perspectives remain in the discussion.

Contextual Notes

The discussion does not resolve the assumptions regarding boundary conditions or the specific context in which the determinant is being calculated, leaving these aspects open for further exploration.

jarowit
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how to calculate determinant of [tex]\frac{\partial}{\partial t} \delta (t-\tau)[/tex]?
 
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Maybe this came from a path integral? I think you should use the discretized version of this operator.
 
weejee said:
Maybe this came from a path integral? I think you should use the discretized version of this operator.

exactly, the source of this determinant is a path integral

is it possible that this det is a constant which can be absorbed by normalization ??
 
Depending on what you are calculating it is possible that it can be included in normalization. Are you looking to integrate out certain degrees of freedom to get an effective generating functional to evaluate correlation functions then I would say yes, it can be absorbed into normalization. On the other hand if you are calculating fluctuations around some classical path then this factor becomes important. The value of the determinant also depends on your boundary conditions, are you using periodic boundary condtions (thermal field theory)?

I think you may find some useful calculations H. Kleinert "Path integrals in quantum mechanics, Statistics, Polymer Physics and Financial markets" It is a great source of detailed calculations of this kind of stuff.

You can download up to 5 chapters from his homepage:

http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=booklist
 

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