Calculating Determinants of Operators: A Step-by-Step Guide

In summary, to calculate the determinant of \frac{\partial}{\partial t} \delta (t-\tau), it is recommended to use the discretized version of this operator. The source of this determinant is a path integral, and it is possible that it can be absorbed into normalization depending on the calculation being performed. The value of the determinant also depends on the boundary conditions being used. For more detailed calculations and information, H. Kleinert's book "Path integrals in quantum mechanics, Statistics, Polymer Physics and Financial markets" is a useful resource.
  • #1
jarowit
4
0
how to calculate determinant of [tex]\frac{\partial}{\partial t} \delta (t-\tau)[/tex]?
 
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  • #2
Maybe this came from a path integral? I think you should use the discretized version of this operator.
 
  • #3
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 ??
 
  • #4
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
 

What is the purpose of calculating determinants of operators?

The purpose of calculating determinants of operators is to determine the properties and behavior of linear operators in a given vector space. It can also be used to solve systems of linear equations and understand the transformations performed by the operators.

What are the steps involved in calculating determinants of operators?

The steps involved in calculating determinants of operators include finding the matrix representation of the operator, performing row operations to reduce the matrix to its upper triangular form, and then multiplying the diagonal elements to get the determinant value.

How do I know if a matrix is invertible?

A matrix is invertible if its determinant is non-zero. If the determinant is zero, the matrix is not invertible and the operator does not have an inverse.

Can I calculate determinants of operators for non-square matrices?

No, determinants of operators can only be calculated for square matrices. This is because non-square matrices do not have the necessary properties to determine an operator's behavior.

Can the determinant of an operator be negative?

Yes, the determinant of an operator can be negative. The sign of the determinant depends on the orientation of the vector space and the order in which the vectors are listed.

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