How to Expand Noncommuting Variables in a Formal Power Series?

In summary, to prove that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}, we need to expand f(a,a^\dagger) in a formal power series by using the fact that [a,a^\dagger]=1. This can be done by using induction on the power of a^\dagger and showing that the desired formula holds for each individual term. Once this is understood, the proof can be extended to more general functions of a and a^\dagger.
  • #1
fuchini
11
0

Homework Statement


Need to show that [tex][a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger}[/tex]

Homework Equations


[tex][a,a^\dagger]=1[/tex]

The Attempt at a Solution


Need to expand [tex]f(a,a^\dagger)[/tex] in a formal power series. However I don´t know how to do it if the variables don´t commute.
 
Last edited:
Physics news on Phys.org
  • #2
##f(a,a^+)= b_0+ b_1 a+b_2 a^+ b_3 a^2 +b_4 aa^+ +b_5 a^+a+b_6 (a^+)^2+\ldots##
So the general term is some product of a and ##a^+## in arbitrary order.
 
  • #3
Thanks for answering, but how would it be in terms of derivatives? Normally It would be:

[tex]f=\sum_{m,n} \frac{a^m a^{\dagger m}}{n!m!}\frac{\partial^{n+m} f}{\partial a^n \partial a^{\dagger m}}[/tex]

But in this case I guess I have to take into account that they're noncommuting.
 
  • #4
Start with a simpler case, i.e., ##f = (a^\dagger)^n## only. Use induction (on ##n##) to show that the desired formula holds. Once you understand the induction method for this problem, you'll probably work it out for more general ##f## more easily.
 
  • #5
Thanks, that did the trick!
 

What is a noncommuting series expansion?

A noncommuting series expansion is a mathematical technique used to express a function in terms of a series of noncommuting operators. This means that the operators do not commute, or do not have the same result regardless of the order in which they are applied. This technique is commonly used in quantum mechanics and other fields of physics.

What is the significance of noncommuting series expansion in quantum mechanics?

In quantum mechanics, noncommuting series expansion is used to express observables, such as position and momentum, in terms of noncommuting operators. This allows for a more accurate and precise mathematical description of the behavior of quantum systems.

How is a noncommuting series expansion different from a regular power series expansion?

A regular power series expansion involves only commuting operators, where the order of operators does not affect the result. However, a noncommuting series expansion involves noncommuting operators, which requires a different approach and mathematical techniques.

What are some applications of noncommuting series expansion?

Noncommuting series expansion is commonly used in quantum mechanics, but it also has applications in other fields such as statistical mechanics, solid state physics, and differential geometry. It is also used in the study of non-Abelian gauge theories and supersymmetry.

What are the limitations of noncommuting series expansion?

Noncommuting series expansion is a powerful mathematical tool, but it has limitations. It can only be used for systems with a finite number of degrees of freedom and it may not always converge for certain systems. Additionally, the calculations involved can become computationally intensive for complex systems.

Similar threads

  • Advanced Physics Homework Help
Replies
3
Views
761
  • Advanced Physics Homework Help
Replies
8
Views
4K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
8
Views
3K
  • Advanced Physics Homework Help
Replies
7
Views
1K
Back
Top