Solving Perplexing Commutator for Simplification

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In summary, when simplifying the given integral, we use the known commutator relations to factor out the laplacian operator. By expanding the commutator and using integration by parts, we can simplify the integral to the desired form. This approach is similar to using functional derivatives in quantum mechanics.
  • #1
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When simplifying this

[tex] \int d^3x' [\pi(x), \frac{1}{2}\pi^2(x') + \frac{1}{2} \phi(x')( -\nabla^2 + m^2)\phi(x')] [/tex]

we know that

[tex] [\pi(x), \pi(x')] = 0 [/tex]

[tex] [\phi(x), \pi(x')] = -i\delta(x-x') [/tex]

how does that simplify to

[tex] \int d^3x' \delta(x-x')( -\nabla^2 + m^2)\phi(x')[/tex]

I know that

[tex] [\pi(x), \pi^2(x')] = 0 [/tex]

but not sure how does the laplacian gets factored out like that and one-half disappears?
 
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  • #2
Have you tried to see what part integration brings you ?
 
  • #3
By expanding the commutator we get,

[tex]
\frac{1}{2} \int d^3x' \pi(x)\phi(x')( -\nabla^2 + m^2)\phi(x') - \phi(x')( -\nabla^2 + m^2)\phi(x')\pi(x)
[/tex]

the laplacian in the second term is sandwiched between two phis, integration by part doesn't seem to help to factor it out.
 
  • #4
I think you're better off using [tex][A,BC] = [A,B]C + B[A,C][/tex] and then integration by parts. Also, it's a good time in your life to realize that a commutator behaves very much like a functional derivative (recall QM 101).
 

What is a commutator?

A commutator is a mathematical tool used to simplify complex algebraic expressions. It involves rearranging the terms in an expression in a specific way to make it easier to solve.

Why is it important to solve perplexing commutators for simplification?

Solving perplexing commutators is important because it allows us to simplify complex expressions and make them easier to work with. This can help us find solutions to problems and understand mathematical concepts more clearly.

What are some common techniques for solving perplexing commutators?

Some common techniques for solving perplexing commutators include using the commutative property to rearrange terms, using the distributive property to simplify expressions, and combining like terms.

How do I know when to use a commutator?

A commutator is typically used when you have a complex expression with multiple terms and you need to simplify it. It can also be used when you need to rearrange terms in a specific way to make the expression easier to solve.

Can commutators be used in other areas of science?

Yes, commutators can be used in various areas of science, including physics, chemistry, and engineering. They are a fundamental tool in mathematics and can be applied to many different scenarios and problems.

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