Is it valid to pull out the -\nabla^2 \phi(x') in this commutator calculation?

In summary, the conversation discusses the calculation of a commutator and the question of whether it is valid to pull out a certain term in order to get the correct answer.
  • #1
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Homework Statement



When calculating this commutator,

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

I almost get the right answer, but not sure if this is valid, or if there is an identity

The Attempt at a Solution



when I get to this point

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

I need to take out [tex] -\nabla^2 \phi(x') [/tex] to form

[tex] (-\nabla^2 \phi(x')) [\pi(x), \phi(x')] [/tex]

that way the rest would follow and give me the correct answer which is

[tex] -i(-\nabla^2 + m^2)\phi(x')) [/tex]
 
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  • #2
[\pi(x), \phi(x')] but I am not sure if this is an identity, or if it is valid to pull out the -\nabla^2 \phi(x') any help would be appreciated
 

FAQ: Is it valid to pull out the -\nabla^2 \phi(x') in this commutator calculation?

What is a commutator in Quantum Field Theory (QFT)?

A commutator in QFT is an operator that measures the difference between the results of two different measurements in quantum mechanics. It is a fundamental concept in QFT and is used to describe the uncertainty principle.

How does a complicated commutator affect QFT calculations?

A complicated commutator can make QFT calculations more challenging and time-consuming. This is because it involves complex mathematical operations and requires a deep understanding of the underlying principles of QFT.

What are some common applications of complicated commutator in QFT?

Complicated commutators are commonly used in QFT to describe the behavior of quantum fields, such as the electromagnetic field or the Higgs field. They are also used to calculate scattering amplitudes and to study the properties of particles in quantum systems.

How do you simplify a complicated commutator in QFT?

Simplifying a complicated commutator in QFT can be achieved through various methods such as using symmetries, approximations, or mathematical transformations. Each approach depends on the specific problem at hand and requires a deep understanding of QFT principles and techniques.

How does a complicated commutator relate to the fundamental principles of QFT?

The concept of commutator is closely related to the fundamental principles of QFT, such as the uncertainty principle and the concept of observables. It is a crucial tool for understanding the behavior of quantum fields and plays a central role in many QFT calculations and theories.

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