# QFT Commutator for spacelike separation

• PeroK
In summary, the conversation discusses the problem of showing that the expression $$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) - \exp(-ip.y))$$ is zero for spacelike ##y##. The speaker suggests that the Lorentz invariance of the integral can be used to simplify the problem by choosing coordinates where ##x## and ##y## are simultaneous, leading to the conclusion that the expression is a function of ##y^2## only.
PeroK
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Homework Statement
Show that the commutator for field operators is zero for spacelike separations
Relevant Equations
$$\hat \phi(x) = \int \frac{d^3p}{(2\pi)^{3/2}(2E_p)^{1/2}}(\hat{a}_{\vec p}exp(-ip\cdot x)+\hat{a}^{\dagger}_{\vec p}exp(ip\cdot x))$$
I got as far as:
$$[\hat \phi(x), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(-ip.(x-y) - \exp(-ip.(y-x))$$
Then I simplified the problem by taking one of the four-vectors to be the origin:
$$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) - \exp(-ip.y))$$
We need to show that this is zero for spacelike ##y##. But, I can't see how the spacelike condition is relevant. The hint in the book is that we can change ##-y## to ##y## in the second term. If ##p## was a three-vector I can see this, but the zeroth component of ##p## is the positive ##E_p##. And, if it's a simple ##p## substitution, then that should work for any ##y##, spacelike or otherwise.

Any help would be appreciated.

Last edited:
JD_PM
I might have a solution. Assuming this whole theory is Lorentz invariant, then we could choose coordinates so that ##x## and ##y## are simultaneous. The zeroth term disappears from the inner product and the result follows.

I guess that's the trick.

Hint: You don't have to assume that the theory is Lorentz invariant, because the integral for sure is, because ##\mathrm{d}^3 p/E_p## is invariant, and ##y \cdot p## is too. That's why the integral is a scalar field operator. Now you can choose ##y## (given that it is space) in the most simple way to evaluate this function. Note that since you only have the four-vector ##y## available it's clear that the expression must be a function of ##y^2=y \cdot y## only!

## 1. What is a QFT commutator for spacelike separation?

A QFT commutator for spacelike separation is an operator used in quantum field theory to describe the relationship between two observables that are separated by a spatial distance, meaning they are not in direct contact with each other. It is a mathematical expression that helps to determine the commutation relations between these observables.

## 2. Why is the QFT commutator important in quantum field theory?

The QFT commutator is important because it helps to understand the fundamental principles of quantum mechanics, such as the uncertainty principle and the concept of non-locality. It also plays a crucial role in determining the dynamics and symmetries of quantum field theories, and is used in various calculations and predictions.

## 3. How is the QFT commutator different from the classical commutator?

The QFT commutator differs from the classical commutator in that it takes into account the principles of quantum mechanics, such as the uncertainty principle and the concept of non-locality. It also involves operators that are described by quantum fields, rather than classical variables. Additionally, the QFT commutator is defined using the Heisenberg picture, whereas the classical commutator is defined using the Schrödinger picture.

## 4. Can the QFT commutator for spacelike separation be measured?

No, the QFT commutator for spacelike separation cannot be directly measured. It is a mathematical expression that helps to describe the relationship between observables in quantum field theory. However, the results of its calculations can be compared to experimental data, providing a way to test the predictions of the theory.

## 5. Are there any applications of the QFT commutator for spacelike separation?

Yes, the QFT commutator for spacelike separation has various applications in theoretical physics and quantum field theory. It is used to calculate scattering amplitudes, determine the symmetries and dynamics of quantum field theories, and investigate the properties of quantum systems. It is also used in quantum information theory and quantum computing.

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