Question on discrete commutation relation in QFT

In summary, the given commutation relation states that the Fourier transforms of the operators ##\phi(t,\vec{x})## and ##\pi(t,\vec{x})## are equal to each other. This can be written as ##\left[\tilde{\phi}(\vec{k}),\tilde{\pi}(\vec{k}')\right]=i\delta_{\vec{k},-\vec{k}'}=i\delta_{-\vec{k},\vec{k}'}##, where the Kronecker deltas are interchangeable. It is important to distinguish between the operators ##\phi(t,\vec{x})## and ##\tilde{\phi}(\vec{k})## to avoid confusion.
  • #1
user1139
72
8
Homework Statement
The statement is in title
Relevant Equations
The equations are given below
Given the commutation relation

$$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$

and define the Fourier transform as

$$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\phi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

$$\tilde{\pi}(t,\vec{k})=\frac{1}{\sqrt{L^{n-1}}}\int_{\,0}^{\,L}\pi(t,\vec{x})e^{-i\vec{k}\cdot\vec{x}}\,\mathrm{d}^{n-1}\vec{x}$$

Is it then correct to say the following?

$$\left[\tilde{\phi}(\vec{k}),\tilde{\pi}(\vec{k}')\right]=i\delta_{\vec{k},-\vec{k}'}=i\delta_{-\vec{k},\vec{k}'}$$

i.e. can I use both Kronecker deltas interchangeably?
 
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  • #2
Have you tried to derive the commutator? You also should destinguish the operators ##\phi(t,\vec{x})## from ##\tilde{\phi}(\vec{k})##, because otherwise it leads to confusion. If so, where is your problem. Of course ##\delta_{\vec{k},-\vec{k}'}=\delta_{-\vec{k},\vec{k}'}##.
 

FAQ: Question on discrete commutation relation in QFT

What is a discrete commutation relation in QFT?

A discrete commutation relation in quantum field theory (QFT) is a mathematical expression that describes the relationship between two operators in a quantum system. It specifies how these operators behave when they are applied to a state in the system, and it is an essential tool for studying the dynamics of quantum systems.

How is a discrete commutation relation different from a continuous commutation relation?

A discrete commutation relation is different from a continuous commutation relation in that it applies to operators that act on discrete states, while a continuous commutation relation applies to operators that act on continuous states. In QFT, operators that act on discrete states are often used to describe particles, while operators that act on continuous states are used to describe fields.

What is the significance of discrete commutation relations in QFT?

Discrete commutation relations play a crucial role in QFT as they allow us to calculate the probabilities of different outcomes of measurements in a quantum system. They also help us understand the behavior of particles and fields in the quantum world and make predictions about their interactions.

How are discrete commutation relations derived in QFT?

Discrete commutation relations are derived using the principles of quantum mechanics and the mathematical framework of QFT. They are based on the fundamental commutation relations between the position and momentum operators, and they can be derived using techniques such as canonical quantization and path integrals.

Can discrete commutation relations be violated in QFT?

No, discrete commutation relations cannot be violated in QFT. They are fundamental laws of quantum mechanics and are supported by experimental evidence. Any violation of these relations would indicate a breakdown of the principles of QFT and would require a new theoretical framework to explain the behavior of quantum systems.

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