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silence11
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If it's a dirac delta doesn't it mean it's infinite when x=y? Or is it a sort of kronecker where it's equal to one but the indices x and y are continuous? I'm confused.
silence11 said:If it's a dirac delta doesn't it mean it's infinite when x=y? Or is it a sort of kronecker where it's equal to one but the indices x and y are continuous? I'm confused.
haushofer said:The second is a distribution, and is there because the fields are really distributions. These kind of relations only make sense if you integrate them with a test function. Otherwise you would naively say that the commutator blows up if x=y.
You can compare it with the commutators in QM; those only make sense if you apply them to a wave function.
Could you elaborate on this?geoduck said:In QM, X and P are distributions.
lugita15 said:Could you elaborate on this?
The delta in the commutation relations of QFT refers to the Dirac delta function, also known as the delta function or impulse function. It is a mathematical function that is zero everywhere except at the origin, where it is infinite. It is commonly used in quantum field theory (QFT) to represent point-like particles or fields.
While both are commonly referred to as "delta," the Dirac delta and Kronecker delta are two distinct mathematical functions. The Dirac delta is a continuous function that is infinite at the origin and zero elsewhere, while the Kronecker delta is a discrete function that takes on a value of 1 when its arguments are equal, and 0 otherwise. In QFT, the Dirac delta is used to represent point-like particles, while the Kronecker delta is used to represent discrete quantum states.
The delta in the commutation relations of QFT is a function, specifically the Dirac delta function. It is not an operator, as it does not act on any functions or operate on any variables. It is simply a mathematical function used to model point-like particles or fields in QFT.
The delta in the commutation relations of QFT plays a crucial role in Heisenberg's uncertainty principle. The uncertainty principle states that it is impossible to know both the position and momentum of a particle simultaneously with absolute precision. The Dirac delta function is used in the commutation relation between position and momentum operators, which is a fundamental concept in QFT and is directly related to the uncertainty principle.
Yes, the delta in the commutation relations of QFT can be generalized to higher dimensions. In QFT, the Dirac delta function can be extended to multiple dimensions to represent point-like particles or fields in higher dimensional space. This allows for the study of QFT in more complex systems and scenarios, such as in curved spacetime or in theories with extra dimensions.