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Results tagged with fourier-transform
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A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.
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Calculating the contraction of a field operator and a creation operator
In Folland's Quantum Field Theory (section 6.4) he considers a field:
$$\phi_\pi = \sum_\tau \int f(\textbf{q})\big[u(\textbf{q}, \tau, \pi)a(\textbf{q}, \tau, \pi) e^{-iq_\mu x^\mu} + v(\textbf{q}, …
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How do we determine momentum when going from position space to momentum space in Feynman dia...
I have read chapter 4 in Peskin & Schroeder on interacting fields and Feynman diagrams. I believe I have some fundamental misunderstanding, so I have outlined what I know so far below and included my …
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Why are test functions always taken to be Schwartz in quantum field theory?
A quantum field theory is defined as a measure over tempered distributions. Why do we take the space of test functions to always be Schwartz space? Why don't we, for example, smear fields using $C^\in …
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How do we know that all quantum fields are Fourier transforms of creation and annihilation o...
In Folland's book Quantum Field Theory, he says
...we start out with classical field equations and a relativistically invariant Lagrangian from which they are derived, then replace the classical fiel …
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What does it mean to apply a creation or annihilation operator to a free field, e.g. $\langl...
I am self studying Quantum Field Theory, and I am starting to get a little lost. So far, I have studied free fields and some basic computations involving them, such as creation and annihilation operat …
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Why is the wave function of a particle with definite momentum $p$ given as $e^{ipx/\hbar}$?
In many textbooks it is stated that, in position space, the wave function of a particle with definite momentum $p$ is given by $e^{ipx/\hbar}$. I know that the $\hbar$ comes from the de Broglie hypoth …
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Understanding quantized vector fields
Let $A^\nu$ be a 4-potential. In Folland's book on quantum field theory (page 117), he quantizes this object in the case where it represents a massive particle as:
$$A^\nu(x) = \int \sum_{j=1}^3 \frac …
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