Abstract
We have learned how the consistency of quantum mechanics with special relativity forces us to abandon the single-particle interpretation of the wave function. Instead we have to consider quantum fields whose elementary excitations are associated with particle states, as we will see below. In this chapter we study the basics of field quantization using both the canonical formalism and the path integral method.
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Notes
- 1.
The identity \(p^{2}=m^{2}\) satisfied by the four-momentum of a real particle will be referred to in the following as the on-shell condition.
- 2.
In momentum space, the general solution to this equation is \(\tilde{\phi}(p)=f(p)\delta(p^{2}-m^{2}),\) with f(p) a completely general function of \(p^{\mu}.\) The solution in position space is obtained by inverse Fourier transform. The step function \(\theta(p^{0})\) enforces positivity of the energy.
- 3.
We remind the reader that in a normal-ordered product all annihilation operators appear to the right.
- 4.
Alternatively, one could introduce any cutoff function \(f(p_{\perp}^{2}+p_{\parallel}^{2})\) going to zero fast enough as \(p_{\perp},\;p_{\parallel}\rightarrow \infty.\) The result is independent of the particular function used in the calculation.
- 5.
For the remaining of this chapter we restore the powers of \(\hbar.\)
- 6.
Here we focus on bosonic fields. Path integrals for fermions will be discussed in Chap. 3 (see page 43).
- 7.
There is a global sign ambiguity associated with the sense in which the integration contour surrounds the branch cut. Here we take it clockwise.
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Álvarez-Gaumé, L., Vázquez-Mozo, M.Á. (2012). From Classical to Quantum Fields. In: An Invitation to Quantum Field Theory. Lecture Notes in Physics, vol 839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23728-7_2
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DOI: https://doi.org/10.1007/978-3-642-23728-7_2
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