37.1 Quantum field operators

37.1 Quantum field operators

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra

The multi-particle formalism developed in chapter 36 works well to describe states of multiple free particles, but does so purely in terms of states with welldefined momenta, with no information at all about their position. Instead of starting with $t=0$ momentum eigenstates $|p⟩$ and corresponding Schr¨odinger solutions $A(p)$ as continuous basis elements of the single-particle space ${\mathcal{H}}_1$ , the position operator $Q$ eigenstates $|x⟩$ could be used. The solution that is such an eigenstate at $t:=:0$ will be denoted $\Psi (x)$ , the conjugate solution will be written $\overline{\Psi }(x)$ . The $\Psi (x),\overline{\Psi }(x)$ can be thought of as the complex coordinates of an oscillator for each value of x.

The corresponding quantum state space would naively be a Fock space for an infinite number of degrees of freedom, with an occupation number for each value of $x.$ This could be made well-defined by introducing a spatial cutof and discretizing space, so that x only takes on a finite number of values. However, such states in the occupation number basis would not be free particle energy eigenstates. While a state with a well-defined momentum evolves as a state with the same momentum, a state with well-defined position at some time does not evolve into states with well-defined positions (its wavefunction immediately spreads out).

One does however want to be able to discuss states with well-defined positions, in order to describe amplitudes for particle propagation, and to introduce local interactions between particles. One approach is to try and define operators corresponding to creation or annihilation of a particle at a fixed position, by taking a Fourier transform of the annihilation and creation operators for momentum eigenstates. Quantum fields could be defined as

$$\hat{\Psi }(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ipx}a(p)dp\text{\hspace{1em}(37.1)}$$

and its adjoint

$${\hat{\Psi }}^{†}(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-ipx}{a}^{†}(p)dp$$

Note that, just like $a(p)$ and $a^{†}(p)$ , these are not self-adjoint operators, and thus not themselves observables, but physical observables can be constructed by taking simple (typically quadratic) combinations of them. As explained in section 36.5, for multi-particle systems the state space $H$ is complicated to describe and work with, it is the operators that behave simply. These will generally be built out of either the field operators $\hat{\Psi }(x),{\hat{\Psi }}^{†}(x)$ , or annihilation and creation operators $a(p),a^{†}(p)$ , with the Fourier transform relating the two possibilities.

In the continuum formalism the annihilation and creation operators satisfy the distributional equation

$$[a(p),a^{†}(p^{\prime })]=\delta (p-p^{\prime })$$

and one can formally compute the commutators

$$[\hat{\Psi }(x),\hat{\Psi }(x^{\prime })]=[{\hat{\Psi }}^{†}(x),{\hat{\Psi }}^{†}(x^{\prime })]=0$$ $$\begin{array}{rl}[\hat{\Psi }(x),{\hat{\Psi }}^{†}(x^{\prime })]& =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{ipx}{e}^{-i{p}^{\prime }{x}^{\prime }}[a(p),a^{†}(p^{\prime })]dpd{p}^{\prime }\\ & =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{ipx}{e}^{-i{p}^{\prime }{x}^{\prime }}\delta (p-p^{\prime })dpd{p}^{\prime }\\ & =\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{e}^{ip(x-x^{\prime })}dp\\ & =\delta (x-x^{\prime })\end{array}$$

getting results consistent with the interpretation of the field operator and its adjoint as operators that annihilate and create particles at a point $x$.

This sort of definition relies upon making sense of the operators $a(p)$ and $a^{†}(p)$ as distributional operators, using the action of elements of ${\mathcal{H}}_1$ and ${\overline{\mathcal{H}}}_1$ on Fock space given by equations 36.12 and 36.13. We could instead more directly proceed exactly as in section 36.3, but with solutions characterized by initial data given by a function $\psi (x)$ rather than its Fourier transform $\alpha (p)$ . One gets all the same objects and formulas, related by Fourier transform. Our notation for these transformed objects will be

$$\begin{array}{rl}\alpha (p)& \to \psi (x),\overline{\alpha }(p)\to \overline{\psi }(x)\\ A(\alpha )& \to \Psi (\psi ),\overline{A}(\alpha )\to \overline{\Psi }(\psi )\\ A(p)& \to \Psi (x),\overline{A}(p)\to \overline{\Psi }(x)\end{array}$$

$\Psi$($\psi$) will be the solution in ${\mathcal{H}}_1$ with initial data $\psi (x)$ and $\overline{\Psi }(\psi )$ the conjugate solution in ${\overline{\mathcal{H}}}_1.\Psi (x)$ can be interpreted as the distributional solution equal to $\delta (x-x^{\prime })$ at $t=0$ and we can write

$$\Psi (\psi )=\int \psi (x)\Psi (x)dx,\overline{\Psi }(\psi )=\int \overline{\psi (x)}\overline{\Psi }(x)dx$$

These satisfy the Poisson bracket relations

$$\{\Psi ({\psi }_1),\Psi ({\psi }_2)\}=\{\overline{\Psi }({\psi }_1),\overline{\Psi }({\psi }_2)\}=0,\{\Psi ({\psi }_1),\overline{\Psi }({\psi }_2)\}=i⟨{\psi }_1,{\psi }_2⟩\text{\hspace{1em}(37.2)}$$

$$\{\Psi (x),\Psi (x^{\prime })\}=\{\overline{\Psi }(x),\overline{\Psi }(x^{\prime })\}=0,\{\Psi (x),\overline{\Psi }(x^{\prime })\}=i\delta (x-x^{\prime })\text{\hspace{1em}(37.3)}$$

Quantization then takes (note the perhaps confusing choice of notational convention due to following the physicist’s convention that $\hat{\Psi }$ is the annihilation operator)

$$\Psi (\psi )\to -i{\hat{\Psi }}^{†}(\psi ),\overline{\Psi }(\psi )\to -i\hat{\Psi }(\psi )$$

The quantum field operators can be defined in terms of tensor products using the same generalization of the finite dimensional case of section 26.4 that we used to define $a(\alpha )$ and $a^{†}(\alpha )$ . Here

$${\hat{\Psi }}^{†}(\psi )P^{+}\left(\Psi \left({\psi }_1\right)\otimes \cdots \otimes \Psi \left({\psi }_n\right)\right)=\sqrt{n+1}{P}^{+}\left(\Psi (\psi )\otimes \Psi \left({\psi }_1\right)\otimes \cdots \otimes \Psi \left({\psi }_n\right)\right)\text{\hspace{1em}(37.4)}$$

$$\begin{array}{l}\hat{\Psi }(\psi )P^{+}(\Psi ({\psi }_1)\otimes \cdots \otimes \Psi ({\psi }_n))=\\ \frac{1}{\sqrt{n}}{\sum }_{j=1}^n⟨\psi ,{\psi }_j⟩P^{+}(\Psi ({\psi }_1)\otimes \cdots \otimes \widehat{\Psi ({\psi }_j)}\otimes \cdots \otimes \Psi ({\psi }_n))\end{array}\text{\hspace{1em}(37.5)}$$

(the $\widehat{\Psi ({\psi }_j)}$ means omit that term in the tensor product, and $P^{+}$ is the symmetrization operator defined in section 9.6). This gives a representation of the Lie algebra relations 37.2, satisfying

$$[\hat{\Psi }({\psi }_1),\hat{\Psi }({\psi }_2)]=[{\hat{\Psi }}^{†}({\psi }_1),{\hat{\Psi }}^{†}({\psi }_2)]=0,[\hat{\Psi }({\psi }_1),{\hat{\Psi }}^{†}({\psi }_2)]=⟨{\psi }_1,{\psi }_2⟩$$

Conventional multi-particle wavefunctions in position space have the same relation to symmetric tensor products as in the momentum space case of section 36.4. Given an arbitrary state $|\psi ⟩$ in the multi-particle state space, the position space wavefunction component with particle number $N$ can be expressed as

$${\psi }_N(x_1,x_2,\dots ,x_N)=⟨0|\hat{\Psi }(x_1)\hat{\Psi }(x_2)\dots \hat{\Psi }(x_N)|\psi ⟩$$

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