43.4 Quantization of the Klein-Gordon theory

43.4 Quantization of the Klein-Gordon theory

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

Given the description we have found in momentum space of real solutions of the Klein-Gordon equation and the choice of complex structure $J_r$ described in the last section, we can proceed to construct a quantum field theory by the Bargmann-Fock method in a manner similar to the non-relativistic quantum field theory case. Quantization takes

$$A(\alpha (\mathbf{p}))\in {\mathcal{M}}_{J_r}^{+}={\mathcal{H}}_1\to a^{†}(\alpha (\mathbf{p}))$$ $$\overline{A}(\alpha (\mathbf{p}))\in {\mathcal{M}}_{J_r}^{-}={\overline{\mathcal{H}}}_1\to a(\alpha (\mathbf{p}))$$

Here $A(\alpha (\mathbf{p}))$ is the positive energy solution of the (complexified) Klein-Gordon equation with initial data given by $\alpha (\mathbf{p})$ and $a^{†}(\alpha (\mathbf{p}))$ is the operator on the Fock space of symmetric tensor products of ${\mathcal{H}}_1$ given by equation 36.12. $\overline{A}(\alpha (\mathbf{p}))$ 1 is the conjugate negative energy solution and $a(\alpha (\mathbf{p}))$ is the operator given by equation 36.13. All these objects are often written in distributional form, where one has

$$A(\mathbf{p})\to a^{†}(p),\overline{A}(\mathbf{p})\to a(\mathbf{p})$$

which one can interpret as corresponding to taking the limit of $\alpha ({\mathbf{p}}^{\prime })\delta (\mathbf{p}-{\mathbf{p}}^{\prime })$

The operators $a(\alpha (\mathbf{p}))$ and $a^{†}(\alpha (\mathbf{p}))$ satisfy the commutation relations

$$[a({\alpha }_1(\mathbf{p})),a^{†}({\alpha }_2(\mathbf{p}))]=⟨{\alpha }_1(\mathbf{p}),{\alpha }_2(\mathbf{p})⟩$$

or in distributional form

$$[a(\mathbf{p}),a^{†}({\mathbf{p}}^{\prime })]={\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$$

For the Hamiltonian we take the normal ordered form

$$\hat{H}={\int }_{{\mathbf{R}}^3}{\omega }_{\mathbf{p}}{a}^{†}(\mathbf{p})a(\mathbf{p})d^3\mathbf{p}$$

Starting with a vacuum state |0⟩, by applying creation operators one can create arbitrary positive energy multi-particle states of free relativistic particles, with single-particle states having the energy momentum relation

$$E(\mathbf{p})={\omega }_{\mathbf{p}}=\sqrt{|\mathbf{p}{|}^2+m^2}$$

This description of the quantum system is essentially the same as that of the non-relativistic theory, which seems to difer only in the energy-momentum relation, which in that case was $\begin{array}{r}E(\mathbf{p})=\frac{|\mathbf{p}{|}^2}{2m}\end{array}$ . The diferent complex structure used for quantization in the relativistic theory changes the physical meaning of annihilation and creation operators:

• Non-relativistic theory. $\mathcal{M}={\mathcal{H}}_1$ is the space of complex solutions of the free particle Schr¨odinger equation, which all have positive energy. ${\overline{\mathcal{H}}}_1$ is the conjugate space. For each continuous basis element $A(\mathbf{p})$ of ${\mathcal{H}}_1$ (these are initial data for a positive energy solution with momentum p), quantization takes this to a creation operator $a^{†}(\mathbf{p})$ , which acts with the physical interpretation of addition of a particle with momentum p. Quantization of the complex conjugate $\overline{A}(\mathbf{p})$ in ${\overline{\mathcal{H}}}_1$ gives an annihilation operator $a(\mathbf{p})$ which removes a particle with momentum p.

• Relativistic theory. ${\mathcal{M}}_{J_r}^{+}={\mathcal{H}}_1$ is the space of positive energy solutions of the Klein-Gordon equation. It has continuous basis elements $A(\mathbf{p})$ which after quantization become creation operators adding a particle of momentum p and energy ${\omega }_{\mathbf{p}}.{\mathcal{M}}_{J_r}^{-}$ is the space of negative energy solutions of the Klein-Gordon equation. Its continuous basis elements $\overline{A}(\mathbf{p})$ after quantization become annihilation operators for antiparticles of momentum −$p$ and positive energy ${\omega }_{\mathbf{p}}$

To make physical sense of the quanta in the relativistic theory, assigning all non-vacuum states a positive energy, we take such quanta as having two physically equivalent descriptions:

• A positive energy particle moving forward in time with momentum p.

• A positive energy antiparticle moving backwards in time with momentum −$p$.

The operator $a^{†}(\mathbf{p})$ adds such quanta to a state, the operator $a(\mathbf{p})$ destroys them. Note that for a theory of quantized real-valued Klein-Gordon fields, the field $\Phi$ has components in both ${\mathcal{M}}_{J_r}^{+}$ and ${\mathcal{M}}_{J_r}^{-}$ so its quantization will both create and destroy quanta.

Just as in the non-relativistic case (see equation 37.1) quantum field operators can be defined using the momentum space decomposition and annihilation and creation operators:

Definition (Real scalar quantum field)

The real scalar quantum field operators are the operator-valued distributions defined by

$$\hat{\Phi }(\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(a(\mathbf{p})e^{i\mathbf{p}\cdot \mathbf{x}}+a^{†}(\mathbf{p})e^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\text{\hspace{1em}(43.19)}$$

$$\hat{\Pi }(\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(-i{\omega }_{\mathbf{p}})(a(\mathbf{p})e^{i\mathbf{p}\cdot \mathbf{x}}-a^{†}(\mathbf{p})e^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\text{\hspace{1em}(43.20)}$$

By essentially the same computation as for Poisson brackets, the commutation relations are

$$[\hat{\Phi }(\mathbf{x}),\hat{\Pi }({\mathbf{x}}^{\prime })]=i{\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }),[\hat{\Phi }(\mathbf{x}),\hat{\Phi }({\mathbf{x}}^{\prime })]=[\hat{\Pi }(\mathbf{x}),\hat{\Pi }({\mathbf{x}}^{\prime })]=0\text{\hspace{1em}(43.21)}$$

These can be interpreted as the distributional form of the relations of a unitary representation of a Heisenberg Lie algebra on $\mathcal{M}\oplus \mathbf{R}$ , where $M$ is the space of solutions of the Klein-Gordon equation.

The Hamiltonian operator will be quadratic in the field operators and can be chosen to be

$$\hat{H}={\int }_{{\mathbf{R}}^3}\frac{1}{2}:(\hat{\Pi }(\mathbf{x}{)}^2+(\nabla \hat{\Phi }(\mathbf{x}){)}^2+m^2\hat{\Phi }(\mathbf{x}{)}^2):d^3\mathbf{x}$$

This operator is normal ordered, and a computation (see for instance chapter 5 of [16]) shows that in terms of momentum space operators this is the expected

$$\hat{H}={\int }_{{\mathbf{R}}^3}{\omega }_{\mathbf{p}}{a}^{†}(\mathbf{p})a(\mathbf{p})d^3\mathbf{p}\text{\hspace{1em}(43.22)}$$

The dynamical equations of the quantum field theory are now

$$\frac{\partial }{\partial t}\hat{\Phi }=[\hat{\Phi },-i\hat{H}]=\hat{\Pi }$$ $$\frac{\partial }{\partial t}\hat{\Pi }=[\hat{\Pi },-i\hat{H}]=(\Delta -m^2)\hat{\Phi }$$

which have as solution the following equation for the time-dependent field operator:

$$\hat{\Phi }(t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}(a(\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}+a^{†}(\mathbf{p})e^{i{\omega }_{\mathbf{p}}t}{e}^{-i\mathbf{p}\cdot \mathbf{x}})\frac{d^3\mathbf{p}}{\sqrt{2{\omega }_{\mathbf{p}}}}\text{\hspace{1em}(43.23)}$$

Unlike the non-relativistic case, where fields are non-self-adjoint operators, here the field operator is self-adjoint (and thus an observable), and has both an annihilation operator component and a creation operator component. This gives a theory of positive energy quanta that can be interpreted as either particles moving forward in time or antiparticles of opposite momentum moving backwards in time.

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