43.2 The symplectic and complex structures on M

43.2 The symplectic and complex structures on M

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · charge operator · complexification

Taking as dual phase space $M$ the space of solutions of the Klein-Gordon equation, one way to write elements of this space is as pairs of functions $(\varphi ,\pi )$ on ${\mathbf{R}}^3$ . The symplectic structure is then given by

$$\Omega (({\varphi }_1,{\pi }_1),({\varphi }_2,{\pi }_2))={\int }_{{\mathbf{R}}^3}({\varphi }_1(\mathbf{x}){\pi }_2(\mathbf{x})-{\pi }_1(\mathbf{x}){\varphi }_2(\mathbf{x}))d^3\mathbf{x}\text{\hspace{1em}(43.12)}$$

and the $(\varphi (\mathbf{x}),\pi (\mathbf{x}))$ can be thought of as pairs of conjugate coordinates analogous to the pairs $q_j,p_j$ but with a continuous index x instead of the discrete index $j.$ . Also by analogy with the finite dimensional case, the symplectic structure can be written in terms of the distributional fields $\Phi (\mathbf{x}),\Pi (\mathbf{x})$ as

$$\{\Phi (\mathbf{x}),\Pi ({\mathbf{x}}^{\prime })\}={\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }),\{\Phi (\mathbf{x}),\Phi ({\mathbf{x}}^{\prime })\}=\{\Pi (\mathbf{x}),\Pi ({\mathbf{x}}^{\prime })\}=0\text{\hspace{1em}(43.13)}$$

We now have a dual phase space $M$ and a symplectic structure on it, so in principle could quantize using an infinite dimensional version of the Schr¨odinger representation, treating the values $\varphi (\mathbf{x})$ as an infinite number of position-like coordinates, and taking states to be functionals of these coordinates. It is however much more convenient, as in the non-relativistic case of chapters 36 and 37, to use the Bargmann-Fock representation, treating the quantum field theory system as an infinite collection of harmonic oscillators. This requires a choice of complex structure, which we will discuss in this section.

By analogy with the non-relativistic case, it is tempting to try and think of solutions of the Klein-Gordon equation as wavefunctions describing a single relativistic particle. A standard physical argument is that a relativistic singleparticle theory describing localized particles is not possible, since once the position uncertainty of a particle is small enough, its momentum uncertainty will be large enough to provide the energy needed to create new particles. If you try and put a relativistic particle in a smaller and smaller box, at some point you will no longer have just one particle, and it is this situation that implies that only a many-particle theory will be consistent. This leads one to expect that any attempt to find a consistent relativistic single-particle theory will run into some sort of problem.

One obvious source of trouble are the negative energy solutions. These will cause an instability if the theory is coupled to other physics, by allowing initial positive energy single-particle states to evolve to states with arbitrarily negative energy, transferring positive energy to the other physical system they are coupled to. This can be dealt with by restricting to the space of solutions of positive energy, taking ${\mathcal{H}}_1$ to be the space of complex solutions with positive energy $(i.e.$ 2 letting $\varphi$ take complex values and setting $f(-{\omega }_{\mathbf{p}},-\mathbf{p})=0$ in equation 43.5). One then has solutions

$$\varphi (t,\mathbf{x})=\frac{1}{(2\pi {)}^{3/2}}{\int }_{{\mathbf{R}}^3}f({\omega }_{\mathbf{p}},\mathbf{p})e^{-i{\omega }_{\mathbf{p}}t}{e}^{i\mathbf{p}\cdot \mathbf{x}}\frac{d^3\mathbf{p}}{2{\omega }_{\mathbf{p}}}\text{\hspace{1em}(43.14)}$$

parametrized by complex functions $f$ of $\mathbf{p}.$ .

This choice of ${\mathcal{H}}_1$ gives a theory much like the non-relativistic free particle Schr¨odinger case (and which has that theory as a limit if one takes the speed of light to $\infty )$ . The factor of ${\omega }_{\mathbf{p}}$ required by Lorentz invariance however leads to the following features (which disappear in the non-relativistic limit):

• There are no states describing localized particles, since one can show that solutions $\varphi (t,\mathbf{x})$ of the form 43.14 cannot have compact support in $x$. The argument is that if they did, the Fourier transforms of both $\varphi$ and its time derivative $\dot{\varphi }$ would be analytic functions of $\mathbf{p}.$ . But the Fourier transforms of solutions would satisfy

$$\frac{\partial }{\partial t}\tilde{\varphi }(t,\mathbf{p})=-i{\omega }_{\mathbf{p}}\tilde{\varphi }(t,\mathbf{p})$$

This leads to a contradiction, since the left-hand side must be analytic, but the right-hand side can’t be (it’s a product of an analytic function, and a non-analytic function, ${\omega }_{\mathbf{p}})$

• A calculation of the propagator (see section 43.5) shows that it is nonzero for space-like separated points. This implies a potential violation of causality once interactions are allowed, with influence from what happens at one point in space-time traveling to another at faster than the speed of light.

To construct a multi-particle theory with this ${\mathcal{H}}_1$ along the same lines as the non-relativistic case, we would need to introduce a complex conjugate space ${\overline{\mathcal{H}}}_1$ 1 and then apply the Bargmann-Fock method. We would then be quantizing a theory with dual phase space a subspace of the solutions of the complex Klein-Gordon equation, those satisfying a condition (positive energy) that is non-local in space-time.

A more straightforward way to construct this theory is to start with dual phase space $\mathcal{M}$ the real-valued solutions of the Klein-Gordon equation. This is a real vector space with no given complex structure, but recall equation 43.11, which describes points in $\mathcal{M}$ by a complex function $\alpha (\mathbf{p})$ rather than a pair of real functions $\varphi (\mathbf{x})$ and $\pi (\mathbf{x})$ . We can take this as a choice of complex structure, defining:

Definition (Relativistic complex structure)

The relativistic complex structure on the space $\mathcal{M}$ is given by the operator $J_r$ that, extended to $\mathcal{M}\otimes \mathbf{C}$ , is $+i$ on the $\alpha (\mathbf{p}),-i$ on the $\overline{\alpha }(\mathbf{p})$

The $A(\mathbf{p})$ are then a continuous basis of ${\mathcal{M}}_{J_r}^{+}$ , the $\overline{A}(\mathbf{p})$ a continuous basis of ${\mathcal{M}}_{J_r}^{-}$ and

$$\mathcal{M}\otimes \mathbf{C}={\mathcal{M}}_{J_r}^{+}\oplus {\mathcal{M}}_{J_r}^{-}$$

A confusing aspect of this setup is that after complexification of $\mathcal{M}$ to get $\mathcal{M}\otimes \mathbf{C},\mathbf{\alpha }(\mathbf{p})$ and $\overline{\alpha }(\mathbf{p})$ are not complex conjugates in general, only on the real subspace $\mathcal{M}$ . We are starting with a real dual phase space $\mathcal{M}$ , which can be parametrized by complex functions

$$f_{+}(\mathbf{p})=f({\omega }_{\mathbf{p}},\mathbf{p}),f_{-}(\mathbf{p})=f(-{\omega }_{\mathbf{p}},\mathbf{p})$$

that satisfy the reality condition (see equation 43.6)

$$f_{-}(\mathbf{p})=\overline{f_{+}(-\mathbf{p})}$$

Complexifying, $\mathcal{M}\otimes \mathbf{C}$ will be given by pairs $f_{+},f_{-}$ ₋ of complex functions, with no reality condition relating them. The choice of relativistic complex structure $J_r$ is such that ${\mathcal{M}}_{J_r}^{+}$ is the space of pairs with $f_{-}=0$ (the complex functions on the positive energy hyperboloid), ${\mathcal{M}}_{J_r}^{-}$ is the space of pairs with $f_{+}=0$ (complex functions on the negative energy hyperboloid). The conjugation map on $\mathcal{M}\otimes \mathbf{C}$ is NOT the map conjugating the values of $f_{-}$ or $f_{+}$ . It is the map that interchanges

$$(f_{+}(\mathbf{p}),f_{-}(\mathbf{p}))⟷(\overline{f_{-}(-\mathbf{p})},\overline{f_{+}(-\mathbf{p})})$$

The $\alpha (\mathbf{p})$ and $\overline{\alpha }(\mathbf{p})$ given by

$$\alpha (\mathbf{p})=\frac{f_{+}(\mathbf{p})}{\sqrt{2{\omega }_{\mathbf{p}}}},\overline{\alpha }(\mathbf{p})=\frac{f_{-}(-\mathbf{p})}{\sqrt{2{\omega }_{\mathbf{p}}}}$$

(see equation $43.7)$ are related by this non-standard conjugation (only on real solutions is $\overline{\alpha }(\mathbf{p})$ the complex conjugate of $\alpha (\mathbf{p}))$ .

Also worth keeping in mind is that, while in terms of the $\alpha (\mathbf{p})$ the complex structure $J_r$ is just multiplication by $i,$ for the basis of field variables $(\varphi (\mathbf{x}),\pi (\mathbf{x}))$ , $J_r$ is not multiplication by $i,$ but something much more complicated. From equation 43.11 one sees that multiplication by $i$ on the $\alpha (\mathbf{p})$ corresponds to

$$(\varphi (\mathbf{x}),\pi (\mathbf{x}))\to \left(\frac{1}{{\omega }_{\mathbf{p}}}\pi (\mathbf{x}),-{\omega }_{\mathbf{p}}\varphi (\mathbf{x})\right)$$

on the $\varphi (\mathbf{x}),\pi (\mathbf{x})$ coordinates. This transformation is compatible with the symplectic structure (preserves the Poisson bracket relations 43.13). As a transformation on position space solutions the momentum is the diferential operator $\mathbf{p}=-i\mathbf{V}$ , so $J_r$ needs to be thought of as a non-local operation that can be written as

$$(\varphi (\mathbf{x}),\pi (\mathbf{x}))\to \left(\frac{1}{\sqrt{-{\nabla }^2+m^2}}\pi (\mathbf{x}),-\sqrt{-{\nabla }^2+m^2}\varphi (\mathbf{x})\right)\text{\hspace{1em}(43.15)}$$

Poisson brackets of the continuous basis elements $A(\mathbf{p}),\overline{A}(\mathbf{p})$ are given by

$$\{A(\mathbf{p}),A({\mathbf{p}}^{\prime })\}=\{\overline{A}(\mathbf{p}),\overline{A}({\mathbf{p}}^{\prime })\}=0,\{A(\mathbf{p}),\overline{A}({\mathbf{p}}^{\prime })\}=i{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })\text{\hspace{1em}(43.16)}$$

Recall from section 26.2 that given a symplectic structure Ω and positive, compatible complex structure $J$ on $\mathcal{M}.$ , we can define a Hermitian inner product on ${\mathcal{M}}_J^{+}$ by

$$⟨u,v⟩=i\Omega (\overline{u},v)$$

for $u,v\in {\mathcal{M}}_J^{+}$ . For the case of $\mathcal{M}$ real solutions to the Klein-Gordon equation we have on basis elements $A(\mathbf{p})$ of ${\mathcal{M}}_{J_{\tau }}^{+}$

$$⟨A(\mathbf{p}),A({\mathbf{p}}^{\prime })⟩=i\Omega (\overline{A}(\mathbf{p}),A({\mathbf{p}}^{\prime }))=i\{\overline{A}(\mathbf{p}),A({\mathbf{p}}^{\prime })\}={\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })\text{\hspace{1em}(43.17)}$$

As a Hermitian inner product on elements $\alpha (\mathbf{p})\in {\mathcal{M}}_{J_r}^{+}$ , which is our singleparticle state space ${\mathcal{H}}_1$ , equation 43.17 implies that

$$⟨{\alpha }_1(\mathbf{p}),{\alpha }_2(\mathbf{p})⟩={\int }_{{\mathbf{R}}^3}\overline{{\alpha }_1(\mathbf{p})}{\alpha }_2(\mathbf{p})d^3\mathbf{p}\text{\hspace{1em}(43.18)}$$

This inner product on ${\mathcal{H}}_1$ will be positive definite and Lorentz invariant.

Note the diference with the non-relativistic case, where one has the same $E(3)$ invariant inner product on the position space fields or their momentum space Fourier transforms. In the relativistic case the Hermitian inner product is only the simple one (43.18) on the momentum space initial data for solutions $\overline{\alpha }(\mathbf{p})$ but another quite complicated one on the position space data $\varphi (\mathbf{x}),\pi (\mathbf{x})$ (due to the complicated expression 43.15 for $J_r$ there). Unlike the Hermitian inner product and $J_r,$ the symplectic form is simple in both position and momentum space versions, see the Poisson bracket relations 43.13 and 43.16.

Digression. Quantum field theory textbooks often contain a discussion of a non-positive Hermitian inner product on the space of Klein-Gordon solutions, given by

$$⟨{\varphi }_1,{\varphi }_2⟩=i{\int }_{{\mathbf{R}}^3}\left({\overline{\varphi }}_1(t,\mathbf{x})\frac{\partial }{\partial t}{\varphi }_2(t,\mathbf{x})-\left(\frac{\partial }{\partial t}{\overline{\varphi }}_2(t,\mathbf{x})\right){\varphi }_1(t,\mathbf{x})\right)d^3\mathbf{x}$$

which can be shown to be independent $oft$ . This is defined on $\mathcal{M}\otimes \mathbf{C}$ , the complexified Klein-Gordon solutions and is zero on the real-valued solutions, so does not provide an inner product on those. It does not use the relativistic complex structure. If we start with a theory of complex Klein-Gordon fields, the function $⟨\varphi ,\varphi ⟩$ will be the moment map for the $U(1)$ action by phase transformations on the fields. It will carry an interpretation as charge, and give after quantization the charge operator. This will be discussed in section $\mathrm{44.1.2.}$ To compare the formula for the charge to be found there (equation $44.4)$ with the formula above, use the equation of motion

$$\Pi =\frac{\partial }{\partial t}\overline{\Phi }$$

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