# Quantum Field Theory – The Most General Procedure for Quantization

hamiltonian-formalismquantizationquantum-field-theoryquantum-gravity

I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me):

A physical system is usually described in terms of states and observable. In the Hamiltonian framework of classical mechanics, the states form a symplectic manifold $(M,\omega)$ and the observables are functions on $M$. The dynamics of a (time invariant) system is a one parameter group of symplectic diffeomorphisms; the generating function is the energy or Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine symplectic space and the motion is by a one-parameter group of symplectic transformations. This general descriptions applies to any system that includes classical particles, fields, strings and other types of objects.

The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:

1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?

2. What is the general procedure for such a construction, given $M$ and $\omega$?

3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)

4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?

5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)

6. What is meant by a 'free' system as given above?

7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?

8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system?

The overall idea is the following. As the symplectic manifold is affine (in the sense of affine spaces not in the sense of the existence of an affine connection), when you fix a point $O$, the manifold becomes a real vector space equipped with a non-degenerate symplectic form. A quantization procedure is nothing but the assignment of a (Hilbert-) Kahler structure completing the symplectic structure. In this way the real vector space becomes a complex vector space equipped with a Hermitian scalar product and its completion is a Hilbert space where one defines the quantum theory. As I shall prove shortly in the subsequent example, symplectic symmetries becomes unitary symmetries provided the Hilbert-Kahler structure is invariant under the symmetry. In this way time evolution in Hamiltonian description gives rise to a unitary time evolution.

An interesting example is the following. Consider a smooth globally hyperbolic spacetime $M$ and the real vector space $S$ of smooth real solutions $\psi$ of real Klein-Gordon equation such that they have compactly supported Cauchy data (on one and thus every Cauchy surface of the spacetime).

A non-degenerate (well defined) symplectic form is given by: $$\sigma(\psi,\phi) := \int_\Sigma (\psi \nabla_a \phi - \phi \nabla_a \psi)\: n^a d\Sigma$$ where $\Sigma$ is a smooth spacelike Cauchy surface, $n$ its normalized normal vector future pointing and $d\Sigma$ the standard volume form induced by the metric of the spacetime. In view of the KG equation the choice of $\Sigma$ does not matter as one can easily prove using the divergence theorem.

There are infinitely many Kahler structures one can build up here. A procedure (one of the possible ones) is to define a real scalar product: $$\mu : S \times S \to R$$ such that $\sigma$ is continuous with respect to it (the factor $4$ arises for pure later convenience): $$|\sigma(\psi, \phi)|^2 \leq 4\mu(\psi,\psi) \mu(\psi,\psi)\:.$$ Under this hypotheses a Hilbert-Kahler structure can be defined as I go to summarize.

It is possible to prove that there exist a complex Hilbert space $H$ and an injective $R$-linear map $K: S \to H$ such that $K(S)+ i K(S)$ is dense in $H$ and, if $\langle | \rangle$ denotes the Hilbert space product: $$\langle K\psi | K\phi \rangle = \mu(\psi,\phi) -\frac{i}{2}\sigma(\psi,\phi) \quad \forall \psi, \phi \in S\:.$$ Finally the pair $(H,K)$ is determined up to unitary isomorphisms form the triple $(S, \sigma, \mu)$.

You see that, as a matter of fact, $H$ is a Hilbertian complexfication of $S$ whose antisymmetric part of the scalar product is the symplectic form. (It is also possible to write down the almost complex structure of the theory that is related with the polar decomposition of the operator representing $\sigma$ in the closure of the real vector space $S$ equipped with the real scalar product $\mu$.)

What is the physical meaning of $H$?

It is that the physicists call one-particle Hilbert space. Indeed consider the bosonic Fock Space, ${\cal F}_+(H)$, generated by $H$.

$${\cal F}_+(H)= C \oplus H \oplus (H\otimes H)_S \oplus (H\otimes H\otimes H)_S \oplus \cdots\:,$$ and we denote by $|vac_\mu\rangle$ the number $1$ in $C$ viewed as a vector in ${\cal F}_+(H)$

One may define of ${\cal F}_+(H)$ a faithful representation of bosonic CCR by defining the field operator:

$$\Phi(\psi) := a_{K\psi} + a^*_{K\psi}$$

where $a_f$ is the standard annihilation operator referred to the vector $f\in H$ and $a_f^*$ the standard creation operator referred to the vector $f\in H$. It turns out that, with that definition the vacuum expectation values: $$\langle vac_\mu| \Phi(\psi_1)\cdots \Phi(\psi_n) |vac_\mu\rangle$$ satisfy the standard Wick's prescription and thus all them can be computed in terms of the two-point function only: $$\langle vac_\mu| \Phi(\psi) \Phi(\phi) |vac_\mu\rangle$$ Moreover they are in agreement with the formula valid for Gaussian states (like free Minkowski vacuum in Minkowski spacetime) $$\langle vac_\mu | e^{i \Phi(\psi)} |vac_\mu \rangle = e^{-\mu(\psi,\psi)/2}$$

Actually, in view of the GNS theorem the constructed representation of the CCR is uniquely determined by $\mu$, up to unitary equivalences.

The field operator $\Phi$ is smeared with KG solutions instead of smooth supportly compacted functions $f$ as usual. However the "translation" is simply obtained. If $E : C_0^{\infty}(M) \to S$ denotes the causal propagator (the difference of the advanced and retarded fundamental solution of KG equation) the usual field operator smeared with $f\in C_0^{\infty}(M)$ is: $$\hat{\phi}(f) := \Phi(Ef)\:.$$

The CCR can be stated in both languages. Smearing fields with KG solutions one has:

$$[\Phi(\psi), \Phi(\phi)] = i \sigma(\psi,\phi)I\:,$$

smearing field operators with functions, one instead has:

$$[\hat{\phi}(f), \hat{\phi}(g)] = i E(f,g) I$$

Every one-parameter group of symplectic diffeomorphisms $\alpha_t :S \to S$ (for instance continuous Killing isometries of $M$) give rise to an action on the algebra of the quantum fields $$\alpha^*_t(\Phi(\psi)) := \Phi(\psi \circ \alpha_t)\:.$$ If the state $|vac_\mu\rangle$ is invariant under $\alpha_t$, namely $$\mu\left(\psi \circ \alpha_t,\psi \circ \alpha_t\right) = \mu\left(\psi ,\psi \right)\quad \forall t \in R,$$ then, essentially using Stone's theorem, one sees that the said continuous symmetry admits a (strongly continuous) unitary representation: $$U_t \Phi(\psi) U^*_t =\alpha^*_t(\Phi(\psi))\:.$$ The self-adjoint generator of $U_t= e^{-itH}$ is an Hamiltonian operator for that symmetry. Actually this interpretation is suitable if $\alpha_t$ arises by a timelike continuous Killing symmetry. Minkowki vacuum is constructed in this way requiring that the corresponding $\mu$ is invariant under the whole orthochronous PoincarĂ© group.

All the picture I have sketched is intermediate between the "practical" QFT and the so-called algebraic formulation. I only would like to stress that choosing different $\mu$ one generally obtain unitarily inequivalent representations of bosonic CCR.