The most general procedure for quantization

Question

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?

I have asked a lot of long questions, so please answer as many as you can and link relevant articles.

Answer 1

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 Poincaré 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.

Answer 2

Here are some comments on the literature, maybe serving to put Valter Moretti's more concrete response into a broader perspective.

The question asked is a surprisingly good question. It is "good", because it is indeed true that there is this very general prescription for quantization; and "surprisingly" because, while the general idea has been around for ages, this has been understood in decent generality only last year!

Namely, on the one hand it is long appreciated in the context of quantum mechanics that what physicists sweepingly call "canonical quantization" is really this: the construction of the covariant phase space as a (pre-)symplectic manifold, and then the quantization this by the prescription of either algebraic deformation quantization or geometric quantization.

In contrast, it has been understood only surprisingly more recently that established methods of perturbative quantization of field theories, especially in the guise of Epstein-Glaser's causal peruturbation theory (such as QED, QCD, and also perturbative quantum gravity, as in Scharf's textbooks) are indeed also examples of this general method.

For free fields (no interactions), this was first understood in

• J. Dito. "Star-product approach to quantum field theory: The free scalar field". Letters in Mathematical Physics, 20(2):125–134, 1990.

and then amplified in a long series of articles on locally covariant perturbative quantum field theory

by Klaus Fredenhagen and collaborators, starting with

• M. Dütsch and K. Fredenhagen. "Perturbative algebraic field theory, and deformation quantization". In R. Longo (ed), "Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects", volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001.

Curiously, despite this insight, these authors continued to treat interacting quantum field theory by the comparatively ad hoc Bogoliubov formula, instead of similarly deriving it from a quantization of the (pre-)symplectic structure of the phase space of the interacting theory.

That last step, to show that the traditional construction of interacting peturbative quantum field theory via time-ordered products and Bogoliubov's formula also follows from the general prescription of deformation/geometric quantization of (pre-)symplectic phase space was made, unbelievably, only last year, in the highly recommendable thesis

• Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory arXiv:1603.09626

Just read the introduction of this thesis, it is very much worthwhile.

(I learned about this article from Igor Khavkine and Alexander Schenkel for which I am grateful.)

In a similar spirit a little later appeared

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory arxiv:1612.09157

which disucsses the situation in a bit more generality than Collini does, but omitting the technical details of renormalization in this perspective.

Answer 3

I will give you a more general answer, in fact, a very mathematical answer that defines quantization.

Setup: k is a characteristic 0 field, and K = k[[h]]

Deformation

A Deformation algebra is a topologically free K-algebra, so by definition if you have some associative K-algebra $A_0$, a deformation of $A_0$ is a deformation algebra $A$ s.t. $A_0 = A/hA$.

Now consider $A_0$ being commutative (A not necessarily). Then, $A_0$ inherits a Poisson Structure from $A$.

Let multiplication in $A$ be *. Then for $f_0,g_0 \in A_0$, by choosing arbitrary listings $f,g$, we have $f *g - g*f = 0\ (mod\ h)$

We set $$\{f_0,g_0\} = \frac{1}{h} (f*g - g*f)\ (mod\ h)$$

You can see that $(A_0,\{,\})$ is indeed a Poisson algebra.

This Poisson algebra is the quasiclassical limit of $A$, and $A$ is the quantization of $A_0$.

Here you can see how quasiclassical limit and quantization are processes that are in a way inverse, with a few caveats:

• The Quasiclassical Limit is unique and well defined
• Quantization is in general non-unique and there exist Poisson algebras that admit no quantization.

Moyal-Weyl Quantization

The famous deformation quantization of Moyal and Weyl is an example of this.

Here $A_0 = [k,p]$ with the classical Poisson Structure, and we have

$$f*g = m(exp(\frac{1}{2}h(\partial_{[x} \otimes \partial_{,p]})), f \otimes f)$$ where $m:A_0 \otimes A_0 \rightarrow A_0$ is the multiplication.

Important Theorems

Theorem 1 (Symplectic Manifold Local Quantization)

Any symplectic Manifold admits a local quantization. (construction via cotangent bundle.

Theorem 2 (Groenwald)

Let $A$ be a quantization of $A_0 =\mathcal{C}^ {\infty}(M)$. There is no homomorphism $$\Phi : SDiff(M)\rightarrow Aut(A)$$

such that $\Phi = \Phi_0 \ (mod\ h)$

The Groenwald theorem is a very famous No go theorem that says that symmetry is broken by quantization.

This is bad news for symplectic manifolds and associative K-algebras, but worry not, as Poisson-Lie Groups do not suffer from this theorem's implications.

Poisson Manifolds

Kontsevich made the brilliant discovery of the theorem that states that any Poisson Manifold, admits a local quantization.

A celebrated example is geometric quantization.

Familiar Notions in Physics

These facts lead to the basic statement that Quantum Mechanics becomes Classical Mechanics in the quasiclassical limit, where now the formal variable h is $\hbar$.

Let me now give an example to make it a bit more clear.

Example

Take $M = T^* \Bbb R$ with coordinates $(x,p)$ and the usual Poisson structure.

The Hamiltonian is given by $H = \frac{p^2}{2m} + V(x)$, and we have the usual Hamilton's equations.

Now from polynomial deformation quantization, we know that the quantization of the Poisson Algebra $C^\infty _{pol} (T^* \Bbb R)$ is

$$A = (\bigoplus_{n \geq 0} h^n \mathcal{D}^n_{\Bbb R})[[h]]$$

This quantization has a polynomial rational form and defines a family of algebras $A(\hbar)$. Such an algebra is an algebra of polynomial differential operators on $C^\infty (\Bbb R)$ when $\hbar \neq 0$, and $A(0) = C^\infty _{pol} (T^* \Bbb R)$. The algebra $A = A(\hbar)$ acts on the Hilbert Space $L^2(\Bbb R)$. The position coordinate and momentum coordinates are the famous quantum operators, and the Hamiltonian takes don't he famous form of

$$\hat{H} = - \frac{\hbar^2}{2} d_x^2 + V(x)$$

and the evolution equations become

$$i\hbar \frac{dQ}{dt} = [- \frac{\hbar^2}{2} d_x^2 + V(x),Q]\ \ Q \in A$$

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Sources: Etingof, Schiffmann: "Lectures on Quantum Groups"