Geometric ﬂavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution

In this series of papers we aim to provide a mathematically comprehensive framework to the hamiltonian pictures of quantum ﬁeld theory in curved spacetimes. Our ﬁnal goal is to study the kinematics and the dynamics of the theory from the point of diﬀerential geometry in inﬁnite dimensions. In this second part we use the tools of Gaussian analysis in inﬁnite dimensional spaces introduced in the ﬁrst part to describe rigorously the procedures of geometric quantization in the space of Cauchy data of a scalar theory. This leads us to discuss and establish relations between diﬀerent pictures of QFT. We also apply these tools to describe the geometrization of the space of pure states of quantum ﬁeld theory as a Kh¨aler manifold. We use this to derive an evolution equation that preserves the geomeric structure and avoid norm losses in the evolution. This leads us to a modiﬁcation of the Schr¨odinger equation via a quantum connection that we discuss and exemplify in a simple case

1 Introduction In the first paper of this series [1] we presented a set of mathematical tools to develop Gaussian integration theory on infinite dimensional spaces.Those tools are important to characterize geometrically the Hilbert space of pure states of a particular quantum field theory.In this paper we aim to clarify the physical interpretation of a quantum field theory of the scalar field described over a Cauchy hypersurface of a generic spacetime.However, not every physical aspect of the theory relies solely on the mathematics of Gaussian integration.In order to fully understand the physics we have to choose a Kähler structure (ω, µ, J) on the set of classical fields.The corresponding Hilbert space of pure states is defined by the Gaussian measure associated with it.Here we also investigate how this choice of a Kähler structure determines very relevant features of the quantum theories.
In this work we try to answer two central questions.What does it mean to quantize a classical field theory?and as a consequence, what is the most efficient way of describing a QFT in geometric terms to couple it to gravity in equal footing?
To answer the first question we thoroughly develop a quantization program.For that matter, we investigate how can we rigorously quantize a scalar field using geometric quantization, how can we issue the ordering problems of the algebra of observables in canonical quantization and how each mathematical tool introduced in [1], reflecting a different aspect of the structure of Gaussian integration, has its implications on the interpretation of the physical counterpart of our theory.
To answer the second we argue that a tool of huge importance to detach the particularities of a particular theory form the general geometric structures underlying it, is the concept of second quantized test function.This concept was introduced in the first part of this series as the algebra of Hida test functions.The domain of those functions are distributions and we use them to model a dense subset of pure states that is common and dense to any particular theory that we present.The main advantage of this procedure is that we can describe a generic quantum theory modelling the manifold P of pure states regardless of any particular choice of Kähler structure at the classical level and encode the particular features of a single theory in a second quantized Kähler structure (G, Ω, J ) P .
In this work we take the gravitational degrees of freedom as a given background.We assume space-time to be globally hyperbolic and consider it as a foliation of spatial hypersurfaces, labelled by a time-like parameter t.On each hypersurface, the background metric induces a 3-metric h ij and the corresponding field-momenta π ij , which encode the extrinsic curvature of each leaf.The transformations from one leaf to the next in the foliation are encoded in the lapse and shift vector fields (N, N i ).Foliation is thus represented by the family (h ij (t), π ij (t), N (t), N i (t)) of geometrical objects parametrized by the time-like parameter.On that foliation, a classical Hamiltonian scalar field theory of matter is considered, with fields and fieldmomenta having domain on the hypersurfaces.These fields are then quantized on each hypersurface, the quantization depending on a suitably chosen complex structure and therefore on the geometry of the leaf.Therefore quantization acquires a time dependence, given the non-constant nature of the hypersurface geometry induced by any generic background spacetime.This additional dependence is inherited by quantum states and operators, and also by the Kähler structure at the second quantized level.
Our final goal is to build a description of the Hamiltonian dynamics of QFT as a Hamiltonian system with respect to a canonical Poisson structure, following the well known construction of Kibble for geometrical quantum mechanics.The geometrical origin of the quantization procedure and the dependence on the geometrical structures defined on the set of classical fields will be of the utmost importance to study the properties of the resulting dynamics.In particular, the second quantized structure will allow us to present an evolution equation that modify the Schrödinger equation and preserves norm in the evolution.This kind of geometric structure is key to understand the interplay between a fully hybrid theory of gravity and quantum field theory as we show in [2] where the space-time is no longer treated as a background but as a fully dynamical entity.
The structure of this second part of the series is as follows.Sections 2 and 3 present a brief summary of how geometric quantization can be used to build a quantization of linear observables and the set of quantum states for different relevant cases as are those of holomorphic, Schrödinger or momentum-space quantizations; and how to define isomorphisms relating them.Then, Section 4 studies the quantization of arbitrary observables in the different approaches and how to relate them.Afterwards, Section 5 presents the construction of geometric quantum field theory, following Kibble's construction.We identify the canonical Kähler structure of the set of quantum states build previously, and study the relations with the one defined on the set of classical fields which determines the quantization of the theory.Then, in Section 6 we use the geometrical formalism to describe the solutions of Schrödinger equation as the integral curves of a Hamiltonian vector field associated with the canonical symplectic structure.But the geometrical construction also requires the introduction of a connection to take into account the dependence of the quantization procedure on the geometrical structures of the set of classical fields.This dependence is reflected in the time-evolution which makes the whole construction to be time-dependent.Finally, Section 7 presents an application of all the ingredients introduced so far, in the case of the dynamics of a free quantum scalar field.In Section 8 we present the main conclusions of the two papers and describe potential applications for the future.

Mathematical summary
Studying integration theory, in [1], we argued that the natural model space for a quantum field is a space of distributions regarded as the strong dual (DFN) of a Fréchet Nuclear (FN) space.Regarding spacetime structure, we are interested in globally hyperbolic spacetimes such that admit a compact space-like Cauchy hypersurfaces Σ.In that case the usual model space for classical fields used in the literature is the space of smooth compactly supported functions N = C ∞ c (Σ), in [1] we made the choice of modelling such fields over its strong dual N ′ = D ′ (Σ).The ultimate reason for that choice is to develop a rigorous quantization program thoroughly explained in the forthcoming sections.It is important to notice that, even though here we will limit our study to the compact Σ case, the only ingredient needed generalize our analysis is the FN topology of N , that amounts to proper boundary conditions.For this reason we may use Minkowsky spacetime as an example of our analysis with the space of tempered functions S(R 3 ) and distributions as model.We will also use the notation N and N ′ for the FN space and its dual instead of any particular example to keep in mind this feature.Let us also recall from [1] the notation that we will employ in this work ξ x ∈ N , ϕ x ∈ N ′ paired by ξ x ϕ x := ϕ x , ξ x ∈ R (C). ( Operators and bilinears are denoted with abstract indices as A x y , A xy or A xy and their distinction gives rise to three notions of Dirac delta, δ x y represents point evaluation while δ xy and δ xy is the dual pairing for test functions and distributions respectively such that δ xz δ zy = δ x y , see [1] for further information.We define operator composition and raise and lower indices following the conventions A particular tool of huge importance in our work is the concept of Hida test function.The domain of those functions are distributions of N ′ and posses a FN topology that relies solely on the structure of N .For this reason those functions are denoted by a parenthesized expression (N ).We will use Hida test functions to model a dense subset of pure states that is common and dense to any particular theory that we present.The main advantage of this procedure is that we can describe a generic quantum theory modelling the manifold P of pure states over (N ) regardless of any particular choice of Kähler structure at the classical level and, as we explained above, describe a particular theory with its second quantized Kähler structure (G, Ω, J ) P .As we will see, the ability to characterize the space of states independently of the choice of classical complex structure is of crucial importance in curved spacetimes, as (N ) will be a common subset to all the Hilbert spaces arising from the parametric family of complex structures similarly to the construction considered in [3].

Geometric quantization I: the general framework
The goal of this section is to build the quantum field theory corresponding to the classical model presented in [1].As our goal is a geometric description of the resulting quantum model, we will consider a geometric quantization framework adapted to our infinite dimensional manifold.The procedure is, roughly speaking, similar to the usual construction on finite dimensional symplectic manifolds (see [4]), and based in two steps.
• a pre-quantization which builds an initial Hilbert space as the set of sections of a complex line bundle defined on the classical phase space and a representation of classical functions as operators on that set, • and the definition of a polarization, which chooses a Lagrangian submanifold on the phase space of fields, defines the Hilbert space as the space of sections restricted to the submanifold and the operators as those preserving it in a suitable way.
The first step is straightforward and relatively simple to define for a general symplectic classical manifold.The definition of the polarization is a more subtle task and introduces arbitrary choices which restrict the set of classical magnitudes that can be quantized.We will present now, first the general construction adapted to our manifold of fields M F .For simplicity we will assume that this manifold is linear and modelled over the DFN space N ′ .Then, we will exploit a Kähler structure defined on its complexification M C to consider one particular type of polarization, the holomorphic polarization, which exhibits several nice properties and largely simplifies the technicalities of the construction.This procedure was explained in [1].Finally, we will compare the resulting construction with the usual approach based on a real polarization.

Prequantization I: The definition of the bundle and the measure
The first step towards quantization is the complex prequantum line bundle B over the classical field phase space.This is an Hermitian line bundle π M C ,B : B → M C , associated with a U (1)-principal bundle on M C .On this bundle we define a principal connection which is required to have, as local curvature form, the symplectic form of M C .This can be done if we consider the local connection one form to be defined by the corresponding symplectic potential θ (as our field manifold is a linear space, this choice is well defined globally).By using this connection, we can define a covariant derivative on the associated bundle B. The next step is to find a prequantized Hilbert space H.To define this space we consider the set of square integrable sections B with respect to a measure Dµ invariant under symplectic transformations.Let us consider for simplicity the case where M C can be covered with a single chart in N ′ C , the complexified space of distributions.In this case we will consider the unique Borel probability measure Dµ c (φ x ) on M C provided by Minlos theorem such that the characteristic functional C obtained as satisfies: • it is continuous in the Fréchet topology.Hence, for each choice of functional C satistfying these conditions there exists only one measure Dµ c .Let us see how there are natural choices of functionals based on physical arguments.
Recall from [1] that, in holomorphic coordinates, the Kähler structure (ω, µ, J) M C is derived from the Hermitian form with trivial complex structure and inverse h −1 M C = ∆ xy ∂ φ x ⊗ ∂φy .From a physical point of view we should ask for the measure to be invariant under symplectic transformations.Thus we select C to correspond to a function of the Hermitian form h −1 M C , acting on ρx dφ x as covectors.This provides the measure whose functional C is equal to This measure can be fully understood from the Gel'fand triple provided by its Cameron-Martin Hilbert space where ).This space represents the allowed directions of translation that leave invariant the space, as is explained below, and has the physical interpretation of a one particle state structure.We can reconstruct the whole space starting with finite dimensional orthogonal discretizations of H ∆ .
In this vision we perform a projective limit of the chain of orthogonal projections p nm : C n → C m , m < n, in each of them we can define Gaussian measures dµ n expressed in the usual form with respect to the canonical Lebesgue measure.If we perform the projective limit in the category of measure spaces we do not recover H ∆ as a measure space, but we can instead perform the limit to obtain the Borel σ-algebra of N ′ C and define the measure given by C on it (see [5,6] for further details).Due to this reason, we will call this measure gaussian.
With this choice, the measure exists and then the scalar product of Ψ s , Φ s ∈ Γ(M C , B), sections of the prequantum bundle, can be written as: 2.2 Prequantization II.The states: introducing the vacuum

A trivializing section
In this way we have introduced a measure on the set of sections of the prequantum bundle and we can define our quantum pure states to be the square-integrable sections with respect to the measure Dµ c , or equivalently, those sections with finite norm.Now, we want to trivialize the bundle and factorize those sections with respect to a preferred one Ψ 0 , which physically represents the vacuum state of the physical theory.We will ask this section to be constant in norm with respect to this measure, i.e., All other sections can then be represented as simple functions, which will be square integrable with respect to a measure defined by using Ψ 0 .As sections are also functions on M C , we can write our pre-quantum Hilbert space as The local expression of the vacuum section can be further determined by the covariant derivative of the bundle.We can define the vacuum section Ψ 0 adapted to the symplectic potential (hence, to the complex structure) requiring that Notice that while condition (8) fixes the modulus of the vacuum state, this condition fixes the phase.We shall get back to this point below, when considering the two types of polarization.In any case, we can already remark that clearly the local expression of the state (or equivalently, the connection) depends on the choice of the complex structure on M C .Notice also that thinking in terms of the projective limit of discretized Gaussian measures, Dµ c includes in its definition the expression of the term |Ψ 0 | 2 which usually appears in the expressions in finite dimensions multiplying the Lebesgue measure.This is the reason for the necessity of Equation ( 8), which only leaves free the vacuum state phase for the chosen measure.Using this section Ψ 0 as a reference, any other section Φ s is given by Φ s = ΦΨ 0 where Φ : ).The action of the covariant derivative is then translated to the function Φ as: Our pre-quantum bundle is then identified with a space of regular functions of L 2 (N ′ C , Dµ c ) referred to a vacuum Ψ 0 rather than sections.The effect of the reference to the vacuum is that covariant derivatives incorporate an extra multiplicative factor −iθ(X) provided by the symplectic potential to the directional derivative.
Another key factor of this construction is that in the quantum theory every magnitude is obtained by integration of the classical degrees of freedom.This turns the classical manifold M C into a rather auxiliary object.Let Ψ be a prequantum state, and consider a translation by Ψ(φ x ) → Ψ(φ x + h x ).Only when h x ∈ H K ≃ H ∆ , the Cameron-Martin Hilbert space, we can reabsorb this into the Gaussian measure (3) with an integrable Radon-Nikodym factor.Hence, we can consider H ∆ as the only well-defined tangent vectors on the linear manifold M C for the quantum counterpart, since they are the only ones that leave the Hilbert space invariant.Nonetheless, in general curved spacetimes this is not the case, and evolution will not preserve L 2 (N ′ C , Dµ c ) unless the spacetime is static or stationary.We will deal with these issues in Section 6.

Ambiguity of the construction
We may consider the existence of another connection on B and the corresponding vacuum section Ψ0 it is related to.As we want the section to be square integrable, we can consider the existence of a function α ∈ C ∞ (M C ) satisfying for α : M C → C a function.Hence, sections of B should satisfy For the two covariant derivatives to coincide on a section of B, we obtain that As this should happen for any vector field X, we conclude that the function α must satisfy dα − i( θ − θ)α = 0 (15) Notice that as the modulus of the vacuum section is fixed to 1, the function must be a phase α = e −if .This implies that the measure does not change but the symplectic potentials are required to satisfy the gauge condition This is the freedom left from Equation (8) once we fixed the measure Dµ c .Notice that if the modulus of the function is equal to one the measure is not changed, since we are just choosing among the different phases of the vacuum state.

Prequantization III: The prequantization of observables
The covariant differentiation is precisely the basic tool to build a quantization mapping Q for the classical functions f ∈ C ∞ (M C ) acting on the set of sections of B. Thus, the operator Q(f ) becomes where X f is the Hamiltonian vector field associated to f .For the sake of simplicity we will take = 1 in the rest of the paper.When we consider the section Ψ 0 , and the corresponding function Φ, we can adapt Q to the new setting.Thus, ∇ X f must become a self-adjoint operator on the Hilbert space of square integrable functions, where Φ is contained.
The classical observables are chosen as functions on the classical phase space Remember that, being weakly symplectic, the equation above may not have a solution in general.But we will consider also that the first Malliavin derivative, see [1] , is defined for them and hence that F ∈ D 2,1 µ .Therefore the expression ) and H ∆ is the natural domain of ω so we can always find It follows that this (pre)quantization procedure Q meets the Dirac quantization (pre)conditions [7] Q1)

The polarization
The prequantization provides a Hilbert space too big to realize the quantum states since, in principle, Φ are functions with domain on the whole manifold while, in regular quantum theories, the domain must have half the degrees of freedom of the latter.
To deal with this discrepancy we must restrict the domain of the functions Φ to a Lagrangian submanifold of M C .Notice, though, that in this setting the concept of Lagrangian submanifold is not as straightforward as in the finite-dimensional case.First, we must remember that ω M C is defined, at most, on a dense subset of the vector fields of M C and therefore that the condition of (co)isotropy must take into account this fact.To bypass this difficulty we will introduce a polarization at the level of the one particle state structure introduced in the complex description of the classical phase space (6).Thus, we will choose a Lagrangian subspace P of the complex Hilbert space P ⊂ H ∆ on which the symplectic form ω M C is well defined.The quantum Hilbert space is given by: As, again, the space is linear, we can choose a symplectic potential −dΘ = ω M C adapted to the polarization such that Under this assumption the definition of the polarization simplifies enormously since, due to (11) we can identify the polarized Hilbert space as: Unitary dynamics is trickier to implement due to the fact that the quantization rule (19) may produce operators that do not respect the polarization.Moreover the situation is even more complicated if we ask the representation of the quantum operators to be irreducible adding a fourth condition to Q1-Q3 above: Q4) (Irreducibility condition) For a given set of classical observables {f i } I such that {f i , g} = 0 ∀i ∈ I implies g constant then if an operator A commutes with every Q(f i ) then A is a multiple of the identity.
Then it is known that no quantization satisfying Q1-Q4 exists.We can partially escape from those two problems choosing to quantize only a subset of classical observables such as the linear ones in Darboux coordinates (f x ϕ x + g x π x ).More complicated functions exhibit the usual ordering problem of quantization.We will deal with this issue in section 4.

Geometric quantization II: types of quantization
Roughly speaking, we can consider two kind of polarizations: real polarizations, where the linear space P is characterized by P = P * (which define what we call Schrödinger quantizations) and holomorphic ones for which P ∩ P * = 0 (which define what we call holomorphic quantizations).Intermediate situations are combinations of those two extreme cases, at least in the finite dimensional setting.As the holomorphic case is easier to handle in many ways so we will begin our discussion with it.

Choosing holomorphic tensors
For the holomorphic case the complex structure J M C provides a projection X → 1 2 (1 − iJ M C )X for the elements of the Cameron Martin tangent space X ∈ H ∆ , i.e., those elements in the tangent space which define integrable translations for the Gaussian measure Dµ c .As M C is a linear space, the polarization is taken to be The adapted symplectic potential in holomorphic coordinates (φ x , φy ) is obtained from the (non unique) Kähler potential The set of states of the quantum Hilbert space given by ( 22) is therefore the space of holomorphic (square integrable) functions Φ : M C ∼ N ′ C → C.These functions represent the excitation with respect to the vacuum state Ψ 0 (Equation 10), in such a way that the physical state corresponds to the section of the complex line bundle ΦΨ 0 , which are square integrable with respect to the measure Dµ c , or, equivalently, to the set of square integrable holomorphic functions if we take the section Ψ 0 as a reference: This is, by construction, a well-defined Hilbert space but it is not the most frequent quantum model in QFT.Instead, it is more common to represent the states as element of a bosonic (or fermionic) Fock space.We will discuss this point later.

Segal isomorphism: defining Fock states
From the properties of Gaussian integration, we know that the Segal isomorphism presented in [1] allows us to identify the vectors in L 2 (N ′ C , Dµ c ) with the vectors of the Fock space constructed on the space of one-particle states H ∆ .As the quantum states correspond to holomorphic functions Ψ ∈ L 2 Hol (N ′ C , Dµ c ), we can write: where ψ (n,0) xn correspond to the coefficients of the state Ψ with respect to the space of Wick complex monomials where S µ the integral transform introduced in [1].This expression allows us to recover the usual description of quantum states in the Literature.

Quantizing linear operators
At the same time, as the connection is also holomorphic, the quantization mapping Q is simple to compute and defines holomorphic linear operators on H Hol .In particular using (19) for (23), we verify that Notice that these operators correspond to the creation and annihilation operators, as it was to be expected a †x = Q(φ x ) while a x = Q( φx ).

Schrödinger quantization
Let us consider now the other most common example, the case of Schrödinger quantization.Now, the quantum states must depend only on the field variables, and not on the field momenta.This is achieved by a real polarization spanned by the momentum directions in M C that are obtained by a natural notion of momenta provided by the Riemannian metric h defined over the Cauchy hypersurface Σ.In any case, we must re-consider the construction above from a perspective based on real manifolds and not complex ones.Here we recover from [1] the basic ingredients of the real representation.For that matter we consider the realification of M C to be M F = N ′ × N ′ .In physical terms it is defined by the space of classical fields ϕ x and their corresponding field momenta π x .Those coordinates are tailored such that ω M F = −ω M C by convention is The complex structure on M F is expressed as With the conventions of (2) we have Notice that D is fixed once ∆ and A are known.Let K x y be the inverse of ∆ x y , i.e. ∆ x z K z y = δ x y then D = (iA t + 1)K(iA − 1).This complex structure, together with the symplectic form (29) recovers a Rie- which, in the coordinates above, reads: Due to the fact that it is Riemannian structure we derive also ∆ xy > 0 > D xy .We recover holomorphic coordinates with a change of coordinates, locally given by depends on the particular expression of the complex structure introduced.

The measure and the connection
The change from complex to real coordinates does not really affect the structure of the line bundle B required to define the prequantization: we construct identical bundles on the two identical base manifolds M C and M F with different coordinates; and for each bundle, we can define a connection.These two connections need not to be identical (since we can consider several different connections for each case), but they must have a curvature proportional to the symplectic form.
Identifying M F and M C also means that the measure on the space of fields can also be considered associated with the functional We conclude thus that the measure Dµ is defined for the real-fields description, and coincides, up to an scale factor, with the measure Dµ c constructed in the previous section for the complex description.
Again, we will consider a pre-quantum Hilbert space of sections of the bundle B, square-integrable with respect to the measure Dµ.We can also identify a vacuum section Ψ S 0 compatible with the connection in the sense of Equation (10).With this trivialization, the space of states becomes the set of square-integrable functions on M F with respect to the measure Dµ.
From the isomorphism discussed above between M C and M F , we can also conclude the equivalence of the connections on both spaces and the corresponding covariant derivatives.Strictly speaking, we should use different notation for the vector fields with respect to which we differentiate, since they are isomorphic and not identical, but we will use the same, to make the manuscript easier to read.We shall use then the notation of Equation (11) to represent the covariant derivative in the real polarization case, keeping in mind the fact that the only relevant vector fields are those tangent to the submanifold spanned by the field directions, as we will see now.

The real polarization
In the Schrödinger case the polarization is spanned by the momentum directions, i.e. the real subspace P S = Im(H ∆ ).We can consider an adapted symplectic potential with respect to P S satisfying This means that θ S must be a linear combination of the set {d φx } of "basical" forms.The coefficients are read from the identification of the Gel'fand triple and the canonical Liouville 1-form of T * N ′ .Writing it in terms of the variables we introduced in the previous section we obtain: Notice that we added the term −i φx K xy d φy (which is just an exact one form equal to the differential of the norm of the φx ) to the usual conventions because in this way 2Θ = θ S + θ M with From this point of view θ S is the restriction of Θ to P S directions with an scale factor of 2 to keep up with the conventions in changes of coordinates.In this way we can stablish the Schrödinger picture as a restriction of the holomorphic case to real directions.Nonetheless, this restriction must be taken with care.
The first conclusion of this analysis on the polarization is that functions on H S (i.e., the quantum states) must be functions depending on the submanifold N ′ with coordinates ϕ x rather on the whole M F , as it was to be expected.Indeed, as for any X ∈ P S , θ S (X) = 0, As the vacuum state Ψ S 0 must satisfy this condition, any other section in the set of square-integrable ones will be obtained as a product by a function which is also annihilated by the momentum directions.Nonetheless, the definition of a measure for this set of functions is not immediate, since the measure in the original scheme above was defined for functions depending on both the fields and their momenta.However, we can repeat the construction for a different Gel'fand triple defined as where H S ∆ is the subspace of H ∆ defined by the field states ξ R x .This triple defines a corresponding dual product ξ x , ϕ x = ξ x ϕ x which allows us to define a measure Dµ S by the functional Notice that, by construction for C S the functional defined in Equation (34).Again, Minlos theorem ensures that there exists a unique measure satisfying this condition and therefore we can define the set of (polarized) quantum states to be the space of square-integrable functions (with respect to the vacuum state)

Quantizing linear operators
We can readily quantize linear operators, such as the field operator ϕ x or the field momentum π x , whose action on the states Φ ∈ L 2 (N ′ , Dµ S ) will be: Analogously, we can also introduce creation and annihilation operators from the field and momentum operators as: These operators satisfy that and Q(π x ) = −iK x y a y −a †y √ 2 .Notice that the complex term of (37) is essential in order for the quantized momentum to be self adjoint and it provides the right prescription for quantization obtained in [8] from an algebraic rather than geometric quantization procedure.

Quantization in the Momentum Field space
Another real polarization is P M = Re(H ∆ ), in this case we could proceed as we have done so far with the Hilbert space associated to the measure Dµ c and the symplectic potential (38) but it soon leads to cumbersome expressions for the quantized operators.This is because the change of coordinates (32) is adapted to deal with the Schrödinger representation in the fields space, we will develop the Schrodinger picture in the momentum-field space.For now on we will denote it simply Momentumfield representation.In order to quantize this theory in a way akin to the one adapted to the Schrödinger picture we should respect the momentum in the change of coordinates we modify (32) to with ).This treatment, following the arguments below (32), modifies (5) and the measure considered for this case is Dν c defined by This also leads to a different Holomorphic representation in L 2 Hol (N ′ C , Dν c ).Following our steps with the Schrödinger picture we define Dν M as the measure choosing as characteristic functional for the momentum representation CM (ξ x ) = e It follows from this prescription that In the momentum field space creation and annihilation operators are These operate dually to creation and annihilation operators in the field space, they satisfy that 3.4 Isomorphisms between Holomorphic, Schrödinger, and Momentum-field pictures.
Once we have shown different representations of the same quantum field theory we must know to what extent they represent the same theory.As we will see, it turns out that we can relate every picture with unitary isomorphisms and therefore we can consider them to represent the same theory.

Holomorphic and Schrodinger pictures: Segal-Bargmann modified transforms
Slightly modifying the Segal-Bargmann transform defined in [1] we can stablish a unitary isomorphism In order to define this modified Segal-Bargmann transform we should deal with the extra 1/2 factor appearing in the characteristic functional C (42).To do so let us first define Ψ Hol = BSch (Ψ Sch ).Then, we choose where e if (ϕ x ) is a phase factor.This is similar to the definition in [9], with the addition of the phase factor.We include it because the isomorphism does not respect the operator representation of the creation and annihilation operators and therefore modifies the representation of the momentum and field operators.For instance if we take f = 0, the corresponding B0 transformation produces Thus, departing from the expressions of a x and a †x shown under (28) we can not recover the expressions of (44).To solve this problem we choose and obtain With this choice we preserve the expected creation and annihilation expressions for each picture.This may be interpreted as a nontrivial phase in the relation of the Schrödinger and holomorphic vacua as is explained in subsubsection 2.2.2.Indeed, if Ψ Hol 0 = 1 represents the vacuum of the Fock space in the holomorphic representation we obtain Ψ Sch 0 = exp(iϕ x (KA) xy ϕ y /2) as the representation of the vacuum in the Schrödinger representation.
Similarly for the momentum space we can stablish a unitary isomorphism with the antiholomorphic representation Holomorphic coordinates adapted to field or momenta representations are related via a holomorphic transformation Using the relation, D = (iA t + 1)K(iA − 1), it is easy to see that is an isometry relating holomorphic and anti-holomorphic functions, with inverse Notice that the transformation acts non-trivially over creation and annihilation operators because Using this mapping we are in disposition to define a Fourier transform where B is the regular Segal-Bargmann transform and B its antiholomorphic counterpart.We skip technicalities about factors 1/2 in covariances that do not affect the discussion.Then, by construction, we get Therefore, ( 43) is transformed into (49) and viceversa.Hence, the quantization mapping Q is respected by the Fourier transform.But this implies that creation and annihilation operators mix, in general, in a nontrivial manner under F: Notice that, from this expression, it is immediate that the mixing depends on the properties of the complex structure.Indeed, whenever A x y = 0 we get D −1 = −∆.Hence, in those cases there will be no mixing between creation and annihilation operators under the Fourier transform.

Quantization of the algebra of observables of a field theory
The holomorphic representation of quantum field theory is particularly well behaved to describe a quantum field theory over a Cauchy hypersurface Σ.In sections above, we showed a detailed account of this representation as well as its relation with several other representations of QFT.Thus, the space of pure states for this theory is considered to be L 2 Hol N ′ C , Dµ c , for a suited choice of Gaussian measure Dµ c .In this space the Wiener Ito decomposition is particularly simple and Skorokhod integral amounts to simple multiplication by φ x , see [1], which is also reflected in the simple form of creation and annihilation operators of (28).In this section we will present a quick summary of quantization in the holomorphic representation.In this case we will use the new tool of reproducing kernels presented in [1] to deal with ordering problems in the quantization of nonlinear operators.

Weyl quantization
The quantum picture is not complete until we prescribe a mapping from classical functions to quantum operators.This is a quantization that assigns a quantum observable for a suitable class of classical observables.Of course, there is an ambiguity on this prescription usually addressed under the name of ordering problems resulting from the nontrivial commutation relation [a x , a †y ] = ∆ xy .The first choice that we make is called Weyl quantization and denote it Q W eyl .It is defined as follows.Firstly, any quantization map must coincide over lineal functions of M C , this is Then the Weyl quantization mapping assigns to each monomial φ n φm an averaged assignment of every possible order.We can then define the Weyl quantization to any polynomial classical functional inductively.For linear operators the quantization procedure must coincide with geometric quantization For higher order monomials, we proceed symbolically as From this symbolic expression it follows that the action of Skorokhod integrals with covariance ∆ xy /2 is cast into Thus this quantization procedure is well suited to quantize classical functions belonging to a Hilbert space O cl defined by Indeed, for functions in this Hilbert space we can consider a chaos decomposition Now, quantizing the functional just constructs the operator with the same chaos coefficients.For one such monomial we obtain It is straightforward to obtain the action on the total classical function by linearity although we must proceed with care because the resulting operators are, in general, unbounded.
An important property of Weyl quantization is the transformation complex conjugation into involution Therefore, real classical functions of O cl are quantized into Hermitian operators.The machinery of reproducing kernels is better suited to deal with the finer details of this quantization we explore it in the next section.

Weyl quantization from reproducing kernels
We can reexpress Weyl prescription, acting over the dense subset of Hida test functions (N C ) ⊂ L 2 Hol N ′ C , Dµ c , using reproducing kernels as explained in the first part of this series.This is, let Ψ ∈ (N C ) and f x ∈ H ∆ then we can define the action of a symbol f on the state Ψ as: This quantization prescription is particularly simple for the algebra trigonometric exponentials T (N ′ C ), this is for E χ = exp i χx φ x + iχ x φx we have We know from [1] that T (N ′ C ) is dense in O cl and therefore we extend Weyl quantization to the whole space by linearity.

Moyal product
To complete the picture of Weyl quantization we need to know how composition of operators is translated as an operation in the space of classical functions.For that matter we define the Moyal star product, and denote it ⋆ m , as the one that preserves operator composition after quantization, this is for an appropriate subset of classical functions The space W must be an algebra with the ⋆ m multiplication and then we denote it the algebra of classical Weyl quantizable functions.We will characterize this algebra below endowing it with a norm.Using (70) together with (69) it follows that E ρ with the Moyal product is a representation of Weyl relations These Weyl relations will play a crucial role in our discussion about canonical quantization, they represent the canonical commutation relations in an exponential form.
Using the identity e χ x ∂ φ x F (φ x , φx ) = F (φ x + χ x , φx ) over trigonometric exponentials E ρ , and because of the density of T (N ′ C ), we can rewrite Weyl relations and extend the Moyal star product of any two functions F, G ∈ W ⊂ O cl with an integral representation This expression is tightly related with the one in the covariant formalism of QFT [10] .

The algebra of Weyl classical quantizable functions
Our goal in this section is to study the algebra W and to prove that it is indeed a subset of O cl .To do so we start by noticing that a dense subalgebra should be (T (N ′ C ), ⋆ m ).Then we must topologize this algebra in such a way that the ⋆ m product is continuous.The natural way to do this is, since ⋆ m is a reflection of the operator product under the map Q W eyl , the topology is the coarsest one making this map continuous.Thus the topology is the one induced by Q W eyl with the operator norm in the image.In more practical terms, we endow W with a norm such that for F ∈ W , It is indeed a norm because Q W eyl is linear and respects involution.We start by studying the operator norm of the generators of the algebra, i.e. the trigonometric exponentials.Then (68) allows us to compute the operator norm in the following way Here we have used the relation This very important result means that trigonometric exponentials are quantized into unitary operators of L 2 Hol N ′ C , Dµ c .In order to compute the operator norm of a generic element of T (N ′ C ) we must understand the action of Q W eyl E χ over a vector Ψ.For that matter it is convenient to decompose trigonometric exponentials in terms of holomorphic (and antiholomorphic) coherent states K χ = exp ( χx φ x ) in the following way Quantized coherent states act in a straightforward way over the vector Ψ then we can compute for later use This expression allows us to compute the Weyl induced norm of a generic element F ∈ T (N ′ C ).By definition it can be written, in general in a non unique manner, as It is easy to see that the norm in O cl of F is simply computed by means of the characteristic functional (65).Then using (77) and the characteristic functional of Dµ c defined in (5), the following equality holds This expression and (73) provides us with the bound F cl ≤ F W eyl and, by density, this is valid for every F ∈ W .The equality does not hold in general.From this fact we have that W ֒→ O cl is a dense subspace such that the inclusion is continuous.We also have that is a C * algebra with complex conjugation as involution.In this fashion Q W eyl is a C * -isomorphism between W and Ŵ = Q W eyl (W ).The latter is the von Neumann algebra of quantum observables which is a representation of Weyl relations (71) as a closed subalgebra of the algebra of bounded operators Ŵ ⊂ B L 2 Hol N ′ C , Dµ c .An important remark to notice is that that in previous sections we obtained an expression for the quantization of linear functions of O cl .The result are linear combinations of creation and annihilation operators which may be unbounded.This means that, even though we can enlarge Q W eyl to make sense over the whole space of classical functions O cl , the norm .cl just provides a lower bound on the operator norm and the Moyal star product ⋆ m is not well defined outside W .In simpler terms, even though we could quantize the whole O cl we cant treat it as an algebra.This fact, even though may seem abstract a priori, is the source of the renormalization program in which, as a first step, the theory should be regularized to ensure that products of operators do not led to divergent integrals.We will not deal with this feature in this work and refer elsewhere [11] for further information in the standard renormalization program.
Finally let us provide an upper bound to F W eyl .In general, it is difficult to obtain an explicit expression of it.Nonetheless, by Hölder inequality we obtain that for an element written as This is the best that we can do, in turn it leads to a criterion of convergence for a function of O cl to be in W that may be used in a regularization program.

Wick Holomorphic quantization
The most common ordering in QFT is not Weyl quantization, but Wick quantization, which assigns to regular monomials the so called normal order prescription.This prescription is designed to quantize regular monomials into normal ordered products of creation and annihilation operators that guarantees null vacuum expectation values.
In order to find an explicit expression for this operation we introduce the Wick operator W ∆ 2 for general functionals.Its action on a monomial must re-order the terms to write the resulting expression in normal ordered.From the properties of Skorokhod integrals [1], acting on the constant function 1, it is straightforward to obtain the expression of this operator as Notice that we use the same symbol as for the case of the Wiener Ito chaos decomposition in [1], since it is the action of these operator on the suitable pair of fields what produces the suitable polynomials.As an operator, it is invertible and Wick quantization is defined from Weyl quantization as . We won't cover this case in full detail, the results of the previous section can be related to this case just by studying the properties of W ∆ 2 .Our main interest in this section will be to find a correct star product ⋆ w to find a representation of the Wick relations.Writing Q W eyl in an integral kernel representation Wick star product follows from its definition Then a representation of the Weyl algebra is obtained by the Wick ordered trigonometric exponentials This procedure is often used in quantum field theory because classical functions representing Hamiltonians are expressed in terms of regular monomials.Nonetheless, the space of classical functions possesses better analytical functions in Weyl quantization and the difference between a Hamiltonian written in terms of regular or Wick monomials often amounts to an infinite constant that is physically interpreted as a trivial shift of the ground energy without physical meaning.

Algebraic quantum field theory: Fock quantization
So far we used geometric quantization to find a particular class of representations of a quantum filed theory.In those cases we obtain a Hilbert space as representation of the class of pure states and the algebra of observables as the image of a quantization mapping from a class of classical observables.In those cases, though, geometric quantization comes with the ambiguity associated with the choice of polarization then we are forced to study separately holomorphic, Schrödinger and fieldmomentum representations and find unitary isomorphisms relating each case.The reason why every representation is equivalent has its roots in the abstract study of the C * -algebra of quantum observables, a program called Algebraic Quantum Field Theory (AQFT).We present here a quick summary of its ingredients in order to introduce the most common QFT representation found in introductory books of the subject, i.e.Fock quantization.We refer to [12,13] for a thorough analysis on the subject.

The C* algebra of quantum observables
Our goal in this section is to find Weyl relations, that correspond to the the abstract axioms that characterize the C * -algebra of quantum observables, from the geometric narrative that has been our guideline in this work.Then we find how AQFT recovers the space of (not necessarily pure) states form this abstract setting.
The first problem that we encountered in order to quantize a classical theory was how to model our phase space T * N where we know that there is a canonical symplectic form.So far, and in the first paper, our approach was to model this space treating both base and fiber as distributions then M F = N ′ × N ′ .This choice, even though hardens the treatment of the classical theory itself, suits the needs of geometric quantization for quantum field theory.In this section we are interested in AQFT thus we start by a different choice of model space for our classical theory.In our model the manifold of fields is M A = N × N with coordinates (ϕ x , π x ) and it is endowed with a symplectic form Notice the difference with (29) because here the coordinates are functions and not densities of weight one.At this stage we can write Weyl relations in a coordinate free manner.Let ρ, α ∈ T M A be vector fields, then the Weyl algebra is a C * algebra with generators R(•) that fulfil Weyl relations Notice that particular representations of this algebra are given by ( 71) and (83).These are particular examples of a representation of Weyl relations in the algebra of bounded operators2 of a particular Hilbert space.
Once the algebra is known we need to recover the space of states of the theory.To carry on with that study we use the GNS construction.Our starting point is a particular state of the algebra, i.e. a linear functional ̟ : A → C which is positive definite (i.e.̟(aa * ) ≥ 0) with ̟(1) = 1.With this state we can define the GNS representation of the C * algebra Theorem 4.1 (GNS construction) Let A be a C * -algebra with unit and let ̟ : A → C be a state.Then there exist a Hilbert space H , a representation π : A → B(H ) and a vector Ψ ∈ H such that, Furthermore, the vector Ψ is cyclic.The triplet (H , π, Ψ) with these properties is unique (up to unitary equivalence).
In this fashion we recovered the usual prescription of quantum theories where we look for a representation of the algebra of observables as a subalgebra of the space of bounded operators acting on a separable Hilbert space.In this fashion the GNS construction ensures that is enough to restrict the study to the physically meaningful states ̟ : A → C.This is, we can study the representation π(A ) ⊂ B(H ) in our physically relevant setting instead of every possible abstract C * −algebra.
We can study many physical aspects of quantum theories by asking properties over ̟.For example the quasi free or Gaussian condition ensures that the states are completely determined by two point correlation functions and the Hadamard condition ensure that causality is preserved [12].In our case we are interested in the theory in a Hamiltonian form and therefore such features are part of the particular Hamiltonian that describes the theory.
Our goal in this section was more modest than the general aim of AQFT as a whole [12], we only pretend to set the groundings to understand the equivalence of the same theory described in different terms.So far we studied quantization mappings Q to stablish particular representations of Weyl relations.This is our C * -algebra of quantum observables always acted over L 2 Hol (N ′ , Dµ c ) or similar.By GNS we now know that this Hilbert space, that seemed the central object in geometric quantization, is in fact irrelevant since it can be reconstructed by means of a particular state ̟.Thus, to recover Holomorphic quantization we set R(α) = Q W eyl (E α ) as our generators of the C * -algebra of observables, then we can reconstruct the Hilbert space knowing that the Hilbert space vector representing the vacuum state is given by the constant function 1.To recover the state of the algebra we look at the GNS construction and by (77) we obtain This construction may seem to involved for our purposes so far but it is crucial to understand the equivalence with Fock quantization.

Fock space representation
In this section, we will consider how the representation of the C * algebra can be chosen for the Hilbert space to be the usual Fock space of QFT.Besides, we will identify the role of the different geometric structures introduced in the manifold of classical fields in the representation.
In a Fock space representation we start by introducing a complex structure compatible with (84), for simplicity we choose the same entries of (30) adapted to the new setting The sign convection is switched to render positive definite the induced Riemannian metric

We can use it to define a Hermitian metric on M
Moreover, by linearity (M A , h M A ) is completed to a complex Hilbert space that we denote H 1ps the Hilbert space of one particle states.We will use this Hilbert space as an starting point to represent our quantum filed theory.The next step is to notice that we must allow states with an arbitrary number of bosonic particles, thus we represent our theory over H F ock = ΓH 1ps , the symmetric Fock space.
With this structure at hand we use a change of variables dual to (32) to obtain, in coordinates, h M A = ∆ xy dψ x ∧ d ψy .In those coordinates it is clear that H 1ps = H ∆ where the latter is the completion of N C with the Hermitian form h M A .
An element of the Fock space is written in coordinates in a very straightforward manner as Ψ = (ψ (0) , ψ (1)  x , ψ (2) xn .Annihilation and creation operators are also easily written as Among all the possible choices we pick 1 ∈ H 0 ∆ as the cyclic vector of the GNS construction because we want to interpret it as the vacuum state of our theory.The last ingredient is the state, a Fock generator of the Weyl algebra is [8,13], in holomorphic coordinates Using BCH formula with [a x , a †,y ] = ∆ xy we put annihilation operators to the right of creation ones, then the Fock state acting on the generator, which is the vacuum expectation value of it, is easily computed by Thus ̟ Hol of (86) and ̟ F ock act in the same way over their respective generators of the Weyl algebra and, because of the GNS construction, they are related by a unitary isomorphism and, as such, they represent the same theory.The same is true for the Schrödinger representation [8].Notice that a unitary isomorphism at the level of the Hilbert space is not enough to stablish the equivalence of the theories.
Indeed we already knew that Fock space coordinates (90) are the coefficients of the chaos decomposition of the holomorphic representation under the Segal isomorphism I : L 2 Hol (M C , Dµ c ) → H F ock .But to stablish the relation in full we must also preserve the C * −Algebra of observables.In this case both theories are equivalent if we quantize the C * −algebra of Weyl quantizable functions (W, ⋆ m ) with the mapping Q W eyl studied on section 4.
In any case it is important to remark thar Fock quantization is fundamentally different from that of holomorphic quantization in that relies upon the notion of one particle space structure from which the Fock space is constructed.Kähler structures in this setting are also fundamentally different from those of holomorphic quantization.In holomorphic quantization this structure is densely defined in the domain of the wave function Ψ(φ x ) while in Fock quantization is defined dually over the coefficients of the one particle states.
The relation between both pictures is also more involved than previously expected.In this section we made the design choice of writing Fock operators and vectors in Holomorphic coordinates, this is because the explicit isomorphism given by the Segal isomorphism is only obvious under the change of coordinates (89).The real power of Fock construction relies in its coordinate free nature depending on the Kähler structure of (ω, µ, J) M A .We will keep this section simple ath the cost of blurring its real power.Nonetheless, this feature is well known in the literature of QFT in curved spacetimes, we refer elsewhere for a thorough discussion [13] 5 Geometric quantum field theory In this section we present a description of a QFT in purely geometric terms based on Kibble's program of geometrization of quantum mechanics [14,15].In that program given a quantum theory, provided by a Hilbert space of pure states H and its algebra of observables Ŵ , we describe each ingredient in geometrical terms with a holomorphic manifold of pure states P and a Kähler structure (G, Ω, J ) P for the former and the subset of quadratic functions W 2 (P) for the latter.
To model the space as a manifold at this second quantized level we choose a space for P that, in our case, will be Hida test functions.Because we do not deal with a particular theory we do not know yet the Kähler structure, thus the discussion is kept in in full generality.
To exploit the geometric insights of this approach we developed a set of tools based on white noise analysis [16] that will allow to write down some expressions in convenient sets of coordinates.

Geometrization of quantum quantum field theory I: The manifold of pure states
Here we present, and adapt to our proposes, a summary Kibble's approach to the geometrization of quantum mechanics [14,15].We want to characterize the description of QFT in terms of tensorial objects defined on the manifold of pure states P. It turns out that P describes a quantum theory by means of a Kähler structure (G, Ω, J ) P .
Later we will use this approach to exploit some properties of Holomorphic quantization at the level of P. To unleash the full power of the formalism we describe the theory with white noise analysis tools that allows to write tensors in closed forms for particular sets of coordinates.

Quantum fields in the geometrized setting: The quantum Kähler structure
Due to the infinite dimensional nature of the problem the model space used to define the manifold structure is of crucial importance.For a well posed geometric framework we will choose to consider as reference a particular Fréchet-Nuclear space the set of Hida test functions N = (N C ).
As we did above, we will consider a suitable rigged Hilbert space As a result, the Hilbert space of the states of quantum fields L 2 Hol (M C , Dµ c ), can be densely modeled as a C ∞ -manifold, considering N as the model space.
As the model space for our manifold is the space of test functions, we can introduce a system of coordinates associated to them on the space of quantum fields.In this case we will denote test functions with a superindex Ψ φ ∈ N while Φ φ ∈ N ′ stands for distributions.This convention is reverse to that of classical fields, see (1) of [1], but turns out to be more convenient in this case because we stick to the geometric convention and test functions will usually be treated as coordinates of the manifold.Of course pointwise meaning holds for test functions while it is lost in the distributional setting.
We endow the manifold with a Hermitian form trivially by using the scalar product of the Hilbert space.Since the topology of N is finer than that of L 2 Hol (M C , Dµ c ) we have that its scalar product , µc , inherited by the embedding, is a sesquilinear positive definite and continuous bilinear form on N .Moreover this form is weakly non degenerate.This means that only the zero vector has zero norm also in N but that the map , µc : N → N ′ that maps every element Ψ φ ∈ N to Ψ φ , • µc ∈ N ′ is not surjective, although it is clearly injective.
In the following we will denote as P the manifold of quantum pure states.As a linear complex manifold, it can be trivially treated as a real manifold endowed with a Kähler structure (G, Ω, J ) P related to the Hermitian tensor above.To recover the quantum theory we follow [14] and choose a preferred point in |0 ∈ P that represents the vacuum and consider the tangent space of this point T 0 P ≃ N .In linear spaces, that is the case that we are treating here, this distinction is blurred by the identification of P with T 0 P.In the following we will identify both spaces, when it is possible, to simplify the discussion.
Notice that in this case the complex structure is completely determined by the definition of the space of quantum states and we do not have freedom to select it as in the case of the manifold of classical fields.This is because we are describing a given theory in geometrical terms.Nonetheless the picture is incomplete without the representation of the operators.We will deal with this aspect, together with the dynamics, later on.In the next section and below we will develop some important tools to express the geometrization of this section in concrete sets of coordinates.

Holomorphic quantization in the geometrized setting
In this section we address the geometrization of the space of pure states P of the Holomorphic quantization procedure.To do so, we look for an efficient way of describing tensors over the manifold P. Thus we follow [16] and choose the white noise analysis to describe our tensors.The main idea of the analysis is the following: in the case of regular test functions and distributions the Riemannian metric h over Σ provides a canonical way of identifying the duality through the Gel'fand triple To generalize this triple to the second quantized setting we use the white noise measure Dβ, defined in [1] by the characteristic functional We choose white noise because it is the immediate generalization of (95).The white noise Hilbert space L 2 (N ′ C , Dβ) has as Cameron-Martin Hilbert space L 2 (Σ, dV ol h ).This means that the White noise space is nothing but the bosonic Fock space constructed from it as the natural measure for the rigging.With this choice we obtain the natural triple identifying functions with distributions In this rigging βφ ,σ is the bilinear form that identifies test functions Φ σ with distributions Φ φ = βφ ,σ Φ σ .In this notation distributions are antiholomorphic functionals.It also admits a weak inverse β φ,σ such that β γ,σ β σ,φ = δ γ φ is the evaluation mapping, continuous for Hida test functions as can be shown using the tools of reproducing kernels [1].
In a nutshell, we will use βφ ,σ and β γ,σ as the auxiliary bilinears to raise and lower indices of our operators and tensors.
Let Ψ φ be a test function, representing a state of the manifold P that is provided by the phase space of holomorphic quantization.This notion of canonicity suggest that we must write every bilinear form or operator as the action of an integral kernel with respect to the white noise.In this manner for Fock quantization we have a complex structure is

and the Hermitian tensor
Where the bilinear ∆ σ,φ is easily written as a bivalued kernel with Following our discussion about reproducing Kernels in [1] these expression, of course, do not have a pointwise meaning since they are bi-valued Hida distributions.
It is convenient to describe the manifold with real coordinates.Let Ψ φ = 1 √ 2 ( Φφ + i Πφ ) such that the set of coordinates Φφ Πφ , are real in the sense that they are holomorphic functions with real coefficients in its chaos decomposition.This can be written as Φ(φ x ) = Φ( φx ).In these coordinates To write down a dynamical system is also interesting to describe the Poisson bracket in this context.To do so we define, associated with these coordinates, the Wirtinger derivatives We then obtain that the inverse bilinear (K∆) σ φ = δ σ φ is K φ,σ = exp(σ x K xy φ y ).This fact can be proved by, recalling the conventions of (2), using the computational rule thus the Poisson bracket is Let F ( Φφ , Πφ ) and G( Φφ , Πφ ) be well behaved real functions over N then the bracket action is The space of Hida test functions is nuclear and Fréchet, among other convenient properties suited for geometry [17], this implies that the notions of Gateaux and Fréchet derivatives coincide and there is no ambiguity in the definition of this bracket.

Antiholomorphic quantization in the geometrized setting
The tool at hand to express the antiholomorphic picture is the Fourier transform of (55).For simplicity, we will identify the points of the linear manifold P with its tangent vectors.We express the Fourier transform with an integral kernel as We can also express the inverse as Expressions with matricial kernels can be easily handled by noticing that the computational rule (103) also holds for those kind of exponents, which leads to an algebra of exponential kernels described in Table 1 in the appendix, and its action on a vector is given by Following these rules it is easy to see that Where we define we define the transpose of an operator as Using F as a change of coordinates we see that this is enough to turn (101) into the appropriate expressions for the antiholomorphic representation even if they are represented by holomorphic functions.We will denote the coordinates of the antiholomorphic representation with Φσ Πσ = Fφ σ Φσ Πσ .

Geometrization in Darboux second quantized coordinates
Our approach in this section is to craft a notion of Darboux second quantized (s.q) coordinates that provides some sense of canonical coordinates in this more involved framework.The choice, consistent with the discussion of the previous sections is to choose as a canonical form for the symplectic structure the one corresponding to the white noise Hilbert space which is a direct generalization of Darboux coordinates in (84).Thus the canonical form is We can simply find a system of Darboux s.q.coordinates with a change of variable Φφ Πφ = C Φ φ Π φ that provides a complex structure Finally, this symplectic structure induces a Poisson bracket over functions in Darboux s.q.coordinates { that over smooth real functions acts like This set of coordinates will be important later to discuss the particular expression for the Hermitian of an operator.

Geometrization of quantum field theory II: The algebra of observables
Once we described the space of pure states of the theory in the geometrized setting and we explained how to describe concrete sets of coordinates using white noise analysis, the remaining task is to describe an algebra of observables that fulfills Weyl relations and is unitary equivalent to the ones described so far.As before, we start in full generality and unveil several interesting properties in particular sets of coordinates using Fock quantization.

Quantum observables in the geometrized setting: quadratic forms
We must address first the issue of how to cast into this formalism a previously existing algebra.The description of the space of pure states P that we want to achieve is geometrical in nature then it follows that a description of the algebra should follow the next three conditions.
• The algebra of observables is a subset of the space of C ∞ (P).
• It contains a dense representation of the Weyl algebra (85).This implies that the product of observables cannot be the pointwise product of functions.
• The Hamiltonian vector field generated from an element of the algebra F given by X F = {•, F } P is such that its flow preserves the Kähler structure.This is L X F h P = 0.This implies that quantum observables generates symmetries that cannot transform the structure (G, Ω, J ) P that defines the quantum phase space.If we want to change this structure with respect to a given evolution it must be introduced as a change in the Kähler manifold from an external source such as an external gravitational field.
To achieve this,we will be considering the sesquilinear forms defined in holomorphic s.q.coordinates as for In the following, we will be using a notation similar to the one used above to represent these objects.In formal terms, we will write where Ĝσ γ represents the integral kernel representation of the operator Q W eyl (G).The product of sesquilinear forms follows directly from an extension of the Moyal product With these quadratic functions we can find a C * -isomorphism from (W, ⋆ m , • W eyl ), described subsubsection 4.2.1, to the set Endowed with the product already described and a norm Then it is clear that W 2 (P) is a representation of Weyl relations.
The most interesting property of this product is that trivializes the Lie algebra representation of the commutator of quantum operators.It can be readily checked Where the l.h.s. of this expression only depends on geometrical properties of the manifold, i.e. on its Kähler structure.In summary, a dense representation of Weyl relations W 2 (P) is contained in the algebra of sesquilinear forms endowed with the product * m .Nonetheless, it is too large to meet the one last condition.This is because for a function f A its Hamiltonian vector field X f A does not satisfy L X f A J = 0 in general.To fix this issue, the algebra of observables must be the set of functions associated with selfadjoint operators, which is a subset of the set of quadratic polynomials on P: In our notation this is ∆ φ,σ Âσ ρ K ρ,γ , = Âγ φ , which is equivalent, at the level of classical functions, to Ā = A ∈ O cl .Thus the algebra of quantum observables might be seen as the quantization of real classical functions.In addition, those were the legit observables of a classical theory in the first place.The condition of self-adjointness ensures that for f A ∈ F 2 (P) we have L X f A J = 0.This fact, together with the conservation of the symplectic structure under the Hamiltonian flow, ensures the condition L X f A h P = 0.
In summary F 2 (P) is the algebra of observables of the quantum theory.We can endow it with a product * that leaves F 2 (P) invariant, introducing the Jordan product is the internal product to the algebra of observables.

Hermitian operators in real coordinates
The † operator in different systems of coordinates has different expressions.In general, the Hermitian of an operator is given by the symplectic form, such that Ω(•, A † ) = Ω(A, •).It is for this reason that the simplest expression of the † operation is in Darboux s.q.coordinates.Here we will distinguish Hermitian operators A † of adjoint operators.We call A † the adjoint of A only if [A, J ] = 0 otherwise it will mix the holomorphic and antiholomorphic subspaces not respecting the definition of the whole scalar product.Thus in real coordinates Then (120) is Our goal in this section it then to describe the † operator.To distinguish the canonical case of Darboux s.q.coordinates we denote the adjoint operator with respect to the white noise canonical symplectic form by .In this operation the ǫ matrix of (101) implements the anti-symmetry of the symplectic form We can represent, using this definition, the adjoint operator in other systems of coordinates.With a change of variables like (111) we get an expression for the holomorphic † H and antiholomorphic † H cases where (D −1 ) φσ = exp(−σ x D −1 xy φ y ).In particular, it is interesting to notice that the Fourier transform is not internal to any of the two systems of coordinates and therefore we obtain F † = KF D. and it is immediate to see that

The algebra of observables of Fock quantization in the geometrized setting
Our aim now is to describe a quantization mapping Q in this setting.In Fock quantization, once we choose the Hilbert space of pure states that we can describe as the holomorphic representation in coordinates (90), the representation of Weyl relation stems from the definition of creation and annihilation operators fulfilling the canonical commutation relations.Thus we will limit our discussion here to the linear case.It turns out that creation and annihilation operators are represented by the Skorokhod integral and its adjoint (27), based on the Malliavin derivative.In particular, for a generic system of coordinates we can define two adjoint operators D †,x and D x that in holomorphic coordinates are just Dx = ∆ xy ∂ φ y and D †,x = φ x (127) they are indeed adjoint and not only Hermitian because the complex structure in those coordinates is equivalent to multiply by the constant matrix ǫ.
Another desirable property for our quantization mapping is to map the imaginary unity to the complex structure Q(i) = J .To see this condition in place we must contract the operators with a direction of χ ∈ T 0 M A ≃ M A , the tangent space of the manifold on which the Hermitian structure (88) is defined.In holomorphic coordinates for M A , according to this quantization rule, it must be expressed as χx = 1 √ 2 ( λx + J ηx ), therefore we have where the bilinears are the ones of the Hermitian structure (88).Then we can treat them as the creation and annihilation operators even though they act on the antiholomorphic part as well.
It is also interesting to express the direction χ in canonical coordinates (λ x , η x ) related with the holomorphic coordinates by the change of variable (89).Lets particularize our example to the case of the holomorphic representation.Operators D, D † H contracted with a direction (λ x , η x ) (and its conjugate respectively) are given by Looking at the (32) we can see that the first coordinate selects directions of the ϕ x canonical coordinate while the second selects the direction π x .Thus, in this system of coordinates we get In this way we get [λ x φx , η x πx ] = ǫλ x δ xy η y which are the canonical commutation relations.
It is interesting to compute the Fourier transform for these operators.Let By computing the operators in the antiholomorphic picture with the corresponding dual to the change of variables (45) we would get the operators D, D † H , it follows that.D(λ x , η x ) = ǫ D(λ x , η x ), and D(λ This is, if we switch from the holomorphic to the antiholomorphic picture with the Fourier transform the roles of λ and η are exchanged and the duality is adapted to the covariance of each picture as described in section 3.3 up to an imaginary global factor ǫ. To complete the quantization procedure the only remaining ingredient is the choice of ordering.The discussion is completely equivalent to that of section 4 and therefore we will skip it in this section.Once this is considered, the algebra of quadratic observables (117) is just given by (122).

Time Evolution: The (modified) Schrödinger equation
At this stage we are interested in postulating a dynamical evolution for the system.In regular quantum mechanics this evolution is just the Schrödinger equation, but in dynamical spacetimes the situation is more complicated.Our task at this point is to derive a dynamical equation from first principles, consistent with our geometrical viewpoint, and constrained to reproduce the Schrödinger equation in the situations in which there is no ambiguity in its derivation.More precisely, we must match the unmodified Schrödinger equation in stationary spacetime.
Our starting point is the a priori assumption that we can tell kinematics from dynamics.This is just the requirement of a consistent definition of the manifold of pure states P, its Kähler structure and its algebra of observables in a foliation Σ t independent way.This distinction is crucial when we want to describe the system in a dynamical spacetime with backreaction of the quantum degrees of freedom on the classical spacetime.
In this section our point of view is that of a parametric theory, in that case the parameter t is treated as a label.This label describes a foliation of spacetime into leaves generated by a different the embedding of the Cauchy hypersurface Σ at each value of t.This is expressed as a parametric dependence of the objects defining the geometric structure at each leaf, the Riemannian metric h ij (t), its associated momenta π ij (t) and lapse and shift functions N (t), N i (t).In our case the Kähler structure depend on those objects and therefore it will acquire a parametric dependence (Ω(t), G(t), J (t)) P .The particular dependence is irrelevant for our discussion in this section, we will show a particular example in section 7.However, results obtained in this section from this approach are somehow limited.To understand the source and nature of the dependence on the time parameter we should go beyond the parametric theory and study the structure of admissible h ij and π ij providing the Hamiltonian structure of General Relativity.This study is the subject of [2] and will not be covered here.

Covariance of the time derivative
The first aspect to take into account is that, because the full Kähler structure (Ω(t), G(t), J (t)) P depends on time, as well as the quantization procedure itself Q(t), we can not assume explicit time independence for every set of s.q.coordinates and as such we must define a covariant derivative for the time parameter ∇ t that preserves the structure and behaves well under changes of s.q.coordinates.
Recall from geometric quantization that the points of the manifold of pure states Ψ represent different sections Ψ s of an Hermitian line bundle π M C ,B : B → M C , associated with a U (1)-principal bundle on the manifold of classical fields φ x ∈ M C .
In this scenario the manifold of classical fields must be enlarged to accommodate time in the base of the bundle.In concordance with the picture in which M C accounts for the Cauchy data of a theory in a globally hyperbolic spacetime, the bundle incorporating time in this theory is π M C ,B:t : B t → M C × R. The covariant derivative in the tangent directions of M C has been discussed in regards to geometric quantization.In this section we must study the covariant derivative in the tangent directions to the time parameter that we denote ∇ t .
Recall also from geometric quantization that we can describe general sections as functions because we factor out an special section Ψ 0 such that Ψ s = Ψ 0 Ψ and the nontrivial structure of the bundle is rephrased in terms of the quantization procedure Q.In that construction we use the twofold nature of line bundles.On one hand they are vector bundles and, as such they are represented as vectors but, on the other, they are also principal bundles using the multiplication of the fiber3 C\{0}.
When we treat the line bundle as a real manifold the twofold nature of the line bundle is lost.In that context we substitute the fiber of the complex line bundle B t by a two dimensional real space keeping only the vector bundle structure.We denote this bundle π M C , Bt : Bt → M C × R.This has implications in the decomposition of the section Ψ s = Ψ 0 Ψ.We choose to describe functions Ψ with vectors Φ φ Π φ but the multiplication by the vacuum section Ψ 0 , that is regarded as the multiplication by a phase, as is implied by (15), should be recovered from a principal bundle structure.Then Ψ 0 is part of the associated O(2)-principal bundle must be represented by a section Ψ0 of the B-associated O(2)-principal bundle.
The quantization procedures introduced in the previous sections are crafted to be blind to this subtlety and treat the points of the manifold of pure states as simple functions.This is nonetheless different in the case of the time derivative.The quantization procedure depends itself on time thorough the dependence of the Kähler structure.We can study its interplay with the covariant time derivative for linear operators using the prequantization procedure (19).Consider thus the connection defined on the product M C × R by addition of the pullbacks of the connection one-forms with respect to the canonical projections.We will use ∇ to represent the covariant derivative with respect to this new connection.By introducing the curvature tensor R In order to get a quantization that assigns the same operators independently of the time parameter that we choose it would be desirable to get but this condition does not hold in general.This condition is further explored in [2].This kind of connection Γ appears from a different bundle structure in [2].As a consistency criterion, we ask for the vacuum section to be parallel transported by this connection Sections Ψ s without dynamical evolution, that we will discuss later on, must be also parallel transported and therefore

The connection in Holomorphic coordinates
In realified holomorphic coordinates Φφ Πφ the complex structure is constant and as such the regular time derivative preserves it.In order to preserve also the symplectic structure, and therefore the whole Kähler structure is enough to respect the adjoint operator.Nonetheless, this condition does not select a unique covariant derivative.To choose a criterion, we will assume that the choice of the holomorphic and antiholomorphic representation is, in last instance arbitrary.Then, with the relation (132) we must write down an expression that, acting on creation and annihilation operators, transforms properly under the action of the Fourier transform.All things considered, let O φ σ be an operator and Ô = FOF −1 , thus we postulate that the covariant derivative acts over this operator in a way such that, in these holomorphic coordinates, it is written as The explicit derivative ∇t Dy is explicitly written in (162).The last equality comes from the fact that Γ = − Γ † H + φ x dKxy dt ∆ xy ∂ φ x as a direct consequence of ( ∇t O) † H = ∇t (O † H ). We derive the explicit expression and compute the derivative of creation and annihilation operator in the Appendix A. Recall the inhomogeneous transformation law of the connection that under a change of coordinates Ψ → CΨ transforms as

Dynamics: The Modified Schrödinger equation
We want the evolution to preserve the symplectic structure.For that matter we opt for a Hamiltonian flow generated by an Hermitian operator Ĥ to postulate evolution.Then, for any function F (t, Ψ, Ψ), its evolution is generated by some f Ĥ where Ĥ is a self adjoint operator.The form of f Ĥ is (122).With the Poisson Bracket {•, •} P , we can readily write down the modified Schrödinger in any set of real coordinates It is clear that this equation does not have a solution on (N )×(N ).In Quantum theory it is desirable to solve this equation over a Hilbert space.Unfortunately for Quantum Field theory this is not possible either, to solve this equation we must look for solutions on (N ) ′ × (N ) ′ , this is our way out to deal with different instantaneous Hilbert spaces.

Application: Klein Gordon theory on curved spacetimes
We have seen above how the choice of a complex structure on the phase space of classical fields is a crucial ingredient of the quantization procedure.Nonetheless, from a physical point of view, given a relativistic classical field theory, it is an open question how to identify the correct complex structure to define the quantization procedure.There exists a successful answer for stationary space times [18,8,19], but the solution for arbitrary models is, to the best of our knowledge, still missing.In this section, we review this treatment form the geometrical perspective developed above, in the simple case of a Klein-Gordon model.

The classical theory and the choice of the complex structure on the classical phase space
To start with, we consider a d + 1 globally hyperbolic spacetime (M, g, ∇ g ) endowed with a pseudo-Riemannian structure g of Lorentzian signature (−, +, • • • , +) and the Levi-Civita connection ∇ g .M is diffeomorphic to R × Σ where Σ is a d−dimensional manifold diffeomorphic to every space-like Cauchy hypersurface of M. A one parameter family of embeddings Σ → M is determined by a scalar function t : M → R such that Σ t = { t(x) = t with x ∈ M} i.e.Σ t are the level sets of t.The structure of the whole spacetime project into (Σ t , h t , D : t) where h is a Riemannian structure and D the corresponding Levi-Cività connection Dh = 0.
We can also define the dynamics directly on (Σ, h, D).To this end we take a coordinate system x = (x i ) i = 1, • • • , d over Σ that induces a coordinate system over M as of (t, x).In this coordinate system we obtain the relation where N n = ∇ t and N = [−g( ∇ t, ∇ t)] − 1 2 is the lapse function while N = N i ∂ ∂x i is the shift vector.Thus, if we provide as data (N, N ) over Σ, the embedding is fully determined.
In the Klein-Gordon classical theory the dynamics of a scalar field can be defined as a parametric curve R → N , where N is a nuclear space of test functions.We denote this solution by ϕ x (t).Associated to the space of solutions, we build a Gel'fand triple with respect to the Hilbert space L 2 (N , dVol), where dVol represents the measure associated with the 3-metric tensor h.
With these ingredients we can introduce a Lagrangian formalism, as it is usually done in QFT.From the usual Lagrangian density in four dimensions, we can write the following bilinear form: Where N, N i and h ij are functions on Σ.In order to define the canonical momentum we must choose between different options.One possible path is to define π x as a density and therefore to build it as an element of N ′ , which is the choice made in [8].Another option is to consider ϕ x as an element of a Rigged Hilbert space and profit from this structure to let π x be an element of the same space as the field ϕ x .With respect to the Gel'fand triple and the injection N ⊂ L 2 (Σ, dV ol) we obtain as canonical momentum The Hamiltonian is defined using the same rigging structure H = δ xy π x φy − L and can be written as Where H is the so called superHamiltonian and H i the supermomenta given by As we saw in [1], with this definition we can endow M F with a Poisson bracket, which in our notation reads Notice that the definition of the Hilbert space, and hence of the rigging, depends on the complex structure and thus on time.Therefore, if we write Liouville equation with the Hamiltonian and the Poisson bracket, it is also necessary to introduce a covariant derivative for time already at the classical level.The usual prescription for classical theory is to consider (ϕ x , π x ) as time independent.In such a case, we get a correction term for π x and the evolution of the classical theory is written in our coordinates as: with y is a symmetric operator.We can connect this classical dynamics with the quantum one considering one particle states, which are those that can be written in holomorphic second-quantized coordinates for the holomorphic quantization as Ψ φ 1P S = Φx φ x + i Πx φ x .The main problem with this representation is that it relies on the rigging provided by L 2 Hol (N ′ C , Dµ c ) that, in turn, depends on the complex structure (30) which is, for now, unknown.Notice that, as the complex structure is built on the phase space of fields defined on the submanifold Σ, which in general evolves in time, it will be, in general, time dependent.In [18] it is proved that, for static and stationary spacetimes, there exists a unique complex structure J once the submanifold Σ, the 3-metric h jk (and hence the covariant derivative D) and operators N 0 and N have been fixed.The expression of such a complex structure in our notation reads for F the classical linear evolution defined by Equation (148).
For simplicity we will take this prescription as well in the general case.Nonetheless, this prescription requires further investigation.In the first place the connection Γ c appearing in (149) does not coincide with the the connection that must appear in the one particle state theory Γ derived from the second quantized framework .
To solve this issue one is tempted to try and add the connection that preserves the Kähler structure already at the classical level.However, this connection Γ(J) stems from the complex structure that we are deriving.Thus this procedure is difficult to implement in the general case because both ingredients have to be determined at the same time.Besides, this procedure would modify the classical equations of motion and it is unclear its interpretation in physical terms.
We can write down an explicit expression for J using the prescription (149) studying first two limiting cases where J can be written down straightforwardly .
Null Shift In this case we set N i = 0 and we get Also, the Hamiltonian in holomorphic coordinates is written H = φx Θ xy φ y and the quantum Hamiltonian acting over this space is obtained with Wick quantization as Ĥ = Θ xy (a † ) x a y (156) In holomorphic s.q.coordinates this expression is just φ y ( √ ΘN ) x y ∂ φ x where N x y ϕ x = N ϕ y The diagonal elements of the complex structure are A = 0, then D −1 = −∆ and the Fourier transform (105) is particularly simple to compute.With this expression it follows that the connection (138) is given by It is important to notice that (134) does not hold with this prescription of complex structure.In the holomorphic setting the Schrodinger equation is particularly simple There are new terms in the equation that involves the time derivative ∆ or δxy .The left hand side term is not self adjoint but is needed in the evolution to preserve the probabilistic nature of quantum mechanics.In the right hand side we get terms that modify the self adjoint part and can be interpreted as modifications of the Hamiltonian.The phenomenology of these equations must be addressed in future works.

Stationary spacetimes
In the particular case of an stationary spacetime there is a time-like Killing vector field and a foliation that in a local region has constant shift function N and null shift.
Where E = Θ − 1 2 N 1 2 Θ.In the particular case Minkowsky space-time we can split space and time with a foliation in which N = 1, in this case E = √ −∇ 2 + m 2 which is the energy operator.

Conclusions
In this paper we have made extensive use of the mathematical tools presented in the first part of this series [1] to study the mathematical formalism underlying different approaches to quantization of a system in quantum field theory.The focus of this work is the geometric interpretation of the tools of Gaussian analysis to describe the Hamiltonian picture of a quantum field theory of a scalar field over the space of Cauchy data.
Following this idea, we started our analysis with geometric quantization.The first step of this program is prequantization.In this context we used Minlos theorem to define the measure of the prequantum Hilbert space and the Malliavin derivative as the central tool to define a connection on such space.The next step is the choice of a polarization.Each choice of polarization amounts to a particular picture of the QFT to be described.We discussed the (anti)holomorphic, Schrödinger and Momentum-Field pictures.In this case the tools of Gaussian analysis suited for the discussion are the Wiener-Ito decomposition theorem that provides the particle interpretation of the theory through the Segal isomorphism with the Bosonic Fock space.But this setting is also suited for the discussion of more general relations among these spaces.In particular we defined a new notion of Fourier transform among infinite dimensional spaces that relates the Schrödinger and momentum-field representations.
General operators other than constant or linear operators fail to be consistently quantized by the procedure of geometric quantization.For this reason we had to study problems of ordering in this picture.We found that trigonometric exponentials and coherent states are algebras of functions specially well suited for the discussion of Weyl and Wick quantization and the study of the star products that arise in these quantization procedures.
In accordance with our geometric approach to this problem, we abandoned the description of the spaces of pure states in terms of Hilbert spaces and treated the as Kähler manifolds.The space of Hida test functions is specially suited for this matter.This point of view led us to a deeper understanding of the different tools described so far, in particular we described the ingredients of Fock quantization in the holomorphic and antiholomorphic representation and its relation through the Fourier transform.
From this geometric perspective we have been able to propose a Hamiltonian evolution that stems from the Kähler structure of the problem.In this case we introduce a quantum connection to preserve it.In turn, this modifies the Schrödinger equation in such a way that losses of norm, found in other approaches to quantization, are not allowed in our prescription.
However, there is an infinite family of connections that achieves this purpose.In our case we found a unique choice based on the indistinguishability of the Holomorphic and Antiholomorphic representations.We illustrated the choice of this connection with an example of Klein Gordon theory in a generic curved spacetime.However, this construction is ambiguous in the election of a complex structure that suits the need of the theory.For simplicity we choose an example that matches the stationary cases but it is known that this prescription leads to some unwanted features.This matter will be the subject of future investigation.
A Explicit expressions for the connection and the quantized operators in s.q.coordinates.
In this appendix we will compute explicitly the expressions of the connection in (137) in holomorphic s.q.coordinates.Let O φ σ be an operator then the expression of the covariant derivative is To compute this term recall the kernels with respect to white noise of the Fourier transform (105) and its inverse.We will start by the linear operators (127).For simplicity consider a real direction ξ x then we see that Let the time derivative be represented by a •, consider also ξ x time independent.We will consider that the coordinates (φ x , φx ) are just placeholders for integration and, as such do not depend on the time parameter.
Lets consider the derivative of the operator O = ξ x Dx .We can compute every term of the derivative using the rule (107) and the definitions of the † operators given by (125).We get Ȯ = ξ x ∆xy ∂ φ y , Ȯ † H = 0.For the other piece of the derivative we get The hardest term to compute is the second one.First notice that And the † H operator is given by substituting φ x by −D xy ∂ φ y and transposing the matrix (in this particular case in which ǫ commutes with the operator).Applying the transformations leads to the desired result.Once this is computed, simply by using the definitions of (30) it follows that For the creation operator we use the fact (∇ t O) † = ∇ t (O † ) and simply substitute ∂ φ x by K xy φ y and transpose the matrices.