A field theory in physics is called a free field theory if it describes standard dynamics of fields without any interaction. Otherwise it is called an interacting field theory.

There is some freedom in formalizing precisely what this means. At the very least the equations of motion of a free field theory should be linear differential equations. In relativistic field theory over a Lorentzian spacetime one typically requires that the linear differential equation of motion is, after gauge fixing, in fact the wave equation or Klein-Gordon equation.

Definition

In covariant phase space geometry/multisymplectic geometry

We describe free field theory in the language of covariant phase spaces of local Lagrangians and their multisymplectic geometry.

Let Σ=(ℝ d−1;1,η)\Sigma = (\mathbb{R}^{d-1;1}, \eta) be Minkowski spacetime. Write the canonical coordinates as

σ i:Σ⟶ℝ. \sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R} \,.

Let (X,g)(X,g) be a vector space XX equipped with a bilinear form gg that makes it a Riemannian manifold. Write its canonical coordinates as

ϕ a:X⟶ℝ. \phi^a \;\colon\; X \longrightarrow \mathbb{R} \,.

Let then X×Σ→ΣX \times \Sigma \to \Sigma be the field bundle. Its first jet bundle then has canonical coordinates

{σ i},{ϕ a},{ϕ ,i a}:j ∞ 1(Σ×X)⟶X. \{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X \,.

Definition

The local Lagrangian for free field theory with this field bundle is

L≔(12g ijη abϕ ,i aϕ ,j a)∧dσ 1∧⋯∧dσ d. L \coloneqq \left( \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \,.

In BV-formalism

Kinematics and dynamics

In the formalization of perturbation theory via BV-quantization as in (Costello-Gwilliam), a free field theory is given by a BV-complex that arises from the following data.

The following appears for instance as (Gwilliam 2.6.2).

Definition

A free field theory (local, Lagrangian) is the following data

kinematics:

A smooth manifold XX (“spacetime”/“worldvolume”);
a ℤ\mathbb{Z}-graded complex vector bundle E→XE \to X (the “field bundle” containing also in general antifields and ghosts);
equipped with a bundle homomorphism (the “antibracket density”)

⟨−,−⟩ loc:E×E→Dens X \langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X

from the fiberwise tensor product of EE with itself to the compex density bundle which is fiberwise
- non-degenerate
- anti-symmetric
- of degree -1

(See also at Verdier duality.)

Write ℰ c≔Γ cp(E)\mathcal{E}_c \coloneqq \Gamma_{cp}(E) for the space of sections of the field bundle of compact support. Write

⟨−,−⟩:ℰ c⊗ℰ c→ℂ \langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}

for the induced pairing on sections

⟨ϕ,ψ⟩=∫ x∈X⟨ϕ(x),ψ(x)⟩ loc. \langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,.

The paring being non-degenerate means that we have an isomorphism E→≃E *⊗Dens XE \stackrel{\simeq}{\to} E^* \otimes Dens_X and we write

E !≔E *⊗Dens X. E^! \coloneqq E^* \otimes Dens_X \,.

dynamics

A differential operator on sections of the field bundle

Q:ℰ→ℰ Q \;\colon\; \mathcal{E} \to \mathcal{E}

of degree 1 such that
1. (ℰ,Q)(\mathcal{E}, Q) is an elliptic complex;
2. QQ is self-adjoint with respect to ⟨−,−⟩\langle -,-\rangle in that for all fields ϕ,ψ∈ℰ c\phi,\psi \in \mathcal{E}_c of homogeneous degree we have ⟨ϕ,Qψ⟩=(−1) |ϕ|⟨Qϕ,ψ⟩\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle.

The classical observables

For (E,Q,⟨−,−⟩ loc)(E, Q, \langle-, -\rangle_{loc}) a free field theory, def. , write

E !≔E *⊗Dens X E^! \coloneqq E^\ast \otimes Dens_X

and accordingly write ℰ !¯\overline{\mathcal{E}^!} for its distributional sections. This is the distributional dual to the smooth sections ℰ\mathcal{E} of EE.

Definition

The complex of global classical observables of the free field theory (E,Q,⟨−,−⟩ loc)(E,Q, \langle-,- \rangle_{loc}) is the classical BV-complex

Obs cl≔(Symℰ c !,Q) Obs^{cl} \coloneqq (Sym \mathcal{E}^!_c, Q)

given by the symmetric algebra of dual sections and quipped with the dual of the differential (which we denote by the same letter) defined on generators and then extended as a graded derivation to the full symmetric algebra.

The factorization algebra of local classical observables is the cosheaf of these observables which assigns to U⊂XU \subset X the complex

Obs cl:U↦(Symℰ c !(U),Q). Obs^{cl} \colon U \mapsto (Sym \mathcal{E}^!_c(U), Q) \,.

in (Gwilliam), this is def. 5.3.6.

The quantum observables

There is a canonical BV-quantization of the above classical observable of a free field theory given by defining the BV Laplacian as follows.

Definition

For (E,Q,⟨−,−⟩ loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. , the standard BV Laplacian

Δ:Symℰ c !→Symℰ c ! \Delta \colon Sym \mathcal{E}^!_c \to Sym \mathcal{E}^!_c

is given on generators a,b∈Sym 1ℰ c !a,b \in Sym^1 \mathcal{E}^!_c of homogeneous degree by

Δ(a⋅b)≔{a,b} \Delta(a \cdot b) \coloneqq \{a,b\}

and then extended to arbitrary elements by the formula

Δ(a⋅b)≔(Δa)⋅b+(−1) deg(a)a⋅(Δb)+{a,b} \Delta(a \cdot b) \coloneqq (\Delta a) \cdot b + (-1)^{deg(a)} a \cdot (\Delta b) + \{a,b\}

In (Gwilliam) this is construction 2.4.9 (also construction 3.1.6, and section 5.3.3).

Definition

For (E,Q,⟨−,−⟩ loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. its complex of quantum observables is then the corresponding quantum BV-complex deformation of the classical BV-complex, def. , given by the standard BV-Laplacian of def.

Obs q:U↦(Sym(ℰ c !(U))[[ℏ]],Q+ℏΔ). Obs^{q} \colon U \mapsto \left( Sym(\mathcal{E}^!_c(U))[ [ \hbar ] ], Q + \hbar \Delta \right) \,.

In (Gwilliam) this is def. 5.3.9.

Properties

Characterization of the quantum observables

We characterize the cochain complex Obs qObs^q of quantum observables of def. by an equivalent but small complex built from just the cochain cohomology of the elliptic complex of fields (ℰ,Q)(\mathcal{E}, Q).

Definition

For (E,Q,⟨−,−⟩ loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. , the global pairing constitutes a dg-symplectic vector space (ℰ,Q,⟨−,−⟩)(\mathcal{E}, Q, \langle -,-\rangle), which descends to the cochain cohomology to a graded symplectic vector space (H •(ℰ,⟨−,−⟩)(H^\bullet(\mathcal{E}, \langle -,-\rangle); hence by def. there is a standard BV-Laplacian Δ H •ℰ\Delta_{H^\bullet\mathcal{E}}. Write

ℬ𝒱𝒬(H •ℰ)≔(Sym(H •(ℰ)) *,Δ H •ℰ) \mathcal{BVQ}(H^\bullet \mathcal{E}) \coloneqq ( Sym (H^\bullet(\mathcal{E}))^*, \Delta_{H^\bullet \mathcal{E}} )

for the corresponding quantum BV-complex.

This is part of (Gwilliam, prop. 2.4.10, prop. 5.5.1),

Handle the following with care for the moment.

Proposition

For a free field theory (E,Q,⟨−,−⟩ loc)(E,Q,\langle-,- \rangle_{loc}), def. , the complex of quantum observables Obs qObs^q, def. is quasi-isomorphic to the BV-quantization of the cohomology of the field complex, given by def.

Obs q≃ℬ𝒱𝒬(H •(ℰ)). Obs^q \simeq \mathcal{BVQ}(H^\bullet(\mathcal{E})) \,.

This is (Gwilliam, prop. 5.5.1).

The proof is supposedly along the lines of (Gwilliam, section 2.5), applying the homological perturbation lemma.

Proposition

The bracket {−,−}\{-,-\} on the complex of quantum observables Obs qObs^q of def. descends to a bracket on cochain cohomology, making (H •(Obs q),{−,−})(H^\bullet (Obs^q), \{-,-\}) into a graded symplectic vector space.

Proof

Let a,b∈Obs qa,b \in Obs^q be closed elements of homogeneous degree. Then by the compatibly of Δ\Delta with {−,−}\{-,-\} also {a,b}\{a,b\} is closed:

Δ{a,b}={Δa,b}±{a,Δb}=0. \Delta \{a,b\} = \{\Delta a, b\} \pm \{a , \Delta b\} = 0 \,.

Let in addition c∈Obs qc \in Obs^q be any element. Then

{a,b+Δc} ={a,b}+{a,Δc} ={a,b}+Δ(a⋅(Δb))−(Δa)⋅b−a⋅(Δ 2b) ={a,b}+Δ(a⋅(Δb)) \begin{aligned} \{a, b + \Delta c\} &= \{a,b\} + \{a, \Delta c\} \\ & = \{a,b\} + \Delta (a \cdot (\Delta b)) - (\Delta a)\cdot b -a \cdot (\Delta^2 b) \\ & = \{a,b\} + \Delta (a \cdot (\Delta b)) \end{aligned}

and hence the cohomology class of {a,b}\{a,b\} is independent of the representative cocycle bb, and similarly for aa.

Examples

0-Dimensional free field theory

A degenerate but instructive class of examples to compare to is the case where X=*X = * is the 0-dimensional connected manifold: the point. (See (Gwilliam 2.3.1)).

In this case

E=V⊕V *[−1] E = V \oplus V^*[-1]

is the direct sum of a vector space and its formal dual shifted in degree. The pairing is the canonical pairing between a vector space and its dual.

If {x i:V→ℝ} i\{x^i \colon V \to \mathbb{R}\}_i is a basis for functional on VV and {ξ i}\{\xi_i\} is the corresponding basis of functions on V *[−1]V^*[-1], then the antibracket in this case is

{X i,x j}=0{ξ i,ξ j}=0{x i,x j}=δ j i. \{ X^i, x^j \} = 0 \;\;\; \{\xi_i, \xi_j\} = 0 \;\;\; \{x^i, \x_j\} = \delta^i_j \,.

The BV-Laplacian in this basis is

Δ=∑ i=1 n∂∂x i∂∂ξ i. \Delta = \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial \xi_i} \,.

The action functional is a Gaussian distribution over VV defined by a matrix A=(a ij)A = (a_{i j}). The corresponding differential is

Q=∑ i,j=1 nx ia ij∂∂ξ i. Q = \sum_{i,j =1}^n x^i a_{i j} \frac{\partial}{\partial \xi_i} \,.

Hence for a field of the form

ϕ=∑ i=1 nϕ iξ i \phi = \sum_{i = 1}^n \phi^i \xi_i

we have the action functional

exp(S(ϕ)) =exp(−⟨ϕ,Qϕ⟩) =exp(−ϕ ia kjϕ j⟨ξ i,x k⟩) =exp(−ϕ ia ijϕ j) \begin{aligned} \exp(S(\phi)) & = \exp(-\langle \phi , Q \phi\rangle) \\ & = \exp( - \phi^i a_{k j} \phi^j \langle \xi_i, x^k \rangle ) \\ & = \exp(- \phi^i a_{i j} \phi^j ) \end{aligned}

Locally free field theories

Some sigma-model quantum field theories have the property that they are free locally on their target spaces. Under good conditions then quantization of free field theory locally yields a sheaf of quantum observables on target space from which the full quantization of the field theory may be reconstructed.

A famous example of this is the topologically twisted 2d (2,0)-superconformal QFT (see there for more, and see (Gwilliam, section 6 for the description in terms of factorization algebras).

References

Discussion of free field theories and their quantization on globally hyperbolic Lorentzian manifolds is in

Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)

Discussion on Euclidean manifolds and in terms of BV-formalism is in

Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory (wiki, pdf)
Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)
Owen Gwilliam, Rune Haugseng, Linear Batalin-Vilkovisky quantization as a functor of ∞-categories (arXiv:1608.01290)

Last revised on May 30, 2022 at 02:08:40. See the history of this page for a list of all contributions to it.