free field theory

This entry is about the concept in physics. For the concept in algebra see at free field.



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Fields and quanta



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

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.


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=( d1;1,η)\Sigma = (\mathbb{R}^{d-1;1}, \eta) be Minkowski spacetime. Write the canonical coordinates as

σ i:Σ. \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. \phi^a \;\colon\; X \longrightarrow \mathbb{R} \,.

Let then X×ΣΣX \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 \,.

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

L(12g ijη abϕ ,i aϕ ,j a)dσ 1dσ 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 \,.

The canonical momentum-densities for the free field local Lagrangian of def. 1 are

p a idσ 1dσ d u i aL =(g ijη abϕ ,j a)dσ 1dσ d \begin{aligned} p_a^i \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d & \coloneqq \frac{\partial}{\partial u^a_i} L \\ & = \left( g^{i j} \eta_{a b} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \end{aligned}

So the boundary term θ\theta in variational calculus, (see this remark at covariant phase space ) is

du aι i(ϕ ,i aL) =p a i(ι σ ivol)du a =p a idq i a, \begin{aligned} \mathbf{d}u^a \wedge \iota_{\partial_i} \left( \frac{\partial}{\partial \phi^a_{,i}}L \right) & = p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a \\ & = p^i_a \wedge dq_i^a \,, \end{aligned}

where in the last line we adopted the notation of this remark at multisymplectic geometry.

This shows that the canonical multisymplectic form is the “covariant symplectic potential current density” which is induced by the free field Lagrangian.

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).


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


(See also at Verdier duality.)

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

,: c c \langle -,-\rangle \;\colon\; \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}

for the induced pairing on sections

ϕ,ψ= xXϕ(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 EE *Dens XE \stackrel{\simeq}{\to} E^* \otimes Dens_X and we write

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


  • A differential operator on sections of the field bundle

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

    of degree 1 such that

    1. (,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 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.


From this data we obtain:

  • The action functional S: cS \colon \mathcal{E}_c \to \mathbb{C} of this corresponding free field theory is

    S:ϕ Xϕ,Qϕ. S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,.
  • The classical BV-complex is the symmetric algebra Sym c !Sym \mathcal{E}^!_c of compactly suppported sections of E !E^! equipped with the induced action of the differential QQ and the pairing

    {α,β} xXα(x),β(x). \{\alpha,\beta\} \coloneqq \int_{x \in X} \langle \alpha(x), \beta(x)\rangle \,.

    See below at The classical observables for more.

  • The quantum BV-complex

    Obs q(Sym c ![[]],Q+Δ) Obs^q \coloneqq (Sym \mathcal{E}^!_c[ [\hbar] ], Q + \hbar \Delta)

    is the deformation of the above to the symmetric algebra tensored with the formal power series in \hbar (“Planck's constant”) Sym( !)[[]]Sym(\mathcal{E}^!)[ [\hbar] ] and differential Q+ΔQ + \hbar \Delta with BV-Laplacian defined to vanish on Sym 1Sym^{\leq 1}, given by

    Δ(αβ){α,β} \Delta (\alpha \cdot \beta) \coloneqq \{\alpha,\beta\}

    for α,β !\alpha,\beta \in \mathcal{E}^! and extended by the formula

    Δ(ab)(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\} \,.

    See below at The quantum observables for more.

A closed element 𝒪Obs q\mathcal{O} \in Obs^q is an observable and its formal path integral expectation value 𝒪\langle \mathcal{O}\rangle is its image in the cochain cohomology H Obs qH^\bullet Obs^q. Via the homological perturbation lemma this may be computed in perturbation theory (order by order in \hbar) in terms of Feynman diagrams.

In a non-free field theory the differential would have an additional perturbation of the complex by an interaction term II to

Q+{I,}+Δ. Q + \{I,-\} + \hbar \Delta \,.
action functionalkinetic actioninteractionpath integral measure
exp(S(ϕ))μ=\exp(-S(\phi)) \cdot \mu = exp((ϕ,Qϕ))\exp(-(\phi, Q \phi)) \cdotexp(I(ϕ))\exp(I(\phi)) \cdotμ\mu
BV differentialelliptic complex +antibracket with interaction +BV-Laplacian
d q=d_q =QQ +{I,}\{I,-\} +Δ\hbar \Delta

The classical observables

For (E,Q,, loc)(E, Q, \langle-, -\rangle_{loc}) a free field theory, def. 2, write

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

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


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(Sym 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 UXU \subset X the complex

Obs cl:U(Sym 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.


For (E,Q,, loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. 2, the standard BV Laplacian

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

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

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

and then extended to arbitrary elements by the formula

Δ(ab)(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).


For (E,Q,, loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. 2 its complex of quantum observables is then the corresponding quantum BV-complex deformation of the classical BV-complex, def. 3, given by the standard BV-Laplacian of def.4

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

In (Gwilliam) this is def. 5.3.9.


Characterization of the quantum observables

We characterize the cochain complex Obs qObs^q of quantum observables of def. 5 by an equivalent but small complex built from just the cochain cohomology of the elliptic complex of fields (,Q)(\mathcal{E}, Q).


For (E,Q,, loc)(E, Q, \langle -,-\rangle_{loc}) a free field theory, def. 2, the global pairing constitutes a dg-symplectic vector space (,Q,,)(\mathcal{E}, Q, \langle -,-\rangle), which descends to the cochain cohomology to a graded symplectic vector space (H (,,)(H^\bullet(\mathcal{E}, \langle -,-\rangle); hence by def. 4 there is a standard BV-Laplacian Δ H \Delta_{H^\bullet\mathcal{E}}. Write

ℬ𝒱𝒬(H )(Sym(H ()) *,Δ H ) \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.


For a free field theory (E,Q,, loc)(E,Q,\langle-,- \rangle_{loc}), def. 2, the complex of quantum observables Obs qObs^q, def. 5 is quasi-isomorphic to the BV-quantization of the cohomology of the field complex, given by def. 6

Obs qℬ𝒱𝒬(H ()). 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.


The bracket {,}\{-,-\} on the complex of quantum observables Obs qObs^q of def. 5 descends to a bracket on cochain cohomology, making (H (Obs q),{,})(H^\bullet (Obs^q), \{-,-\}) into a graded symplectic vector space.


Let a,bObs 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 cObs qc \in Obs^q be any element. Then

{a,b+Δc} ={a,b}+{a,Δc} ={a,b}+Δ(a(Δb))(Δa)ba(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.


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=VV *[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} 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 nx iξ 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 ijξ 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 jξ 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 fee 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 twisted2d (2,0)-superconformal QFT (see there for more, and see (Gwilliam, section 6 for the description in terms of factorization algebras).


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

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

Revised on November 7, 2017 13:10:42 by Urs Schreiber (