nLab free field theory

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

---

Idea

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

$$\sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R} \,.$$

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

$$\phi^a \;\colon\; X \longrightarrow \mathbb{R} \,.$$

Let then $X \times \Sigma \to \Sigma$ be the field bundle. Its first jet bundle then has canonical coordinates

$$\{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X \,.$$

Definition

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

$$L \coloneqq \left( \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j} \right) \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d \,.$$

Remark

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

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

Remark

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

$$\begin{aligned} \mathbf{d}u^a \wedge \iota_{\partial_i} \left( \frac{\partial}{\partial \phi^a_{,i}}L \right) & = p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a \\ & = p^i_a \wedge dq_i^a \,, \end{aligned}$$

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).

Definition

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

kinematics:

• A smooth manifold $X$ (“spacetime”/“worldvolume”);

• a $\mathbb{Z}$-graded complex vector bundle $E \to X$ (the “field bundle” containing also in general antifields and ghosts);

• equipped with a bundle homomorphism (the “antibracket density”)

  $$\langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X$$

  from the fiberwise tensor product of $E$ with itself to the compex density bundle which is fiberwise

  • non-degenerate

  • anti-symmetric

  • of degree -1

(See also at Verdier duality.)

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

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

for the induced pairing on sections

$$\langle \phi, \psi\rangle = \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc} \,.$$

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

$$E^! \coloneqq E^* \otimes Dens_X \,.$$

dynamics

• A differential operator on sections of the field bundle

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

  of degree 1 such that

  1. $(\mathcal{E}, Q)$ is an elliptic complex;

  2. $Q$ is self-adjoint with respect to $\langle -,-\rangle$ in that for all fields $\phi,\psi \in \mathcal{E}_c$ of homogeneous degree we have $\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle$.

Remark

From this data we obtain:

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

  $$S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle \,.$$

• The classical BV-complex is the symmetric algebra $Sym \mathcal{E}^!_c$ of compactly suppported sections of $E^!$ equipped with the induced action of the differential $Q$ and the pairing

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

  See below at The classical observables for more.

• The quantum BV-complex

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

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

  $$\Delta (\alpha \cdot \beta) \coloneqq \{\alpha,\beta\}$$

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

  $$\Delta(a \cdot b) \coloneqq (\Delta a) \cdot b + (-1)^{deg(a)} a \cdot (\Delta b) + \{a,b\} \,.$$

  See below at The quantum observables for more.

A closed element $\mathcal{O} \in Obs^q$ is an observable and its formal path integral expectation value $\langle \mathcal{O}\rangle$ is its image in the cochain cohomology $H^\bullet Obs^q$. 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 $I$ to

$$Q + \{I,-\} + \hbar \Delta \,.$$

action functional	kinetic action	interaction	path integral measure
$\exp(-S(\phi)) \cdot \mu =$	$\exp(-(\phi, Q \phi)) \cdot$	$\exp(I(\phi)) \cdot$	$\mu$
BV differential	elliptic complex +	antibracket with interaction +	BV-Laplacian
$d_q =$	$Q$ +	$\{I,-\}$ +	$\hbar \Delta$

The classical observables

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

$$E^! \coloneqq E^\ast \otimes Dens_X$$

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

Definition

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

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

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 $U \subset X$ the complex

$$Obs^{cl} \colon U \mapsto (Sym \mathcal{E}^!_c(U), Q) \,.$$

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

$$\Delta \colon Sym \mathcal{E}^!_c \to Sym \mathcal{E}^!_c$$

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

$$\Delta(a \cdot b) \coloneqq \{a,b\}$$

and then extended to arbitrary elements by the formula

$$\Delta(a \cdot b) \coloneqq (\Delta a) \cdot b + (-1)^{deg(a)} a \cdot (\Delta b) + \{a,b\}$$

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

Definition

For $(E, Q, \langle -,-\rangle_{loc})$ 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.

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

In (Gwilliam) this is def. 5.3.9.

Properties

Characterization of the quantum observables

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

Definition

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

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

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 $(E,Q,\langle-,- \rangle_{loc})$, def. , the complex of quantum observables $Obs^q$, def. is quasi-isomorphic to the BV-quantization of the cohomology of the field complex, given by def.

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

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 $Obs^q$ of def. descends to a bracket on cochain cohomology, making $(H^\bullet (Obs^q), \{-,-\})$ into a graded symplectic vector space.

Proof

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

$$\Delta \{a,b\} = \{\Delta a, b\} \pm \{a , \Delta b\} = 0 \,.$$

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

$$\begin{aligned} \{a, b + \Delta c\} &= \{a,b\} + \{a, \Delta c\} \\ & = \{a,b\} + \Delta (a \cdot (\Delta b)) - (\Delta a)\cdot b -a \cdot (\Delta^2 b) \\ & = \{a,b\} + \Delta (a \cdot (\Delta b)) \end{aligned}$$

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

Examples

0-Dimensional free field theory

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

In this case

$$E = V \oplus V^*[-1]$$

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

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

The BV-Laplacian in this basis is

$$\Delta = \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial \xi_i} \,.$$

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

$$Q = \sum_{i,j =1}^n x^i a_{i j} \frac{\partial}{\partial \xi_i} \,.$$

Hence for a field of the form

$$\phi = \sum_{i = 1}^n \phi^i \xi_i$$

we have the action functional

$$\begin{aligned} \exp(S(\phi)) & = \exp(-\langle \phi , Q \phi\rangle) \\ & = \exp( - \phi^i a_{k j} \phi^j \langle \xi_i, x^k \rangle ) \\ & = \exp(- \phi^i a_{i j} \phi^j ) \end{aligned}$$

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 twisted2d (2,0)-superconformal QFT (see there for more, and see (Gwilliam, section 6 for the description in terms of factorization algebras).

Related concepts

• Wick algebra

• Gaussian probability distribution, Gaussian integral

• Wick's lemma

• gauge theory

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)