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Contents

Idea

In physics, a scalar field is a field on spacetime/worldvolume which is simply a function with values in the field of scalars, typically the real numbers \mathbb{R} or complex numbers \mathbb{C}, sometimes the quaternions \mathbb{H}. Hence it is a field encoded by a field bundle which is a trivial line bundle.

One fundamental (complex, charged) scalar field is seen in experiment, the Higgs field, which is one component of the standard model of particle physics. A widely hypothesized scalar field is the inflaton field in models of cosmic inflation, which however remains speculative and might in any case be an effective compound of more fundamental fields.

But scalar fields also serve as a key toy example in theoretical studies of field theory, such as in phi^4 theory or in the Ising model. The usefulness of the scalar field as a toy example of classical field theory and perturbative quantum field theory is due to it already exhibiting much of the core structure of field theory. For instance the general formulas for propagators and the S-matrix of general local field theories are structurally those of the scalar field, just with some more fairly evident representation theoretic structure thrown in.

Definition

Free scalar field

We discuss here the free scalar field on general spacetimes, hence the scalar field subject to the force of a background field of gravity, but not interacting with itself. This means that its local Lagrangian density is quadratic in the fields and its first derivatives (def. 3) and its equation of motion is the Klein-Gordon equation (hence the wave equation in the case of vanishing mass) (prop. 1 below).

The Poisson bracket on the covariant phase space of this system (prop. 1 below) turns out to have as integral kernel the causal propagator of the Klein-Gordon operator (i.e. the Green function whose support is inside the light cone). Accordingly, the other associated Green functions of the Klein-Gordon operator (the “propagators”, such as the Feynman propagator) govern the perturbative quantum field theory of the scalar field (see at S-matrix for more).

Covariant phase space

Recall that a classical local field theory is for some prescribed class of manifolds Σ\Sigma of given dimension p+1p+1 \in \mathbb{N} interpreted as spacetimes/worldvolumes:

  1. a choice of fiber bundle EΣE \to\Sigma, called the field bundle;

  2. a choice LΩ p+1,0(J (E))L \in \Omega^{p+1,0}(J^\infty(E)) of horizontal differential form of degree p+1p+1 on the jet bundle of the field bundle, called the local Lagrangian density.

Given a classical local field theory defined by a local Lagrangian density LΩ p+1,0(J (E))L \in \Omega^{p+1,0}(J^\infty(E)), then

  1. the configuration space is the smooth space of sections Γ X(E)\Gamma_X(E) of the field bundle;

  2. the equations of motion is the partial differential equation on elements ϕΓ X(E)\phi \in \Gamma_X(E) given by

    (δ ELL)(j ϕ)=0, (\delta_{EL} L)(j^\infty \phi) = 0 \,,

    where

    1. δ EL:Ω n,0(J X (E)) 1(J X (E))\delta_{EL} \;\colon\; \Omega^{n,0}(J^\infty_X(E)) \to \mathcal{F}^1(J^\infty_X(E)) denotes the Euler-Lagrange operator

    2. j :Γ X(E)Γ X(J X (E))j^\infty \;\colon\; \Gamma_X(E) \longrightarrow \Gamma_X(J^\infty_X(E)) denotes jet prolongation.

  3. The covariant phase space (Sol δ ELL=0,dθ)(Sol_{\delta_{EL}L = 0}, d\theta) is the subspace Sol δ ELL=0Γ X(E)Sol_{\delta_{EL} L = 0} \subset \Gamma_X(E) of solutions to the equations of motion, equipped with the canonical presymplectic form.

On Minkowski spacetime

We discuss the covariant phase space of the free scalar field on Minkowski spacetime. For a more detailed exposition see at geometry of physics -- A first idea of quantum field theory.

Definition

(local Lagrangian density for free scalar field on Minkowski spacetime)

For pp \in \mathbb{N}, let spacetime Σ p,1=( d,η)\Sigma \coloneqq \mathbb{R}^{p,1} = (\mathbb{R}^d, \eta) be Minkowski spacetime of dimension p+1p + 1, where η\eta denotes the Minkowski metric tensor of signature (,+,,+)(-,+,\cdots, +). We write dvol ΣΩ p+1(Σ)\mathrm{dvol}_\Sigma \in \Omega^{p+1}(\Sigma) for the corresponding volume form and {x μ:Σ} μ=0 p\{x^\mu \colon \Sigma \to \mathbb{R}\}_{\mu = 0}^p for the canonical coordinate functions.

Let the field bundle EΣE \to \Sigma be the trivial real line bundle over Σ\Sigma.

Then its jet bundle J EJ^\infty E has canonical coordinates

{{x μ},ϕ,{ϕ ,μ},{ϕ ,μν},}. \{ \{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu \nu}\}, \cdots \} \,.

In these coordinates, the local Lagrangian density

LΩ p+1,0(Σ) L \in \Omega^{p+1,0}(\Sigma)

defining the free scalar field of mass m[0,)m \in [0,\infty) on Σ\Sigma is

L12(η μνϕ ,μϕ ,ν+m 2ϕ 2)dvol Σ. L \coloneqq \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} + m^2 \phi^2 \right) \mathrm{dvol}_\Sigma \,.
Proposition

(covariant phase space of free scalar field)

In the situation of def. 1 for Φ:Σ\Phi : \Sigma \to \mathbb{R} the ϕ\phi-component of a section of the field bundle, its equation of motion is the Klein-Gordon equation

(η μν μ ν+m 2)Φ=0. \left(\eta^{\mu \nu} \partial_\mu \partial_\nu + m^2 \right) \Phi = 0 \,.

Moreover, the induced pre-symplectic current ωΩ p1,2(E)\omega \in \Omega^{p-1,2}(E) is, in local coordinates,

ω=(η μνd Vϕ ,μd Vϕ)ι νdvol Σ \omega = \left(\eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma}

and hence the induced symplectic form on the covariant phase space of the free scalar field takes two smooth function w 1,w 2C (Σ)w_1,w_2 \in C^\infty(\Sigma), regarded as tangent vectors at zero to

Ω Σ p(w 1,w 2)= Σ p(( nw 1)w 2w 1 nw 2)dvol Σ p, \mathbf{\Omega}_{\Sigma_{p}}(w_1, w_2) \;=\; \int_{\Sigma_{p}} \left( (\partial_n w_1) w_2 - w_1 \partial_n w_2 \right) dvol_{\Sigma_{p}} \,,

where Σ pΣ\Sigma_{p} \hookrightarrow \Sigma is any Cauchy surface and where nNΣ pn \in N \Sigma_{p} denotes its time-like normal vector field.

Proof

We need to show that Euler-Lagrange operator δ EL:Ω p+1,0(Σ)Ω S p+1,1(Σ)\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma) takes the local Lagrangian density for the free scalar field to

δ ELL=(η μνϕ ,μν+m 2ϕ)d Vϕdvol Σ. \delta_EL L \;=\; \left( \eta^{\mu \nu} \phi_{,\mu \nu} + m^2 \phi \right) d_V \phi \wedge \mathrm{dvol}_\Sigma \,.

First of all, the result of applying the vertical differential to the local Lagrangian density is

d VL=(η μνϕ ,μd Vϕ ,ν+m 2ϕd Vϕ)dvol Σ. d_V L = \left( \eta^{\mu \nu} \phi_{,\mu} d_V \phi_{,\nu} + m^2 \phi d_V \phi \right) \wedge \mathrm{dvol}_\Sigma \,.

By definition of the Euler-Lagrange operator, in order to find EL\mathrm{EL} and θ\theta, we need to exhibit this as the sum of the form ()d Vϕd Hθ(-) \wedge d_V \phi - d_H \theta.

The key to find θ\theta is to realize d Vϕ ,νdvol Σd_V \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma as a horizontal derivative. Since d Hϕ=ϕ ,μdx μd_H \phi = \phi_{,\mu} d x^\mu this is accomplished by

d Vϕ ,νdvol Σ=d Vd Hϕι νdvol Σ d_V \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma = d_V d_H \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma

Hence we may set

θη μνϕ ,μd Vϕι νdvol Σ, \theta \coloneqq \eta^{\mu \nu} \phi_{,\mu} d_V \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \,,

because with this we have

d Hθ=η μν(ϕ ,μνd Vϕη μνϕ ,μd Vϕ ,ν)dvol Σ. d_H \theta = \eta^{\mu \nu} \left( \phi_{,\mu \nu} d_V \phi - \eta^{\mu \nu} \phi_{,\mu} d_V \phi_{,\nu} \right) \wedge \mathrm{dvol}_\Sigma \,.

In conclusion this yields the decomposition of the vertical differential of the Lagrangian density

d VL=(η μνϕ ,μν+m 2ϕ)d Vϕdvol Σ=δ ELLd Hθ, d_V L = \underset{ = \delta_{EL} L }{ \underbrace{ \left( \eta^{\mu \nu} \phi_{,\mu \nu} + m^2 \phi \right) d_V \phi \wedge \mathrm{dvol}_\Sigma } } - d_H \theta \,,

which shows that δ ELL\delta_{EL} L is as claimed, and that θ\theta is a presymplectic potential current. Hence the presymplectic current itself is

ω d Vθ =d V(η μνϕ ,μd Vϕι νdvol Σ) =(η μνd Vϕ ,μd Vϕ)ι νdvol Σ. \begin{aligned} \omega &\coloneqq d_V \theta \\ & = d_V \left( \eta^{\mu \nu} \phi_{,\mu} d_V \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right) \\ & = \left(\eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \end{aligned} \,.

For Σ pΣ\Sigma_p \hookrightarrow \Sigma a Cauchy surface, the transgression of this presymplectic current to the infinitesimal neighbourhood of Σ\Sigma is

ω Σ p(w 1,w 2) =( Σ p(η μνd μϕdϕ)ι νdvol Σ)(w 1,w 2) = Σ p(( nw 1)w 2w 1 nw 2)dvol Σ p \begin{aligned} \omega_{\Sigma_{p}}(w_1, w_2) & = \left( \int_{\Sigma_p} \left( \eta^{\mu \nu} \mathbf{d} \partial_\mu \phi \wedge \mathbf{d} \phi \right) \iota_{\partial_\nu} dvol_\Sigma \right) (w_1, w_2) \\ & = \int_{\Sigma_{p}} \left( (\partial_n w_1) w_2 - w_1 \partial_n w_2 \right) dvol_{\Sigma_{p}} \end{aligned}
Example

(Poisson brackets over Minkowski spacetime)

Consider the covariant phase space over Minkowski spacetime of dimension p+1p+1 as in def. 1, with pre-symplectic current according to prop. 1 given by

ω=η μνd Vϕ ,μd Vϕι μdvol Σ \omega = \eta^{\mu \nu} d_V \phi_{,\mu} \wedge d_V \phi \wedge \iota_{\partial_\mu} dvol_\Sigma \,

The corresponding Poisson bracket Lie (p+1)-algebra has in degree 0 Hamiltonian forms such as

Qϕι 0dvol ΣΩ p,0(E) Q \coloneqq \phi \iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E)

and

Pη μνϕ ,μι νdvol ΣΩ p,0(E). P \coloneqq \eta^{\mu \nu} \phi_{,\mu} \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E) \,.

The corresponding Hamiltonian vector fields are

v P= ϕ ,0 v_P = \partial_{\phi_{,0}}

and

v Q= ϕ. v_Q = - \partial_{\phi} \,.

Hence the corresponding bracket is

{Q,P}=ι v Qι v Pω=ι 0dvol Σ. \{Q,P\} = \iota_{v_Q} \iota_{v_P} \omega = \iota_{\partial_0} dvol_\Sigma \,.

More generally for b 1,b 2C c (Σ)b_1, b_2 \in C^\infty_c(\Sigma) two bump functions then

{b 1Q,b 2P}=±b 1b 2ι 0dvol Σ. \{ b_1 Q, b_2 P \} = \pm b_1 b_2 \iota_{\partial_0} dvol_\Sigma \,.

Upon transgression to the Cauchy surface Σ p t{xΣ|x 0=t}\Sigma^t_{p} \coloneqq \{x \in \Sigma \vert x^0 = t \} this yields the Poisson bracket

{ Σ pb 1(x)ϕ(t,x)ι 0dvol Σ(x)d px, Σ pb 2(x) 0ϕ(t,x)ι 0dvol Σ(x)}= Σ pb 1(x)b 2(x)ι 0dvol Σ(x)d px. \left\{ \int_{\Sigma_p} b_1(\vec x) \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(x) d^p \vec x \;,\; \int_{\Sigma_p} b_2(\vec x) \partial_0 \phi(t,\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x) \right\} \;=\; \int_{\Sigma_p} b_1(\vec x) b_2(\vec x) \iota_{\partial_0} dvol_\Sigma(\vec x) d^p \vec x \,.

where now

ϕ(x), 0ϕ(x):PhaseSpace(Σ p t) \phi(x), \partial_0 \phi(x) : PhaseSpace(\Sigma_p^t) \to \mathbb{R}

are the point-evaluation functions (functionals), which act on a field configuration ΦΓ Σ(E)=C (Σ)\Phi \in \Gamma_\Sigma(E) = C^\infty(\Sigma) as

ϕ(x)(Φ)Φ(x)AAAAAAAA 0ϕ(x)(Φ) 0Φ(x). \phi(x)(\Phi) \coloneqq \Phi(x) \phantom{AAAAAAAA} \partial_0 \phi(x) (\Phi) \coloneqq \partial_0 \Phi(x) \,.

Notice that these point-evaluation functions themselves do not arise as the transgression of elements in Ω p,0(E)\Omega^{p,0}(E), only their smearings such as Σ pb 1ϕdvol Σ p\int_{\Sigma_p} b_1 \phi dvol_{\Sigma_p} do. Nevertheless we may express the above Poisson bracket conveniently via the integral kernel

(1){ϕ(t,x),ϕ(t,y)}=δ(xy). \{\phi(t,\vec x), \phi(t,\vec y) \} = \delta(\vec x - \vec y) \,.

More generally one may express the integral kernel for the Poisson bracket of evaluation functions for different values of tt. Notice that for each time interval [t 1,t 2][t_1, t_2] we have a Lagrangian correspondence

Trajectories([t 1,t 2]) Φ()(Φ(t 1,), 0Φ(t 1,)) Φ()(Φ(t 2,), 0Φ(t 2,)) PhaseSpace(Σ p t 1) PhaseSpace(Σ p t 2) Ω t 1 Ω t 2 Ω cl 2, \array{ && Trajectories([t_1,t_2]) \\ & {}^{\mathllap{ \Phi(-) \mapsto ( \Phi(t_1,-), \partial_0 \Phi(t_1,-) ) }}\swarrow && \searrow^{\mathrlap{ \Phi(-) \mapsto ( \Phi(t_2,-), \partial_0 \Phi(t_2,-) ) }} \\ PhaseSpace(\Sigma^{t_1}_p) && && PhaseSpace(\Sigma^{t_2}_p) \\ & {}_{\mathllap{\Omega^{t_1}}}\searrow && \swarrow_{\mathrlap{\Omega^{t_2}}} \\ && \mathbf{\Omega}_{cl}^2 } \,,

where Trajectories([t 1,t 2])Trajectories([t_1,t_2]) is the space of solutions to the Klein-Gordon equation on [t 1,t 2]× p p,1[t_1,t_2] \times \mathbb{R}^p \subset \mathbb{R}^{p,1}.

There is a unique function on PhaseSpace(Σ p t 1)PhaseSpace(\Sigma^{t_1}_p) whose pullback to Trajectories([t 1,t 2])Trajectories([t_1,t_2]) is the evaluation function ϕ(x)\phi(x) for any x 0[t 1,t 2]x^0 \in [t_1,t_2]. By convenient abuse of notation, we also call that function ϕ(x)\phi(x).

Proposition

(integral kernel for Poisson bracket on Minkowski spacetime is the causal propagator)

The integral kernel on p,1\mathbb{R}^{p,1} for the Poisson bracket of the scalar field over Minkowski spacetime (example 1) is the causal propagator

Δ𝒟(Σ×Σ) \Delta \;\in\; \mathcal{D}'(\Sigma \times \Sigma)

(also known as the Pauli-Jordan distribution or Peierls bracket) on Minkowski spacetime:

{ϕ(x),ϕ(y)} =Δ(x,y) i(2π) p12E(k)(e iE(k)(xy) 0k(xy)e +iE(k)(xy) 0+k(xy))d pk =i(2π) pδ(k μk μ+m 2)sgn(k 0)e ik μ(xy) μd p+1k. \begin{aligned} \{ \phi(x), \phi(y) \} & = \Delta(x,y) \\ & \coloneqq -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)}\left( e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)} - e^{+ i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \,. \end{aligned}

(e. g. Scharf 01 (1.1.13))

Proof

By Fourier transform the general solution to the Klein-Gordon equation may be expressed in the form

Φ(x)=(2π) pAmp(k)e ik μx μδ(k μk μ+m 2)d p+1k, \Phi(x) \;=\; (2\pi)^{-p} \int Amp(k) e^{- i k_\mu x^\mu } \delta( k_\mu k^\mu + m^2 ) d^{p+1} k \,,

where Amp(k)Amp(k) \in \mathbb{C} is the complex amplitude of the kkth mode (k p+1)(k \in \mathbb{R}^{p+1}).

We may split this into the contributions with positive and those with negative energy k 0k_0 by decomposing the integral over k 0k_0 as

Φ(x) =+(2π) p/2 0 Amp(k 0,k)e ik 0x 0ikxδ(k 0 2+k+m 2)d pkdk 0 =+(2π) p/2 0 Amp(k 0,k)e +ik 0x 0ikxδ(k 0 2+k 2+m 2)d pkdk 0. \begin{aligned} \Phi(x) & = \phantom{+} (2\pi)^{-p/2} \int_0^\infty \int Amp(k_0, \vec k) e^{- i k_0 x^0 - i \vec k \cdot \vec x} \delta(- k_0^2 + \vec k + m^2) d^p \vec k \, d k_0 \\ & \phantom{=} + (2\pi)^{-p/2} \int_0^\infty \int Amp(-k_0, \vec k) e^{+ i k_0 x^0 - i \vec k \cdot \vec x} \delta(-k_0^2 + \vec k^2 + m^2) d^p \vec k \, d k_0 \,. \end{aligned}

By changing integration variables via k 0=+hk_0 = +\sqrt{ h } this yields

Φ(x) =+(2π) p/2 0 Amp(h,k)e ihx 0ikxδ(h+k+m 2)d pkdh2h =+(2π) p/2 0 Amp(h,k)e +ihx 0ikxδ(h+k 2+m 2)d pkdh2h =+(2π) p/2Amp(E(k),k)e iE(k)x 0ikxd pk =+(2π) p/2Amp(E(k),k)e +iE(k)x 0ikxd pk \begin{aligned} \Phi(x) & = \phantom{+} (2\pi)^{-p/2} \int_0^\infty \int Amp(\sqrt{h}, \vec k) e^{- i \sqrt{h} x^0 - i \vec k \cdot \vec x} \delta(- h + \vec k + m^2) d^p \vec k \, \frac{d h}{2\sqrt{h}} \\ & \phantom{=} + (2\pi)^{-p/2} \int_0^\infty \int Amp(- \sqrt{h}, \vec k) e^{+ i \sqrt{h} x^0 - i \vec k \cdot \vec x} \delta(- h + \vec k^2 + m^2) d^p \vec k \, \frac{d h}{2 \sqrt{h}} \\ & = \phantom{+} (2\pi)^{-p/2} \int Amp(E(\vec k), \vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k \\ & \phantom{=} + (2\pi)^{-p/2} \int Amp(-E(\vec k), \vec k) e^{+ i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k \end{aligned}

where we defined the on-shell energy

E(k)+k 2+m 2. E(\vec k) \coloneqq + \sqrt{ \vec k^2 + m^2 } \,.

It is convenient to also change variables kk\vec k \mapsto - \vec k in the second integral. This yields

Φ(x) =+(2π) p/2Amp(E(k),k)e iE(k)x 0ikxd pk =+(1) p(2π) p/2Amp(E(k),k)e +iE(k)x 0+ikxd pk. \begin{aligned} \Phi(x) &= \phantom{+} (2\pi)^{-p/2} \int Amp(E(\vec k), \vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k \\ & \phantom{=} + (-1)^p (2\pi)^{-p/2} \int Amp(-E(\vec k), \vec k) e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x} d^p \vec k \end{aligned} \,.

Since Φ\Phi is real-valued, it follows that under complex conjugation () *(-)^\ast the amplitudes are related by

Amp(E(k),k) *=(1) pAmp(E(k),k). Amp(E(\vec k), \vec k)^\ast = (-1)^p Amp(-E(\vec k), \vec k) \,.

We abbreviate (cf. Scharf 01 (1.1.18))

A(k)E(k)Amp(E(k),k), A(\vec k) \coloneqq E(\vec k) Amp(E(\vec k), \vec k) \,,

where the prefactor just serves to make some of the following formulas come out conveniently.

With this the general solution to the Klein-Gordon equation is finally of the form

(2)Φ(x)=(2π) p/212E(k)(A(k)e iE(k)x 0ikx+A(k) *e +iE(k)x 0+ikx)d pk. \Phi(x) = (2\pi)^{-p/2} \int \frac{1}{\sqrt{2 E(\vec k)}} \left( A(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} + A(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x} \right) d^p \vec k \,.

and hence its time derivative is

0Φ(x)=(2π) p/2iE(k)/2(A(k)e iE(k)x 0ikxA(k) *e +iE(k)x 0+ikx)d pk. \partial_0 \Phi(x) = (2\pi)^{-p/2} \int -i \sqrt{E(k)/2} \left( A(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} - A(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x} \right) d^p \vec k \,.

This allows to express the modes in terms of the value of the field and its time derivative at t=0t = 0:

A(k)=(2π) p/2(E(k)/2Φ(0,x)+i12E(k) 0Φ(0,x))exp(ikx)d px. A(\vec k) = (2 \pi)^{-p/2} \int \left( \sqrt{E(k)/2}\Phi(0,\vec x) + i \frac{1}{2\sqrt{E(k)}} \partial_0 \Phi(0,\vec x)\right) \exp( i \vec k \cdot \vec x ) d^p \vec x \,.

As in example 1 we denote the corresponding evaluation functional

a(k):C (Σ) a(\vec k) \;\colon\; C^\infty(\Sigma) \longrightarrow \mathbb{C}

by the corresponding lower case symbol:

a(k)(2π) p/2(E(k)/2ϕ(x)+i12E(k) 0ϕ(x))exp(ikx)d px. a(\vec k) \;\coloneqq\; (2 \pi)^{-p/2} \int \left( \sqrt{E(k)/2}\phi(\vec x) + i \frac{1}{\sqrt{2 E(k)}} \partial_0 \phi(\vec x)\right) \exp( i \vec k \cdot \vec x ) d^p \vec x \,.

With the Poisson bracket kernel {ϕ(x),ϕ(y)}=δ(xy)\{\phi(\vec x), \phi(\vec y)\} = \delta(\vec x - \vec y) from example 1 (1), it follow that the (integral kernel for the) Poisson bracket of these mode functionals is that of the canonical commutation relations:

(3){a(k 1),a(k 2) *} =i(2π) pδ(x 1x 2)e i(k 1x 1k 2x 2)dx 1dx 2 =i(2π) pe i(k 1k 2)xdx = =iδ(k 1k 2), \begin{aligned} \{ a(\vec k_1), a(\vec k_2)^\ast \} & = -i (2\pi)^{-p} \int \delta(\vec x_1 - \vec x_2) e^{i ( \vec k_1 \cdot \vec x_1 - \vec k_2 \cdot \vec x_2) } d \vec x_1 d\vec x_2 \\ & = -i (2\pi)^{-p} \int e^{i (\vec k_1 - \vec k_2) \cdot \vec x } d \vec x & = \\ & = -i \delta(\vec k_1 - \vec k_2) \,, \end{aligned}

where in the last step we used the Fourier transform representation of the delta distribution (this prop.).

In order to finally compute {ϕ(x),ϕ(y)}\{\phi(x), \phi(y)\}, it is convenient to break this up into two contributions: Write

ϕ (+)(x)(2π) p/212E(k)a(k) *e +iE(k)x 0+ikxd pkAAAAϕ ()(x)(2π) p/212E(k)a(k)e iE(k)x 0ikxd pk \phi^{(+)}(x) \;\coloneqq\; (2\pi)^{-p/2} \int \frac{1}{\sqrt{2 E(\vec k)}} a(\vec k)^\ast e^{+ i E(\vec k) x^0 + i \vec k \cdot \vec x} d^p \vec k \phantom{AAAA} \phi^{(-)}(x) \;\coloneqq\; (2\pi)^{-p/2} \int \frac{1}{\sqrt{2 E(\vec k)}} a(\vec k) e^{- i E(\vec k) x^0 - i \vec k \cdot \vec x} d^p \vec k

for the positive and negative energy contributions from the Fourier expansion in (2), so that

ϕ(x)=ϕ ()(x)+ϕ (+)(x). \phi(x) = \phi^{(-)}(x) + \phi^{(+)}(x) \,.

Using the canonical commutation relation of the mode functions (3), we find

(4)iω(x,y) {ϕ ()(x),ϕ (+)(y)} =i(2π) p12E(k)e iE(k)(xy) 0k(xy)d pk =i(2π) pδ(k μk μ+m 2)Θ(k 0)e ik μ(xy) μd p+1k, \begin{aligned} -i \omega(x,y) & \coloneqq \{ \phi^{(-)}(x), \phi^{(+)}(y) \} \\ &= -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)} d^{p} \vec k \\ & = -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) \Theta( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \,, \end{aligned}

where in the last line we again applied change of integration variables. This ω\omega is known as the 2-point function or Hadamard propagator on Minkowski spacetime (see def. 2 below).

Similarly

{ϕ (+)(x),ϕ ()(y)} =+i(2π) p12E(k)e iE(k)(xy) 0+k(xy)d pk =+i(2π) pδ(k μk μ+m 2)Θ(k 0)e ik μ(xy) μd p+1k . \begin{aligned} \{ \phi^{(+)}(x), \phi^{(-)}(y) \} & = + i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)} d^{p} \vec k \\ & = + i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) \Theta( -k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \\ \end{aligned} \,.

In particular this says that

{ϕ (+)(x),ϕ ()(y)}=ω(y,x){ϕ ()(y),ϕ (+)(x)}. \{ \phi^{(+)}(x), \phi^{(-)}(y) \} = - \omega(y,x) \coloneqq - \{ \phi^{(-)}(y), \phi^{(+)}(x) \} \,.

With this we finally obtain the expression for the causal propagator as the skew-symmetrization of the 2-point function:

{ϕ(x),ϕ(y)} ={ϕ ()(x),ϕ (+)(y)}+{ϕ (+)(x),ϕ ()(y)} =ω(x,y)ω(y,x) =i(2π) p12E(k)(e iE(k)(xy) 0k(xy)e +iE(k)(xy) 0+k(xy))d pk =i(2π) pδ(k μk μ+m 2)sgn(k 0)e ik μ(xy) μd p+1k. \begin{aligned} \{ \phi(x), \phi(y) \} & = \{ \phi^{(-)}(x), \phi^{(+)}(y) \} + \{\phi^{(+)}(x), \phi^{(-)}(y)\} \\ & = \omega(x,y) - \omega(y,x) \\ & = -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)}\left( e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)} - e^{+ i E(\vec k) (x-y)^0 + \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \,. \end{aligned}

We record the 2-point function that appeared in this computation:

Definition

(2-point function of scalar field on Minkowski spacetime)

The 2-point function or Hadamard propagator of the scalar field on Minkowski spacetime

ω𝒟(Σ×Σ) \omega \;\in\; \mathcal{D}'(\Sigma \times \Sigma)

is (4)

ω(x,y)i{ϕ ()(x),ϕ (+)(y)}. \omega(x,y) \;\coloneqq\; i \{\phi^{(-)}(x), \phi^{(+)}(y)\} \,.
On general spacetimes

We discuss the covariant phase space of the free scalar field on general spacetimes.

Definition

(local Lagrangian density for free scalar field on general spacetime)

As a classical local field the relativistic free scalar field in dimension p+1p+1 \in \mathbb{N} of mass m[0,)m \in [0,\infty) is

  1. the field bundle given by the trivial line bundle X×kpr 1XX \times k \overset{pr_1}{\to} X over XX;

  2. the local Lagrangian density LΩ p+1,0(J X (X×k))L \in \Omega^{p+1,0}(J^\infty_X(X \times k)) (a horizontal differential form on the jet bundle of the trivial line bundle over XX) given by

    L(|ϕ| 2+m 2ϕ 2)dvol Σ L \;\coloneqq\; \left( \vert \nabla \phi\vert^2 + m^2 \phi^2\right) dvol_ \Sigma

    where

    1. |ϕ| 2\vert \nabla \phi\vert^2 denotes the norm square of the first order jets with respect to the given metric gg, hence in a local coordinate chart J (E| U){{x μ},ϕ,{ϕ ,μ}}J^\infty(E\vert_U) \coloneqq \left\{ \{x^\mu\}, \phi, \{\phi_{,\mu} \cdots\} \right\} of J (X×k)J^\infty(X \times k) the function

      |ϕ| 2| J (E| U)=g μνϕ ,μϕ ,ν \vert \nabla \phi\vert^2\vert_{J^{\infty}(E\vert_U)} \;=\; g^{\mu \nu} \phi_{,\mu} \phi_{,\nu}
    2. dvoldvol denotes the volume form of (Σ,e)(\Sigma,e), canonically regarded as a horizontal differential form on J (Σ×k)J^\infty(\Sigma \times k).

The analogue of prop. 2 holds true for general spacetimes:

(e.g. Bär-Ginoux-Pfäffle 07, corollary 3.4.3)

Proposition

The induced Poisson bracket on the covariant phase space of the free scalar field (def. 3) is given by the Peierls bracket. (…)

By this prop. (e.g. Khavkine 14, Collini 16, lemma 21). See also Fredenhagen-Rejzner 15, 3.3 Example

Interacting scalar field

We discuss perturbative quantum field theory of the free scalar field perturbed by an interaction term via locally covariant perturbative algebraic quantum field theory.

On Minkowski spacetime

We disscuss the interacting scalar field on Minkowski spacetime via causal perturbation theory.

By the discussion at S-matrix we need to determine the Feynman propagator, which is a linear combination of the 2-point function with the advanced propagator:

Definition

advanced propagator

Δ A(x.y)θ((xy) 0)Δ(x,y) \Delta_A(x.y) \coloneqq \theta((x-y)^0) \Delta(x,y)

retarded propagator

Δ R(x.y)θ((yx) 0)Δ(x,y) \Delta_R(x.y) \coloneqq \theta((y-x)^0) \Delta(x,y)

References

For instance

Most of the literatur on causal perturbation theory and perturbative AQFT focuses on the scalar field, for ease of exposition. See the references there.

The standard perturbative quantum field theory (made rigorous via causal perturbation theory) of the interacting scalar field is shown to be Fedosov deformation quantization of the corresponding covariant phase space in

For references on the construction of perturbative scalar field theory in causal perturbation theory see at locally covariant perturbative quantum field theory.

Discussion of scalar fields in cosmology includes

Last revised on September 22, 2017 at 15:06:32. See the history of this page for a list of all contributions to it.