Appendix B — Free-Field and Rindler Primer

Free-Field and Rindler Primer

This appendix collects the QFT-side prerequisites used in Block C and beyond. It is designed for a student who knows quantum mechanics but has limited prior exposure to QFT. The course’s modular conventions (from Week 4 §6.1 and the note-quality template) are used throughout.

B.1 Spacetime Conventions

We work in $(d+1)$-dimensional Minkowski space ${\mathbb{R}}^{1,d}$ with signature $(-,+,\dots ,+)$, metric $\eta =\mathrm{diag}(-1,+1,\dots ,+1)$, coordinates $x^{\mu }=(x^0,x^1,\dots ,x^d)$. The course’s two main examples:

• $d=1$: 2D Minkowski (one space dim + time). Most explicit calculations are here.
• $d=3$: 4D Minkowski (three space dims + time). Physical case.

For most of the course we set $\hslash =c=1$ and work in natural units.

B.2 Distributions and Test Functions

A test function $f\in \mathcal{S}({\mathbb{R}}^{1,d})$ is a Schwartz function: smooth, with all derivatives decaying faster than any polynomial. The dual space ${\mathcal{S}}^{\prime }({\mathbb{R}}^{1,d})$ consists of tempered distributions.

The smeared field $\varphi (f)=\int d^{d+1}xf(x)\varphi (x)$ pairs an operator-valued distribution $\varphi$ with a test function $f$ to produce an honest operator.

Fourier transform conventions (used in Weeks 8–11):

$$\tilde{f}(p):=\int d^{d+1}xf(x)e^{ip\cdot x},f(x)=\int \frac{d^{d+1}p}{(2\pi {)}^{d+1}}\tilde{f}(p)e^{-ip\cdot x},$$

with $p\cdot x=-p^0{x}^0+\vec{p}\cdot \vec{x}$ (Minkowski inner product).

For free fields, the on-shell measure restricts integration over $p^0$ to ${\omega }_p=\sqrt{|\vec{p}{|}^2+m^2}$:

$$\int \frac{d^{d+1}p}{(2\pi {)}^{d+1}}(2\pi )\delta (p^2+m^2)\Theta (p^0)\tilde{f}(p)=\int \frac{d^d\vec{p}}{(2\pi {)}^{d}2{\omega }_p}\tilde{f}({\omega }_p,\vec{p}).$$

B.3 The Free Scalar Field

The free scalar field of mass $m\ge 0$ on Minkowski is the operator-valued distribution

$$\varphi (x)=\int \frac{d^d\vec{p}}{(2\pi {)}^d\sqrt{2{\omega }_p}}\left[a_{\vec{p}}{e}^{-i{\omega }_p{x}^0+i\vec{p}\cdot \vec{x}}+a_{\vec{p}}^{†}{e}^{+i{\omega }_p{x}^0-i\vec{p}\cdot \vec{x}}\right],$$

acting on Fock space $\mathcal{F}={⨁}_n{\mathrm{Sym}}^n(L^2({\mathbb{R}}^d))$, with $[a_{\vec{p}},a_{{\vec{p}}^{\prime }}^{†}]=(2\pi {)}^d{\delta }^d(\vec{p}-{\vec{p}}^{\prime })$ and vacuum $a_{\vec{p}}|0_M⟩=0$.

Smeared: on the finite-particle domain, $\varphi (f)$ is symmetric for real $f\in \mathcal{S}$ and is essentially self-adjoint there. Its closure is the self-adjoint operator used to define $W(f)=e^{i\varphi (f)}$.

Klein–Gordon equation: $(\square -m^2)\varphi =0$. As an operator identity on smeared fields: $\varphi ((\square -m^2)f)=0$ for $f\in \mathcal{S}$. Test functions related by $(\square -m^2)$ give the same smeared field; this is the “test functions modulo the equation of motion” structure.

B.4 Wightman Two-Point Function

On the complexified test-function space, define the positive-frequency wavefunction

$$f_{+}(\vec{p}):=\int d^{d+1}xf(x)e^{i{\omega }_p{x}^0-i\vec{p}\cdot \vec{x}}=\tilde{f}(-{\omega }_p,-\vec{p}).$$

The positive sesquilinear two-point form is then

$$W(f,g):=⟨0_M|\varphi (f{)}^{\ast }\varphi (g)|0_M⟩=\int \frac{d^d\vec{p}}{(2\pi {)}^{d}2{\omega }_p}\overline{f_{+}(\vec{p})}{g}_{+}(\vec{p}).$$

For real $f$ and $g$, this is the usual smeared Wightman function $⟨0_M|\varphi (f)\varphi (g)|0_M⟩$. Naming $f_{+}$ avoids hiding a Fourier-sign choice in later calculations.

For unsmeared fields:

$$W(x-y):=⟨0_M|\varphi (x)\varphi (y)|0_M⟩=\int \frac{d^d\vec{p}}{(2\pi {)}^{d}2{\omega }_p}{e}^{-i{\omega }_p(x^0-y^0)+i\vec{p}\cdot (\vec{x}-\vec{y})}.$$

Two-dimensional massless form, with an infrared qualification. There is no ordinary Poincaré-invariant vacuum for the unsmeared massless scalar field including its zero mode. On the zero-integral test-function subspace (or for the derivative field), one may use the boundary value

$$W_{ϵ}(x-y)=-\frac{1}{4\pi }\log \left[{\mu }^2(-(\Delta t-iϵ{)}^2+(\Delta x{)}^2)\right],$$

up to an additive infrared-scale-dependent constant that drops out on that subspace. Here $\Delta t=x^0-y^0$ and $\Delta x=x^1-y^1$.

Four-dimensional massless form:

$$W_{ϵ}(x-y)=\frac{1}{4{\pi }^2}\frac{1}{-(\Delta t-iϵ{)}^2+|\Delta \vec{x}{|}^2}.$$

The limit $ϵ\downarrow 0$ defines a tempered-distribution boundary value. It does not turn the light-cone singularity into an ordinary pointwise function; smearing remains essential.

B.5 Pauli–Jordan Distribution

The commutator $[\varphi (x),\varphi (y)]=i\Delta (x-y)\cdot 1$ is a $c$-number (the Pauli–Jordan distribution).

For real test functions $f,g$, the symplectic form is

$$\sigma (f,g):=-i⟨0_M|[\varphi (f),\varphi (g)]|0_M⟩=\int d^{d+1}x{d}^{d+1}yf(x)\Delta (x-y)g(y).$$

(B-R convention; matches Week 8 §3.2.)

Properties:

• $\sigma$ is real, bilinear, antisymmetric.
• Spacelike vanishing: $\sigma (f,g)=0$ if $\mathrm{supp}(f)$ and $\mathrm{supp}(g)$ are spacelike separated.
• $\sigma (f,g)=2\mathrm{Im}W(f,g)$ for real $f,g$.

2D massless explicit form:

$$\Delta (x-y)=-\frac{1}{2}\mathrm{sgn}(x^0-y^0)\Theta (-(x-y{)}^2).$$

Sign $-\frac{1}{2}$ inside the future light cone of $y$, $+\frac{1}{2}$ inside the past, zero outside.

4D massless explicit form:

$$\Delta (x-y)=-\frac{1}{2\pi }\mathrm{sgn}(x^0-y^0)\delta ((x-y{)}^2).$$

The minus sign agrees with $[\varphi (x),\varphi (y)]=i\Delta (x-y)$ and the equal-time condition ${\partial }_{x^0}\Delta (0,\vec{x}-\vec{y})=-{\delta }^d(\vec{x}-\vec{y})$.

2D massive form ($m>0$):

$$\Delta (x-y)=-\frac{1}{2}\mathrm{sgn}(x^0-y^0)J_0(m\sqrt{-(x-y{)}^2})\Theta (-(x-y{)}^2).$$

Bessel function; oscillates inside the light cone.

B.6 Weyl Operators (Week 8 §4)

For real $f$ in the Klein—Gordon symplectic test-function space, the Weyl operator is $W(f):=e^{i\varphi (f)}$. In the two-dimensional massless example we also impose the zero-mode restriction described in §B.4.

Weyl relation:

$$W(f)W(g)=e^{-i\sigma (f,g)/2}W(f+g).$$

Consequences:

• $W(f{)}^{\ast }=W(-f)$, $W(0)=1$.
• If $\sigma (f,g)=0$ (spacelike support), $W(f)$ and $W(g)$ commute.
• Vacuum expectation (Gaussian): $⟨0_M|W(f)|0_M⟩=e^{-W(f,f)/2}$ for real $f$.

B.7 Local Algebras

For open $\mathcal{O}\subset {\mathbb{R}}^{1,d}$,

$$\mathcal{A}(\mathcal{O}):=\{W(f):\mathrm{supp}(f)\subset \mathcal{O}{\}}^{\prime \prime }.$$

This is the local von Neumann algebra.

Net structure (Week 9; Haag–Kastler axioms):

1. Isotony: ${\mathcal{O}}_1\subset {\mathcal{O}}_2\Rightarrow \mathcal{A}({\mathcal{O}}_1)\subset \mathcal{A}({\mathcal{O}}_2)$.
2. Locality: ${\mathcal{O}}_1,{\mathcal{O}}_2$ spacelike $\Rightarrow [\mathcal{A}({\mathcal{O}}_1),\mathcal{A}({\mathcal{O}}_2)]=0$.
3. Covariance under ${\mathcal{P}}_{+}^{\uparrow }$.
4. Vacuum invariance.
5. Spectral condition: the joint energy—momentum spectrum lies in the closed future light cone, $\mathrm{sp}(P)\subset {\overline{V}}_{+}$.

B.8 Rindler Coordinates

The right Rindler wedge:

$$W_R:=\{x:x^1>|x^0|\}.$$

Rindler coordinates on $W_R$:

$$x^1=\xi \cosh \eta ,x^0=\xi \sinh \eta ,\xi >0,\eta \in \mathbb{R}.$$

• $\xi$: “radial” coordinate, fixed-$\xi$ worldlines are uniformly accelerated.
• $\eta$: “Rindler time,” $\eta \to \eta +s$ is the Lorentz boost.

Metric in Rindler: $d{s}^2=-{\xi }^{2}d{\eta }^2+d{\xi }^2+d{\vec{x}}_{\perp }^2$. Lapse $\xi$.

Observer at $\xi ={\xi }_0$:

• Proper acceleration $a=1/{\xi }_0$.
• Proper time $\tau ={\xi }_0\eta$.
• Modular time $t=\eta /(2\pi )$ relates to proper time as $\tau =2\pi {\xi }_{0}t=2\pi t/a$.

B.9 Boost Generator

The boost subgroup ${\Lambda }^{\mathrm{boost}}(u):(x^0,x^1)\mapsto (\cosh u{x}^0+\sinh u{x}^1,\sinh u{x}^0+\cosh u{x}^1)$ preserves $W_R$. We fix the active covariance convention

$$U(\Lambda (u))\varphi (x)U(\Lambda (u){)}^{\ast }=\varphi (\Lambda (u)x),U(\Lambda (u))=e^{iuK},$$

where, at $x^0=0$,

$$K={\int }_{x^1>0}{x}^1{T}^{00}(0,\vec{x})d^d\vec{x}-{\int }_{x^1<0}(-x^1)T^{00}(0,\vec{x})d^d\vec{x},$$

the global boost generator. The two displayed integrals are simply ${\int }_{{\mathbb{R}}^d}{x}^1{T}^{00}(0,\vec{x})d^d\vec{x}$ split at $x^1=0$. $K{\Omega }_M=0$ (vacuum invariance), while $K$ has two-sided spectrum.

A boost changes $\eta$ and leaves $\xi$ fixed. It moves an observer along one uniformly accelerated orbit; it does not move that observer toward the bifurcation surface. Approaching the horizon or bifurcation surface requires the separate limit $\xi \downarrow 0$.

B.10 Bisognano–Wichmann (Week 10)

In our convention,

$${\Delta }_{W_R}=e^{-2\pi K},{\sigma }_t^{W_R}(a)=U({\Lambda }^{\mathrm{boost}}(2\pi t))aU({\Lambda }^{\mathrm{boost}}(2\pi t){)}^{\ast }.$$

Modular flow = boost subgroup at rapidity $+2\pi t$. Unruh temperature $T_U=a/(2\pi )$ for an observer at $\xi =1/a$.

B.11 Bogoliubov Transformation in 2D Massless (Week 10 §4)

Decompose $\varphi ={\varphi }_R(x^{-})+{\varphi }_L(x^{+})$ into right/left movers ($x^{\pm }=x^0\pm x^1$). For right-movers in $W_R$:

Minkowski modes: ${\varphi }_R(x^{-})={\int }_0^{\infty }\frac{dk}{\sqrt{4\pi k}}[a_k{e}^{-ik{x}^{-}}+h.c.]$.

Rindler modes in $W_R$ (positive-frequency with respect to $\eta$): for the right-moving sector one may take

$$u_{\omega }^R(x^{-})=\frac{\Theta (-x^{-})}{\sqrt{4\pi \omega }}(-a{x}^{-}{)}^{i\omega /a},x^{-}=-\xi e^{-\eta }.$$

Along a fixed-$\xi$ orbit this is a $\xi$-dependent phase times $e^{-i\omega \eta /a}$. Setting $a=1$ gives the dimensionless convention used in Week 10. The associated annihilation operators are denoted $b_{\omega }^R$.

Bogoliubov ratio:

$$\frac{|{\beta }_{\omega k}{|}^2}{|{\alpha }_{\omega k}{|}^2}=e^{-2\pi \omega /a}.$$

In the dimensionless convention $a=1$, this is $e^{-2\pi \omega }$.

Thermal occupation:

$$⟨0_M|(b_{\omega }^R{)}^{†}{b}_{\omega }^R|0_M⟩\propto \frac{1}{e^{2\pi \omega /a}-1}.$$

This is Bose—Einstein at proper-time temperature $a/(2\pi )$. With $a=1$, it is equivalently thermal at dimensionless Rindler-time inverse temperature $2\pi$.

B.12 Two-Sided Rindler (TFD; Week 15)

The left wedge $W_L=\{x:x^1<-|x^0|\}$ is spacelike to $W_R$, and $[\mathcal{A}(W_R),\mathcal{A}(W_L)]=0$.

With a mode regulator, the Minkowski vacuum takes the familiar TFD form:

$$|0_M⟩=\prod _{\omega }\frac{1}{\sqrt{Z_{\omega }}}\sum _{n_{\omega }}{e}^{-\pi n_{\omega }\omega }|n_{\omega }{⟩}_R\otimes |n_{\omega }{⟩}_L.$$

This is the TFD of the regulated Rindler-mode Hamiltonian at dimensionless temperature $T_R=1/(2\pi )$. In continuum AQFT the product and tensor factorization are formal mnemonic devices: the wedge algebras are type III and the vacuum is handled directly as a cyclic-separating vector.

In the regulated bipartite model, the modular operator for the right algebra is

$${\Delta }_R=e^{-2\pi (K_R-K_L)}.$$

This is one-sided modular data acting in a purification; it is not a thermal density matrix for the joint algebra.

B.13 What’s Used Where

Topic	Week
Smeared fields and Weyl operators	8
Wightman function and Pauli–Jordan	8
Local algebras and Haag–Kastler axioms	9
Reeh—Schlieder cyclicity; separation from locality when the causal complement has interior	9
Rindler coordinates and boost generator	10
Bisognano–Wichmann + Bogoliubov verification	10
Bell–CHSH on wedges	11
Type III${}_1$ classification of local algebras	12
Free-field Rindler crossed product	13
Dressed entropy on Rindler crossed product	14
Regulated TFD picture and its type-III qualification	15

B.14 Further Reading

• Streater & Wightman, PCT, Spin and Statistics, and All That (Wightman axioms, ch. 3).
• Haag, Local Quantum Physics (the standard AQFT reference).
• Birrell & Davies, Quantum Fields in Curved Space (Rindler quantization, ch. 4).
• Witten, “Notes on some entanglement properties of QFT,” arXiv:1803.04993 (modern algebraic introduction; consistent with our conventions).
• Crispino, Higuchi, Matsas, “The Unruh effect and its applications,” Rev. Mod. Phys. 80 (2008) 787 (Unruh effect review).