Free Fields, Wightman Functions, and the iε Prescription

The previous page ended with a very simple fact: the normalization of a free oscillator mode is not cosmetic. It is what makes the canonical commutator come out correctly. We now use that normalization to compute the first genuinely Lorentzian objects in the CFT part of the course: Wightman functions, time-ordered functions, and their $i\epsilon$ prescriptions.

The point of this page is not merely to remember where the symbol $i\epsilon$ goes. The real lesson is analytic. A vacuum correlator such as $\langle0|\phi(t)\phi(0)|0\rangle$ is generally a distribution on the real time axis. Because the spectrum is bounded below, it is the boundary value of a function analytic in one half of the complex time plane. Different operator orderings arise from different analytic functions and different sides of approach. The familiar Feynman prescription is the compact way of keeping track of these boundary values.

This becomes especially sharp in conformal field theory. Euclidean correlators are ordinary power laws. Lorentzian correlators are distributions obtained by approaching their singular light cones from specified sides. The distinction between

$${1\over (\mathbf x^2-t^2)^\Delta}, \qquad {1\over (\mathbf x^2-(t-i0)^2)^\Delta}, \qquad {1\over (\mathbf x^2-t^2+i0)^\Delta}$$

is the distinction between a formal expression, a Wightman function, and a time-ordered Green function.

One oscillator already contains the prescription

Lorentzian boundary values

We use Lorentzian time $t$ and spatial coordinate $\mathbf x$. The mostly-minus invariant is

$$x_L^2=t^2-\mathbf x^2.$$

It is often cleaner on this page to write

$$\rho=\mathbf x^2-t^2=-x_L^2,$$

because free and conformal scalar correlators naturally appear with denominators of the form $\mathbf x^2-(t\mp i0)^2$. The symbol $0$ means a positive infinitesimal. Thus

$$t-i0$$

means that the real time $t$ is approached from the lower half of the complex $t$-plane.

Start with a single harmonic oscillator of frequency $\omega$,

$$q(t)={1\over\sqrt{2\omega}} \left(ae^{-i\omega t}+a^\dagger e^{i\omega t}\right), \qquad [a,a^\dagger]=1, \qquad a|0\rangle=0.$$

Then

$$\begin{aligned} W_+(t) &=\langle0|q(t)q(0)|0\rangle \\ &={1\over2\omega}\langle0| \left(ae^{-i\omega t}+a^\dagger e^{i\omega t}\right)(a+a^\dagger)|0\rangle \\ &={e^{-i\omega t}\over2\omega}. \end{aligned}$$

The opposite ordering gives

$$W_-(t)=\langle0|q(0)q(t)|0\rangle ={e^{+i\omega t}\over2\omega}=W_+^*(t),$$

for real $t$. Because the oscillator commutator is a c-number, their difference is the commutator itself (times the identity operator),

$$[q(t),q(0)]=W_+(t)-W_-(t) =-{i\sin\omega t\over\omega}.$$

The two orderings of a free oscillator give the two Wightman functions $W_+$ and $W_-$. Their difference equals the oscillator’s c-number commutator. Free-field formulas are continuum sums of this elementary identity; for a general interacting operator, the analogous difference is a vacuum expectation value of the operator-valued commutator.

This baby calculation already displays three important features.

First, a Wightman function is not time ordered. It is an ordinary vacuum expectation value in a specified operator order.

Second, $W_+(t)$ contains only positive energies propagating forward in the chosen ordering. It oscillates as $e^{-i\omega t}$, not as a symmetric cosine.

Third, the commutator is not the same object as the Wightman function. The Wightman function knows about vacuum fluctuations. The commutator supplies the spectral kernel from which causal response is built; a retarded correlator also includes the appropriate step function and convention-dependent factor of $i$.

For a free scalar field, the oscillator label becomes momentum:

$$\phi(t,\mathbf x) =\int {d^d\mathbf p\over(2\pi)^d}{1\over\sqrt{2\omega_{\mathbf p}}} \left( a_{\mathbf p}e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x} +a_{\mathbf p}^\dagger e^{i\omega_{\mathbf p}t-i\mathbf p\cdot\mathbf x} \right),$$

where

$$\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}, \qquad [a_{\mathbf p},a_{\mathbf q}^\dagger]=(2\pi)^d\delta^{(d)}(\mathbf p-\mathbf q).$$

The positive-frequency Wightman function is therefore

$$\boxed{ W_+(t,\mathbf x) =\langle0|\phi(t,\mathbf x)\phi(0)|0\rangle =\int {d^d\mathbf p\over(2\pi)^d}{1\over2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x}. }$$

Similarly,

$$W_-(t,\mathbf x) =\langle0|\phi(0)\phi(t,\mathbf x)|0\rangle =W_+(-t,-\mathbf x).$$

For the parity-even free scalar used here—or whenever spatial inversion is a symmetry—$W_+(-t,-\mathbf x)=W_+(-t,\mathbf x)$. Ordinary rotations already imply this for $d\ge2$; in one spatial dimension parity is an additional assumption. Thus

$$W_-(t,\mathbf x)=W_+(-t,\mathbf x).$$

In momentum space the same object can be written invariantly as

$$W_+(x) =\int {d^D p\over(2\pi)^D} (2\pi)\theta(p^0)\delta(p^2-m^2)e^{-ip\cdot x}, \qquad D=d+1.$$

The step function $\theta(p^0)$ is the spectral arrow: $W_+$ only contains positive-energy on-shell modes.

Spectral positivity and half-plane analyticity

The oscillator result is not an accident of free fields. Let $O(t)=O(t)^\dagger$ be a Hermitian Heisenberg operator in a theory whose Hamiltonian has an exact vacuum $|\Omega\rangle$ and a spectrum bounded below. Write $H|\Omega\rangle=E_\Omega|\Omega\rangle$ and insert a complete set of energy eigenstates (with the sum understood to include continuum integrals):

$$\begin{aligned} \langle\Omega|O(t)O(0)|\Omega\rangle &=\sum_n\langle\Omega|O(t)|n\rangle \langle n|O(0)|\Omega\rangle \\ &=\sum_n |\langle n|O(0)|\Omega\rangle|^2 e^{-i(E_n-E_\Omega)t}. \end{aligned}$$

For a non-Hermitian operator, the positive-type correlator is instead $\langle\Omega|O(t)O^\dagger(0)|\Omega\rangle$; its coefficients are again absolute squares. One may also replace $O$ by $O-\langle\Omega|O|\Omega\rangle$ when the vacuum contribution is not wanted.

Since $E_n-E_\Omega\ge0$, the exponential becomes

$$e^{-i(E_n-E_\Omega)(t_R+i t_I)} =e^{-i(E_n-E_\Omega)t_R}e^{(E_n-E_\Omega)t_I}.$$

The sum is damped when $t_I<0$. It therefore defines the analytic function

$$F_+(z)=\sum_n |\langle n|O|\Omega\rangle|^2 e^{-i(E_n-E_\Omega)z}, \qquad \operatorname{Im}z<0.$$

The reversed ordering similarly comes from

$$F_-(z)=\sum_n |\langle n|O|\Omega\rangle|^2 e^{+i(E_n-E_\Omega)z}, \qquad \operatorname{Im}z>0.$$

The real-time Wightman distributions are their boundary values:

$$\boxed{ \begin{aligned} W_+(t)&:=\langle\Omega|O(t)O(0)|\Omega\rangle =\lim_{\epsilon\downarrow0}F_+(t-i\epsilon),\\ W_-(t)&:=\langle\Omega|O(0)O(t)|\Omega\rangle =\lim_{\epsilon\downarrow0}F_-(t+i\epsilon). \end{aligned} }$$

These limits are generally limits of distributions, not pointwise limits. For finite $\epsilon>0$, the first spectral sum contains the genuine damping factor $e^{-\epsilon(E_n-E_\Omega)}$. The notation $i0$ records the side from which the distributional boundary is taken after $\epsilon\downarrow0$; it is not itself a finite convergence factor.

The spectral representation defines $F_+(z)$ for $\operatorname{Im}z<0$ and $F_-(z)$ for $\operatorname{Im}z>0$. The Wightman distributions $W_+$ and $W_-$ are their boundary values; $t-i0$ and $t+i0$ remember the side of approach.

This is the first clean way to understand the $i\epsilon$ prescription. It is not a small imaginary fudge factor added to make integrals converge. It encodes the spectrum condition and the operator ordering.

A useful boundary-value mnemonic is:

$$\begin{array}{ccl} \langle\Omega|O(t)O(0)|\Omega\rangle &:& z\to t-i0,\\ \langle\Omega|O(0)O(t)|\Omega\rangle &:& z\to t+i0. \end{array}$$

The sign is easy to remember from damping. Positive energies $e^{-iEt}$ decay when $t$ is moved downward.

Euclidean continuation

Set

$$t=-i\tau.$$

For $\tau>0$, this lies in the lower half-plane, so the analytic function whose boundary is $W_+$ gives

$$F_+(-i\tau,\mathbf x) =\int {d^d\mathbf p\over(2\pi)^d}{1\over2\omega_{\mathbf p}} e^{-\omega_{\mathbf p}\tau+i\mathbf p\cdot\mathbf x}.$$

This is the Euclidean two-point function for positive Euclidean time separation. In fully Euclidean momentum notation,

$$G_E(\tau,\mathbf x) =\int {d^D p_E\over(2\pi)^D} {e^{ip_E^0\tau+i\mathbf p\cdot\mathbf x}\over (p_E^0)^2+\mathbf p^2+m^2}.$$

The integral over $p_E^0$ can be done by closing the contour. It gives

$$G_E(\tau,\mathbf x) =\int {d^d\mathbf p\over(2\pi)^d}{1\over2\omega_{\mathbf p}} e^{-\omega_{\mathbf p}|\tau|+i\mathbf p\cdot\mathbf x}.$$

Thus Euclidean reflection symmetry in $\tau$ packages both Wightman orderings:

$$\tau>0 \quad \leftrightarrow \quad \langle\phi(t)\phi(0)\rangle,$$

while

$$\tau<0 \quad \leftrightarrow \quad \langle\phi(0)\phi(t)\rangle.$$

A Euclidean power law is an ordinary function away from coincident points. Lorentzian continuation turns it into a distributional boundary value. For the ordering $\langle\Omega|O(t,\mathbf x)O(0)|\Omega\rangle$, one approaches real time from $t-i0$.

For a massless scalar in $D$ Euclidean dimensions,

$$G_E(x_E)=\int {d^D p_E\over(2\pi)^D}{e^{ip_E\cdot x_E}\over p_E^2} ={\Gamma(D/2-1)\over4\pi^{D/2}} {1\over |x_E|^{D-2}}, \qquad D>2.$$

The restriction $D>2$ matters. In $D=2$ the massless scalar Green function is logarithmic rather than a power law, and the undifferentiated scalar has an infrared zero-mode subtlety. Its derivatives and suitably neutral vertex operators are the better-defined conformal observables.

This is exactly the form expected for a scalar operator of dimension

$$\Delta_\phi={D-2\over2}.$$

To obtain the first ordering, set $\tau=\epsilon+it$ with $\epsilon>0$, and only then take $\epsilon\downarrow0$. This gives

$$\boxed{ W_+(t,\mathbf x) ={\Gamma(D/2-1)\over4\pi^{D/2}} {1\over \left(\mathbf x^2-(t-i0)^2\right)^{(D-2)/2}}. }$$

For $D=4$, this reduces to

$$W_+(t,\mathbf x) ={1\over4\pi^2}{1\over \mathbf x^2-(t-i0)^2}.$$

The reversed ordering is the other boundary value:

$$W_-(t,\mathbf x) ={1\over4\pi^2}{1\over \mathbf x^2-(t+i0)^2}.$$

Time ordering and the Feynman prescription

The time-ordered two-point function is

$$G_F(t,\mathbf x) =\langle0|T\phi(t,\mathbf x)\phi(0)|0\rangle =\theta(t)W_+(t,\mathbf x)+\theta(-t)W_-(t,\mathbf x).$$

For the free scalar field, this equals

$$\boxed{ G_F(x)=\int {d^D p\over(2\pi)^D}{i e^{-ip\cdot x}\over p^2-m^2+i0}. }$$

At fixed spatial momentum,

$$G_F(t,\mathbf p) =\int {dp^0\over2\pi}{i e^{-ip^0t}\over (p^0)^2-\omega_{\mathbf p}^2+i0}.$$

For a finite regulator $\epsilon>0$ and $\omega_{\mathbf p}>0$, the denominator

$$D_\epsilon(p^0)=(p^0)^2-\omega_{\mathbf p}^2+i\epsilon$$

has the exact roots

$$p^0=\pm\sqrt{\omega_{\mathbf p}^2-i\epsilon} = \begin{cases} +\omega_{\mathbf p}-i\epsilon/(2\omega_{\mathbf p}) +O(\epsilon^2/\omega_{\mathbf p}^3),\\ -\omega_{\mathbf p}+i\epsilon/(2\omega_{\mathbf p}) +O(\epsilon^2/\omega_{\mathbf p}^3). \end{cases}$$

Thus the positive-energy pole approaches the real axis from below and the negative-energy pole from above. The useful statement at the boundary is the distributional partial-fraction identity

$$\boxed{ {1\over(p^0)^2-\omega_{\mathbf p}^2+i0} ={1\over2\omega_{\mathbf p}} \left( {1\over p^0-\omega_{\mathbf p}+i0} -{1\over p^0+\omega_{\mathbf p}-i0} \right). }$$

This identity, rather than a literal algebraic product of independently shifted factors, encodes the limiting prescription. In the usual shorthand, the pole locations are

$$p^0=+\omega_{\mathbf p}-i0, \qquad p^0=-\omega_{\mathbf p}+i0.$$

The Feynman prescription places the positive-energy pole below the real $p^0$ axis and the negative-energy pole above it. Closing the contour below for $t>0$ selects positive-frequency propagation; closing above for $t<0$ selects the opposite ordering.

For $t>0$, the exponential $e^{-ip^0t}$ decays on the large semicircle in the lower half-plane, so the contour encloses the pole at $+\omega_{\mathbf p}-i0$. The residue gives

$$G_F(t,\mathbf p)={e^{-i\omega_{\mathbf p}t}\over2\omega_{\mathbf p}}.$$

For $t<0$, the contour closes in the upper half-plane and encloses the pole at $-\omega_{\mathbf p}+i0$, giving

$$G_F(t,\mathbf p)={e^{+i\omega_{\mathbf p}t}\over2\omega_{\mathbf p}}.$$

Therefore

$$G_F(t,\mathbf p)={1\over2\omega_{\mathbf p}}e^{-i\omega_{\mathbf p}|t|},$$

which is precisely the oscillator time-ordered correlator.

Time ordering glues together the two Wightman boundary values. For a conformal scalar two-point function, the result is the compact Feynman prescription $C/(\mathbf x^2-t^2+i0)^\Delta$.

For a scalar primary of dimension $\Delta$, the Euclidean correlator is

$$G_E(x_E)={C\over |x_E|^{2\Delta}}.$$

The Wightman formulas below mean distributional limits. For example, the first is the limit of the smooth expression with $t-i\epsilon$ against test functions as $\epsilon\downarrow0$; it is not a pointwise function on the light cone. With the branch inherited from Euclidean signature, the boundary values are

$$\boxed{ W_+(t,\mathbf x)={C\over \left(\mathbf x^2-(t-i0)^2\right)^\Delta}, \qquad W_-(t,\mathbf x)={C\over \left(\mathbf x^2-(t+i0)^2\right)^\Delta}. }$$

The time-ordered correlator is

$$\boxed{ G_F(t,\mathbf x)={C\over\left(\mathbf x^2-t^2+i0\right)^\Delta}. }$$

This last formula is often the easiest one to remember, but it hides the more primitive statement: the $i0$ came from spectral analyticity and operator ordering.

The prescription fixes the Lorentzian boundary value away from coincidence. If the power law is too singular to extend uniquely through $x=0$, ultraviolet renormalization may still add contact terms supported at coincidence. Those local ambiguities do not change the noncoincident Wightman ordering or the causal support discussed next.

Vacuum commutator expectations as discontinuities

For a Hermitian scalar operator, the two Wightman functions determine the vacuum expectation value of the commutator:

$$C_O(t,\mathbf x) :=\langle\Omega|[O(t,\mathbf x),O(0)]|\Omega\rangle =W_+(t,\mathbf x)-W_-(t,\mathbf x).$$

This is not, in general, an identity between the operator-valued commutator and two c-number correlators. Microcausality is the separate, stronger operator statement that local bosonic operators commute at spacelike separation. A free-field commutator is exceptional: it is a c-number times the identity, so its vacuum expectation also gives the full commutator.

For a scalar primary two-point function, define

$$\rho=\mathbf x^2-t^2.$$

Then

$$W_+(t,\mathbf x)={C\over(\rho+i0\operatorname{sgn}t)^\Delta}, \qquad W_-(t,\mathbf x)={C\over(\rho-i0\operatorname{sgn}t)^\Delta}.$$

Thus the vacuum commutator expectation is the discontinuity across the branch cut of $\rho^{-\Delta}$:

$$\boxed{ C_O(t,\mathbf x) =C\left[ {1\over(\rho+i0\operatorname{sgn}t)^\Delta} -{1\over(\rho-i0\operatorname{sgn}t)^\Delta} \right]. }$$

If the separation is spacelike, then $\rho>0$. There is no branch cut, and the two boundary values agree. Therefore

$$C_O(t,\mathbf x)=0, \qquad \mathbf x^2>t^2.$$

This equality is the two-point reflection of causality. In a local theory, microcausality gives the operator identity $[O(t,\mathbf x),O(0)]=0$ throughout the same spacelike region (with a graded commutator for fermionic fields).

If the separation is timelike, then $\rho<0$. For non-integer $\Delta$, the two boundary values differ by a phase:

$$(\rho+i0)^{-\Delta}=e^{-i\pi\Delta}|\rho|^{-\Delta}, \qquad (\rho-i0)^{-\Delta}=e^{+i\pi\Delta}|\rho|^{-\Delta}.$$

Hence, pointwise in the timelike region away from the light cone,

$$C_O(t,\mathbf x) =-2iC\sin(\pi\Delta)\operatorname{sgn}(t) {\theta(t^2-\mathbf x^2)\over(t^2-\mathbf x^2)^\Delta}.$$

Globally, this expression means the distribution obtained by analytic continuation (equivalently, an appropriate Hadamard finite part), including light-cone-supported distributional terms. For $\Delta\ge1$, the displayed power times the step function is not by itself an ordinary locally integrable function at the light cone.

For the Lorentzian power law, the vacuum commutator expectation is the discontinuity across the timelike branch cut in $\rho=\mathbf x^2-t^2$. In the spacelike region $\rho>0$, the two boundary values coincide. Microcausality upgrades this c-number equality to an operator identity for local fields.

All of these boundary values are distributions. Define the generalized principal value

$$\operatorname{Pf}\frac1{u^n} :=\frac{(-1)^{n-1}}{(n-1)!}\, \partial_u^{\,n-1}\operatorname{PV}\frac1u.$$

At a positive integer $n$, the terms supported exactly at the branch point are displayed by the one-variable identity

$$\boxed{ (u\pm i0)^{-n} =\operatorname{Pf}{1\over u^n} \mp i\pi {(-1)^{n-1}\over(n-1)!}\,\delta^{(n-1)}(u). }$$

Consequently,

$$(u+i0)^{-n}-(u-i0)^{-n} =-2\pi i\,{(-1)^{n-1}\over(n-1)!}\,\delta^{(n-1)}(u).$$

In particular, for the free massless scalar in four spacetime dimensions, $\Delta=1$ and the sine factor vanishes away from the light cone. The free commutator is nevertheless nonzero as a distribution; it is a c-number times the identity supported on the light cone:

$$[\phi(t,\mathbf x),\phi(0)] =-{i\over2\pi}\operatorname{sgn}(t)\delta(t^2-\mathbf x^2) \,\mathbf 1$$

for the standard four-dimensional normalization. This is the familiar sharp propagation of the massless free wave equation. For interacting conformal fields with anomalous dimensions, the vacuum commutator expectation generally has support throughout the timelike region.

Conformal covariance and Lorentzian distributions

On the previous page, a primary field in two Euclidean dimensions was defined by the finite transformation rule

$$O(z,\bar z) \mapsto (f'(z))^h(\bar f'(\bar z))^{\bar h}O(f(z),\bar f(\bar z)).$$

Fix the Euclidean coordinate convention

$$x_E=(\tau,x), \qquad z=x+i\tau, \qquad \bar z=x-i\tau.$$

For a self-conjugate primary, the Euclidean two-point function on the plane is

$$\langle O(z,\bar z)O(0,0)\rangle ={C\over z^{2h}\bar z^{2\bar h}}.$$

For a charged or otherwise non-Hermitian primary, the operator at the origin is $O^\dagger$ instead. In either case, the complex powers require a specified branch.

This formula is single-valued only if the spin

$$s=h-\bar h$$

is compatible with the chosen spin structure. For the Ising Majorana fermion,

$$(h,\bar h)=\left({1\over2},0\right)$$

or

$$(h,\bar h)=\left(0,{1\over2}\right),$$

so the correlator is chiral:

$$\langle\psi(z)\psi(0)\rangle={1\over z}.$$

To pass to Lorentzian signature, introduce the light-cone coordinates

$$x^+=t+x, \qquad x^-=t-x.$$

For the ordering with the displayed operator first, continue from positive Euclidean time by setting $\tau=\epsilon+it$, with $\epsilon>0$. Then

$$z=-x^-+i\epsilon, \qquad \bar z=x^+-i\epsilon.$$

Thus, with branches inherited from the Euclidean correlator,

$$\boxed{ W_+(t,x) =\lim_{\epsilon\downarrow0} {C\over(-x^-+i\epsilon)^{2h}(x^+-i\epsilon)^{2\bar h}}. }$$

For a bosonic self-conjugate primary, the reversed ordering is obtained from $\tau=-\epsilon+it$:

$$\boxed{ W_-(t,x) =\lim_{\epsilon\downarrow0} {C\over(-x^--i\epsilon)^{2h}(x^++i\epsilon)^{2\bar h}}. }$$

For fermionic or semilocal fields, the coordinate-side prescriptions are the same, but the relative exchange or monodromy phase is fixed by the Euclidean graded-ordering and spin-structure conventions; it must not be inferred from the displayed bosonic formula alone.

For this convention, the holomorphic Majorana correlator is proportional to $(-x^-+i0)^{-1}$, whereas the antiholomorphic correlator is proportional to $(x^+-i0)^{-1}$ in the $W_+$ ordering. A convention that interchanges $z$ and $\bar z$, or defines the light-cone coordinates differently, relabels these prescriptions. Stating the continuation convention is therefore part of the Lorentzian formula, not an optional detail.

The same moral holds in any dimension. A Euclidean conformal two-point function is constrained by symmetry to be a power law. A Lorentzian conformal two-point function is a specified boundary value of that power law. That small phrase—“specified boundary value”—does a huge amount of work. It controls time ordering, causal commutators, retarded response, and Wick rotation.

Example: recovering the equal-time canonical commutator

For the free scalar field, start from

$$[\phi(t,\mathbf x),\phi(0,\mathbf 0)] =\int {d^d\mathbf p\over(2\pi)^d}{1\over2\omega_{\mathbf p}} \left(e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x}-e^{i\omega_{\mathbf p}t-i\mathbf p\cdot\mathbf x}\right).$$

Change $\mathbf p\mapsto-\mathbf p$ in the second term:

$$[\phi(t,\mathbf x),\phi(0,\mathbf 0)] =-i\int {d^d\mathbf p\over(2\pi)^d} {\sin(\omega_{\mathbf p}t)\over\omega_{\mathbf p}} e^{i\mathbf p\cdot\mathbf x}.$$

At equal time this vanishes:

$$[\phi(0,\mathbf x),\phi(0,\mathbf 0)]=0.$$

Differentiating with respect to $t$ gives

$$[\dot\phi(t,\mathbf x),\phi(0,\mathbf 0)] =-i\int {d^d\mathbf p\over(2\pi)^d} \cos(\omega_{\mathbf p}t)e^{i\mathbf p\cdot\mathbf x}.$$

At $t=0$,

$$[\dot\phi(0,\mathbf x),\phi(0,\mathbf 0)] =-i\delta^{(d)}(\mathbf x).$$

Equivalently,

$$[\phi(0,\mathbf x),\pi(0,\mathbf 0)] =i\delta^{(d)}(\mathbf x), \qquad \pi=\dot\phi.$$

Thus the same Wightman functions whose boundary values define the Feynman prescription also encode the canonical equal-time algebra.

Summary

A free field is a continuum of oscillators. Each oscillator gives two Wightman functions,

$$W_+(t)=\langle0|q(t)q(0)|0\rangle, \qquad W_-(t)=\langle0|q(0)q(t)|0\rangle,$$

and the field-theory correlators are momentum integrals of these elementary objects.

The spectrum condition defines $F_+$ in the lower half of the complex time plane and $F_-$ in the upper half-plane. The Wightman distributions $W_+$ and $W_-$ are their respective boundary values. The prescriptions $t-i0$ and $t+i0$ are therefore fixed by operator ordering.

The Feynman propagator glues the two Wightman functions with time-ordering step functions. In momentum space this is the pole prescription

$${ i\over p^2-m^2+i0}.$$

In conformal field theory, Euclidean power laws become Lorentzian distributions. With their boundary limits and branches understood, the scalar Wightman and Feynman forms are

$$W_+(t,\mathbf x)={C\over(\mathbf x^2-(t-i0)^2)^\Delta}, \qquad G_F(t,\mathbf x)={C\over(\mathbf x^2-t^2+i0)^\Delta}.$$

The difference of the two Wightman functions is the vacuum expectation value of the commutator. For a general interacting field this is not the full operator-valued commutator; local microcausality separately requires that operator to vanish at spacelike separation. Free-field commutators are the special c-number case.

Common pitfalls

The most common mistake is to identify the Feynman propagator with causal response. The Feynman propagator is time ordered; it is the object naturally produced by perturbation theory and path integrals. With the source-sign convention used here, the causal response function is retarded:

$$G_R(x)=i\theta(t)\langle\Omega|[O(x),O(0)]|\Omega\rangle.$$

A second mistake is to write Lorentzian power laws without the $i0$. For non-integer powers this is not a minor omission; without a branch prescription the expression is not a well-defined distribution.

A third mistake is to think the sign of $i0$ is arbitrary. For Wightman functions it is fixed by spectral positivity. Positive energies make the analytic function for $\langle\Omega|O(t)O(0)|\Omega\rangle$ converge below the real-time axis, so its boundary is approached from $t-i0$. A finite $\epsilon$ supplies exponential damping; $i0$ denotes the distributional limit after that regulator is removed.

A fourth mistake is to assume that the $i0$ prescription alone fixes all contact terms at $x=0$. It fixes the side of the Lorentzian boundary value. Extending a sufficiently singular correlator through coincidence is a separate ultraviolet-renormalization problem.

Exercises

Exercise 1: Oscillator time ordering

Let

$$q(t)={1\over\sqrt{2\omega}}(ae^{-i\omega t}+a^\dagger e^{i\omega t}).$$

Compute $\langle0|Tq(t)q(0)|0\rangle$ and show that it equals

$${1\over2\omega}e^{-i\omega|t|}.$$

Solution

For $t>0$,

$$\langle0|Tq(t)q(0)|0\rangle =\langle0|q(t)q(0)|0\rangle ={e^{-i\omega t}\over2\omega}.$$

For $t<0$,

$$\langle0|Tq(t)q(0)|0\rangle =\langle0|q(0)q(t)|0\rangle ={e^{i\omega t}\over2\omega}.$$

Since $t<0$ implies $|t|=-t$,

$$e^{i\omega t}=e^{-i\omega|t|}.$$

Thus in both cases

$$\langle0|Tq(t)q(0)|0\rangle ={1\over2\omega}e^{-i\omega|t|}.$$

Exercise 2: The Feynman energy contour

Use contour integration to prove

$$\int {dp^0\over2\pi}{i e^{-ip^0t}\over (p^0)^2-\omega^2+i0} ={1\over2\omega}e^{-i\omega|t|}.$$

Solution

Begin with a finite $\epsilon>0$ and define

$$\Omega_\epsilon=\sqrt{\omega^2-i\epsilon}, \qquad \operatorname{Re}\Omega_\epsilon>0, \quad \operatorname{Im}\Omega_\epsilon<0.$$

The exact poles are $p^0=\Omega_\epsilon$ in the lower half-plane and $p^0=-\Omega_\epsilon$ in the upper half-plane. For $t>0$, close clockwise below. The residue is

$$\operatorname{Res}_{p^0=\Omega_\epsilon} {i e^{-ip^0t}\over (p^0-\Omega_\epsilon)(p^0+\Omega_\epsilon)} ={i e^{-i\Omega_\epsilon t}\over2\Omega_\epsilon}.$$

The clockwise contour therefore gives

$$I_\epsilon(t>0) ={e^{-i\Omega_\epsilon t}\over2\Omega_\epsilon}.$$

For $t<0$, close counterclockwise above. The residue at $p^0=-\Omega_\epsilon$ is

$$\operatorname{Res}_{p^0=-\Omega_\epsilon} {i e^{-ip^0t}\over (p^0-\Omega_\epsilon)(p^0+\Omega_\epsilon)} =-{i e^{i\Omega_\epsilon t}\over2\Omega_\epsilon},$$

so

$$I_\epsilon(t<0) ={e^{i\Omega_\epsilon t}\over2\Omega_\epsilon}.$$

Taking $\epsilon\downarrow0$ sends $\Omega_\epsilon\to\omega$ and gives $I(t)=e^{-i\omega|t|}/(2\omega)$ in both cases.

Exercise 3: Equality at spacelike separation

Let

$$F_\Delta(t,\mathbf x)={1\over\left(\mathbf x^2-(t-i0)^2\right)^\Delta}.$$

Show that, for spacelike separation $\mathbf x^2>t^2$, the two Wightman boundary values

$${1\over(\mathbf x^2-(t-i0)^2)^\Delta} \quad\text{and}\quad {1\over(\mathbf x^2-(t+i0)^2)^\Delta}$$

are equal.

Solution

Write

$$\rho=\mathbf x^2-t^2.$$

The two denominators are

$$\mathbf x^2-(t-i0)^2=\rho+i0\operatorname{sgn}t,$$

and

$$\mathbf x^2-(t+i0)^2=\rho-i0\operatorname{sgn}t.$$

For spacelike separation, $\rho>0$. The function $\rho^{-\Delta}$ has no branch cut on the positive real axis, so approaching from above or below gives the same boundary value:

$$(\rho+i0)^{-\Delta}=(\rho-i0)^{-\Delta}=\rho^{-\Delta}.$$

Thus their discontinuity vanishes for spacelike separation. When these are two-point functions, this is the vanishing vacuum commutator expectation; microcausality is the corresponding operator statement.

Exercise 4: The timelike branch-cut discontinuity

For non-integer $\Delta$, compute the discontinuity

$$(\rho+i0)^{-\Delta}-(\rho-i0)^{-\Delta}$$

for $\rho<0$, using the principal branch of the logarithm.

Solution

For $\rho<0$, write $\rho=-|\rho|$. On the principal branch,

$$\rho+i0=|\rho|e^{i\pi}, \qquad \rho-i0=|\rho|e^{-i\pi}.$$

Therefore

$$(\rho+i0)^{-\Delta}=|\rho|^{-\Delta}e^{-i\pi\Delta},$$

and

$$(\rho-i0)^{-\Delta}=|\rho|^{-\Delta}e^{+i\pi\Delta}.$$

Their difference is

$$(\rho+i0)^{-\Delta}-(\rho-i0)^{-\Delta} =|\rho|^{-\Delta}\left(e^{-i\pi\Delta}-e^{i\pi\Delta}\right) =-2i\sin(\pi\Delta)|\rho|^{-\Delta}.$$

If the boundary values are instead $\rho\pm i0\operatorname{sgn}t$, this result is multiplied by $\operatorname{sgn}t$.

Exercise 5: Spectral analyticity

Starting from the analytic spectral sum

$$F_+(z)=\sum_n c_n e^{-iE_nz}, \qquad c_n\ge0, \qquad E_n\ge0,$$

explain why $F_+(z)$ is analytic for $\operatorname{Im}z<0$ but not generally for $\operatorname{Im}z>0$, and identify the real-time Wightman distribution.

Solution

Let $z=t_R+i t_I$. Then

$$e^{-iE_n z}=e^{-iE_n t_R}e^{E_n t_I}.$$

If $t_I<0$, the factor $e^{E_n t_I}$ damps high-energy contributions. This makes the spectral sum, or its continuum version, well behaved in the lower half-plane under the usual assumptions on the growth of spectral weights.

If $t_I>0$, the same factor grows exponentially with $E_n$. There is no reason for the spectral sum to converge there. The reversed ordering has phases $e^{+iE_n z}$ and is analytic in the upper half-plane instead. The real-time Wightman distribution is the lower-half-plane boundary value

$$W_+(t)=\lim_{\epsilon\downarrow0}F_+(t-i\epsilon)$$

in the sense of distributions.

References and further reading

• M. Srednicki, Quantum Field Theory, chapters 3, 5, 8, and 13, for canonical free fields, propagators, path integrals, and spectral representations.
• S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, lectures on scalar fields, Green functions, and spectral representations.
• S. Weinberg, The Quantum Theory of Fields, volume I, chapters 5 and 6, for causal fields, propagators, and the analytic structure of free-field Green functions.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, chapters 4–6, for Euclidean conformal correlators and primary fields.
• P. Ginsparg, “Applied Conformal Field Theory,” for a compact account of Euclidean-to-Lorentzian continuation and two-dimensional CFT conventions.
• A. M. Polyakov, Gauge Fields and Strings, for the broader route from conformal fields to random surfaces and strings.

Further reading

This lesson preserves the oscillator-to-CFT lecture sequence. For a broader structural account of ordered vacuum correlators, see Wightman functions and spectral support; for vacuum selection, causal prescriptions, and Wick rotation, see Lorentzian boundary conditions and the iε prescription.