Finite Conformal Maps, Gauge Symmetry, and Free Fields

How does a primary field transform under a finite conformal map? Its local scale and rotation determine a Jacobian factor, but using that factor in a correlator also requires the correct state, domain and, for spinful fields, a compatible choice of frame. This lesson derives the Möbius covariance of separated Euclidean primary correlators and distinguishes a global symmetry from a map between backgrounds.

In dimensions $d>2$, the connected conformal group is finite-dimensional. In two Euclidean dimensions, however, orientation-preserving local conformal transformations are holomorphic maps. This gives a much larger local symmetry and eventually leads to the Virasoro algebra. The present page is the bridge: first we write the finite transformation law for primary fields, then we separate global Möbius transformations from general local maps, and finally we compare conformal covariance with two other local-symmetry ideas—diffeomorphism invariance and gauge invariance. The page ends by returning to the simplest dynamical fields, free oscillator modes, because the next pages will use their Wightman functions and $i\epsilon$ prescriptions as the basic analytic examples.

Required background. Lesson 15 supplies primary weights, conformal correlators, and the two-dimensional notation used here.

Helpful background. Lesson 14 develops finite conformal transformations and inversion in general dimension, while QFT I, Lesson 37 introduces covariant derivatives, gauge redundancy, and Wilson-line logic.

Primary fields under finite maps

Local components and global transformations

We use Euclidean complex coordinates

$$z=x+iy, \qquad \bar z=x-iy, \qquad ds^2=dx^2+dy^2=dz\,d\bar z.$$

A local conformal map is

$$w=f(z), \qquad \bar w=\bar f(\bar z), \qquad f'(z)\ne0.$$

A primary field of weights $(h,\bar h)$ has the passive component law below. For a general local map, “primary” means primary under both Virasoro algebras. A field primary only under the global conformal subalgebra obeys the Möbius law but can acquire extra terms under other maps.

$$O'(w,\bar w)=(f'(z))^{-h}(\bar f'(\bar z))^{-\bar h}O(z,\bar z), \qquad w=f(z).$$

Here the same insertion is expressed in the transported conformal frame. Conversely, the active pullback takes a field evaluated at $f(z)$ back to $z$ and multiplies by the positive powers. When $f$ is an implemented global conformal symmetry, write this action as

$$U_f O(z,\bar z)U_f^{-1} =(f'(z))^h(\bar f'(\bar z))^{\bar h}O(f(z),\bar f(\bar z)).$$

Its scaling dimension and spin are

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

Thus the inverse powers describe new components at the new coordinate, while the positive powers describe the pullback at the old coordinate. A general local holomorphic map need not define a global operator $U_f$ preserving the original Hilbert space, vacuum or boundary conditions. Its component law is still meaningful on an invertible chart with the background and conformal frame transported. This is the distinction in Di Francesco, Mathieu and Sénéchal 1997, §§5.1.4–5.1.5, pp.115–116.

A holomorphic map rescales lengths locally. Since

$$dw=f'(z)dz, \qquad d\bar w=\bar f'(\bar z)d\bar z,$$

the metric transforms as

$$ds'^2=dw\,d\bar w=|f'(z)|^2 dz\,d\bar z.$$

Thus, to linear order in its radius, a small circle near $z$ is mapped to a small circle near $w=f(z)$, multiplied by the local scale $|f'(z)|$ and rotated by the phase of $f'(z)$. Nonlinear terms in $f$ can distort a finite circle, but that distortion vanishes relative to its radius in the local limit. A primary field responds to the linear scale and rotation with weights $(h,\bar h)$.

For a pure dilation and rotation,

$$f(z)=\lambda e^{i\theta}z, \qquad \lambda>0,$$

the transformation factor is

$$(f')^h(\bar f')^{\bar h} =\lambda^{h+\bar h}e^{i\theta(h-\bar h)} =\lambda^\Delta e^{i\theta s}.$$

This is the cleanest way to remember the meanings of $\Delta$ and $s$. The sum $h+\bar h$ measures response to local scale, while the difference $h-\bar h$ measures response to local rotation. For a single-valued bosonic local field on the plane, $s$ is an integer. Fermions have half-integer spin and require a spin structure; the Ising Majorana fields have weights $(1/2,0)$ and $(0,1/2)$.

For noninteger weights, these powers require compatible analytic branches. Choose the frame continuously along the map and insertion configuration, avoiding poles and coincident insertions; do not choose independent principal powers at each point. For half-integer spin this is a choice of lift to the spin frames. A $2\pi$ frame rotation then has its appropriate fermionic sign even though it is the identity on coordinates. The Möbius example below makes this qualification concrete.

If the vacuum is invariant under the transformation, correlation functions obey

$$\boxed{ \langle O_1(z_1,\bar z_1)\cdots O_n(z_n,\bar z_n)\rangle =\prod_{i=1}^n (f_i')^{h_i}(\bar f_i')^{\bar h_i} \langle O_1(w_1,\bar w_1)\cdots O_n(w_n,\bar w_n)\rangle, }$$

where

$$w_i=f(z_i), \qquad f_i'=f'(z_i).$$

The conformal plane vacuum is invariant under the global generators, with the appropriate cover acting on spinful fields; see Di Francesco, Mathieu and Sénéchal 1997, §6.2.2, p.157, Eq.6.26. The displayed identity assumes that vacuum and uses separated insertions. For a general local map, compare the transported states and domains instead of asserting invariance of the original correlator. The stress tensor is not a primary under arbitrary local maps: it acquires the Schwarzian term studied later.

Global maps and Möbius transformations

The infinitesimal holomorphic transformations are

$$z\mapsto z+\epsilon(z).$$

Locally, $\epsilon(z)$ can be any holomorphic function. Globally on the Riemann sphere, however, regularity of the vector field at every finite point and at $z=\infty$ permits only a quadratic polynomial:

$$\epsilon(z)=\alpha+\beta z+\gamma z^2.$$

These three terms generate translations, dilations plus rotations, and special conformal transformations. Exponentiating them gives the Möbius maps

$$\boxed{ f(z)={az+b\over cz+d}, \qquad ad-bc=1. }$$

The matrices

$$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$

and its negative define the same coordinate map, so the group is $PSL(2,\mathbb C)$ on the Riemann sphere. The matrix action on spin frames can distinguish the two lifts. The vector-field derivation is completed in Exercise 2; the finite maps are discussed in Di Francesco, Mathieu and Sénéchal 1997, §§5.1.1–5.1.2, pp.113–114.

The group preserving the upper half-plane is $PSL(2,\mathbb R)$: choose real coefficients and positive determinant, then rescale to $ad-bc=1$. For a real projective matrix before that normalization,

$$\operatorname{Im}f(z)=\frac{(ad-bc)\operatorname{Im}z}{|cz+d|^2}.$$

Setwise preservation of the extended real line alone also permits negative determinant. Those transformations exchange the upper and lower half-planes: $f(z)=-z$ is an example. The full real-line-preserving group is $PGL(2,\mathbb R)$, with these two components. A common complex rescaling can normalize either component’s determinant, but it cannot make a negative-determinant real representative both real and determinant one.

It is useful to record the elementary identities

$$f'(z)={ad-bc\over(cz+d)^2},$$

and

$$f(z_1)-f(z_2)={(ad-bc)(z_1-z_2)\over(cz_1+d)(cz_2+d)}.$$

These identities explain why two-dimensional two-point functions are Möbius covariant. For a primary of weights $(h,\bar h)$ with a nonzero identical-field pairing,

$$\langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle ={C\over z_{12}^{2h}\bar z_{12}^{2\bar h}}, \qquad z_{12}=z_1-z_2.$$

For complex fields, use the corresponding nonzero paired two-point function. Using the two identities above with the compatible powers just specified,

$${(f_1')^h(f_2')^h\over(f(z_1)-f(z_2))^{2h}} ={1\over z_{12}^{2h}},$$

and similarly for the antiholomorphic part. Thus

$$\prod_{i=1}^2(f_i')^h(\bar f_i')^{\bar h} \langle O(f(z_1),\bar f(\bar z_1))O(f(z_2),\bar f(\bar z_2))\rangle = \langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle.$$

For a scalar primary with $h=\bar h=\Delta/2$, this becomes the familiar rotationally invariant form

$$\langle O(z_1,\bar z_1)O(z_2,\bar z_2)\rangle ={C\over |z_{12}|^{2\Delta}}.$$

For a chiral field with $\bar h=0$, the correlator is holomorphic away from coincident points:

$$\langle \psi(z_1)\psi(z_2)\rangle={C\over z_{12}^{2h}}.$$

The Ising fermion corresponds to $h=1/2$, so its chiral two-point function is proportional to $1/z_{12}$.

For example, take $f(z)=-1/z$. Although $f'(z)=1/z^2$, its spin-frame square root must be chosen as a continuous lift such as $\sqrt{f'(z)}=1/z$. At $z_1=-1$ and $z_2=1$, the lifted Jacobian product is $-1$, and

$$\frac{(1/z_1)(1/z_2)}{f(z_1)-f(z_2)}=-\frac12 =\frac1{z_1-z_2}.$$

Taking both numerical principal square roots to be $+1$ instead gives $+1/2$ and destroys covariance. The missing sign records incompatible frames, not a failure of the fermion two-point function.

Domains, boundaries, and the meaning of covariance

A conformal map may act in two related but conceptually different ways. It may be an automorphism of the same background, such as a Möbius transformation of the plane vacuum. Or it may map one domain to another domain. In the second case, the correlator in the original domain $D$ is related to a correlator in the image domain $\Gamma=f(D)$ by local Jacobian factors:

$$\boxed{ \left\langle\prod_i O_i(z_i,\bar z_i)\right\rangle_D = \prod_i (f_i')^{h_i}(\bar f_i')^{\bar h_i} \left\langle\prod_i O_i(w_i,\bar w_i)\right\rangle_\Gamma. }$$

Here $f$ is a conformal bijection between the domains, the brackets denote normalized correlators, and the boundary condition, state and spin structure are carried from $D$ to $\Gamma$. An unnormalized partition function can additionally contain Weyl-anomaly factors. A concrete example is the upper-half-plane-to-strip map with corresponding boundary conditions in Di Francesco, Mathieu and Sénéchal 1997, §11.2.3, pp.419–420.

This distinction between a change of description and a symmetry also appears for general coordinate maps. A scalar’s active pullback is

$$\phi(x)\mapsto \phi(f(x)),$$

so a coordinate-labeled correlator transforms covariantly:

$$\langle \phi(x_1)\phi(x_2)\rangle \mapsto \langle \phi(f(x_1))\phi(f(x_2))\rangle.$$

In an ordinary QFT on a fixed background, this is a useful symmetry statement when $f$ preserves the relevant background structure. In a gravitational theory, local diffeomorphisms treated as gauge redundancies, such as transformations supported away from a boundary, make a coordinate-labeled local field insufficient by itself as a physical observable. Physical observables must be invariant under that redundancy or relationally dressed. Transformations acting nontrivially at a physical or asymptotic boundary can instead carry charges and require a separate analysis.

The group $PSL(2,\mathbb R)$ is therefore relevant to an upper-half-plane background, provided its boundary condition is also preserved. A map that exchanges the half-planes, or changes the boundary state, is not a symmetry of that specified problem. The need to preserve both the geometric boundary and its condition is emphasized in Di Francesco, Mathieu and Sénéchal 1997, §11.2.1, pp.413–414.

Gauge symmetry and dressed correlators

The same warning appears in gauge theory in a simpler, more familiar form. Consider a charged scalar field in Abelian gauge theory, with convention

$$\phi(x)\mapsto e^{+iq\alpha(x)}\phi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x), \qquad D_\mu=\partial_\mu-iqA_\mu.$$

The local field $\phi(x)$ is charged. Therefore the naive two-point function

$$\langle \phi^\dagger(x_1)\phi(x_2)\rangle$$

is not a gauge-invariant observable: its phase changes by

$$e^{-iq\alpha(x_1)}e^{+iq\alpha(x_2)}.$$

There are two standard cures. The first is to make a local neutral composite, such as

$$\phi^\dagger(x)\phi(x).$$

The second is to dress the separated charged fields by a Wilson line along a path $\gamma$ from $x_1$ to $x_2$:

$$W_\gamma(x_1,x_2)=\exp\left(-iq\int_\gamma A_\mu dx^\mu\right).$$

This is the inverse of the site-wide forward transporter along the same oriented path:

$$W_\gamma(x_1,x_2)=U_\gamma(x_2,x_1)^{-1} =U_{\bar\gamma}(x_1,x_2),$$

where $U_\gamma(x_2,x_1)=\exp(+iq\int_{x_1}^{x_2}A)$ and $\bar\gamma$ has the opposite orientation. The reversed sign is therefore a path-orientation map, not a different covariant-derivative convention.

Since

$$W_\gamma(x_1,x_2) \mapsto \exp\left(-iq\alpha(x_2)+iq\alpha(x_1)\right)W_\gamma(x_1,x_2),$$

the bilocal operator

$$\boxed{ \phi^\dagger(x_1)W_\gamma(x_1,x_2)\phi(x_2) }$$

is gauge invariant. In the figure, follow the three phase factors from the left endpoint to the right: the line cancels the two endpoint transformations. Different paths generally define different dressed operators. In a continuum quantum theory, the local neutral product and the line-dressed bilocal also require their appropriate composite-operator and line renormalization; gauge invariance alone does not establish finiteness.

A separated charged pair acquires two independent endpoint phases. Along the chosen path $\gamma:x_1\to x_2$, $W_\gamma=\exp(-iq\int_\gamma A)$ supplies their inverse product, making the dressed bilocal gauge invariant. The path is schematic; it is not a propagator or a claim of path independence.

This is the gauge-theory analogue of the gravity warning above. Local charged fields are perfectly useful inside a fixed gauge or as ingredients in correlation functions, but physical questions must be expressed in gauge-invariant terms. In later pages Wilson loops will become central order parameters for confinement.

Free fields as oscillator modes

We now return to a simpler system: the free scalar field. The reason is strategic. The next page studies Wightman functions, commutators, and the $i\epsilon$ prescription. All of those analytic structures can be seen already in the free oscillator decomposition.

Let a real free scalar of mass $m>0$ live in $D$-dimensional Minkowski spacetime, with $D-1$ spatial momentum components. The positive mass makes every mode below a genuine oscillator. With periodic boundary conditions in a finite spatial volume $V$, write

$$\phi(t,\mathbf x) =\sum_{\mathbf p} \left(a_{\mathbf p}u_{\mathbf p}(t,\mathbf x) +a_{\mathbf p}^\dagger u_{\mathbf p}^*(t,\mathbf x)\right),$$

with

$$u_{\mathbf p}(t,\mathbf x) ={1\over\sqrt V}e^{i\mathbf p\cdot\mathbf x}f_{\mathbf p}(t), \qquad f_{\mathbf p}(t)={e^{-i\omega_{\mathbf p}t}\over\sqrt{2\omega_{\mathbf p}}}, \qquad \omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}.$$

The creation and annihilation operators obey

$$[a_{\mathbf p},a_{\mathbf q}^\dagger]=\delta_{\mathbf p\mathbf q}, \qquad [a_{\mathbf p},a_{\mathbf q}]=[a_{\mathbf p}^\dagger,a_{\mathbf q}^\dagger]=0.$$

The time-dependent oscillator mode is normalized by its Wronskian:

$$\boxed{ f_{\mathbf p}\dot f_{\mathbf p}^{*}-\dot f_{\mathbf p}f_{\mathbf p}^{*}=i. }$$

Indeed, for $f=e^{-i\omega t}/\sqrt{2\omega}$,

$$f\dot f^*-\dot f f^* ={e^{-i\omega t}\over\sqrt{2\omega}} {i\omega e^{i\omega t}\over\sqrt{2\omega}} - {-i\omega e^{-i\omega t}\over\sqrt{2\omega}} {e^{i\omega t}\over\sqrt{2\omega}} =i.$$

This Wronskian is the mode-by-mode version of the canonical commutation relation

$$[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta_V^{(D-1)}(\mathbf x-\mathbf y), \qquad \pi=\dot\phi.$$

Indeed, inserting the mode expansion, relabeling $\mathbf p\mapsto-\mathbf p$ in the second term, and using $f_{-\mathbf p}=f_{\mathbf p}$ gives

$$\begin{aligned} [\phi(t,\mathbf x),\pi(t,\mathbf y)] &={1\over V}\sum_{\mathbf p}e^{i\mathbf p\cdot(\mathbf x-\mathbf y)} \left(f_{\mathbf p}\dot f_{\mathbf p}^* -\dot f_{\mathbf p}f_{\mathbf p}^*\right)\\ &={i\over V}\sum_{\mathbf p}e^{i\mathbf p\cdot(\mathbf x-\mathbf y)} =i\delta_V^{(D-1)}(\mathbf x-\mathbf y), \end{aligned}$$

where $\delta_V^{(D-1)}$ is the periodic delta function. This displays directly why the sign and normalization of the Wronskian matter. If $m=0$ in a periodic box, the spatially constant mode has $\omega_{\mathbf0}=0$ and cannot use $1/\sqrt{2\omega_{\mathbf0}}$. Its canonical variables $Q_0=V^{-1/2}\int_V\phi$ and $P_0=V^{-1/2}\int_V\pi$ have the free-particle Hamiltonian $H_0=P_0^2/2$ and must be quantized separately. The two-dimensional massless example is worked out in Di Francesco, Mathieu and Sénéchal 1997, §6.3.1, pp.159–160; after rescaling its scalar to a canonical kinetic term, it has this same zero-mode structure.

In infinite volume,

$$\sum_{\mathbf p}{1\over V}\longrightarrow \int{d^{D-1} p\over(2\pi)^{D-1}},$$

and the expansion becomes

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

The continuum operators are rescaled relative to their box counterparts, $a_{\mathbf p}^{\mathrm{cont}}=\sqrt V\,a_{\mathbf p}^{\mathrm{box}}$ before the limit. Suppressing those labels in the continuum expansion gives

$$[a_{\mathbf p},a_{\mathbf q}^\dagger]=(2\pi)^{D-1}\delta^{(D-1)}(\mathbf p-\mathbf q).$$

In this limit $\delta_V$ becomes the ordinary spatial delta distribution. For Euclidean momentum $p$, define $\phi(p)=\int d^Dx\,e^{-ip\cdot x}\phi(x)$. The infinite-volume vacuum covariance is

$$\langle\phi(p)\phi(q)\rangle_E =(2\pi)^D\delta^{(D)}(p+q)D_E(p), \qquad D_E(p)=\frac1{p^2+m^2},$$

where $p^2$ is the Euclidean norm squared. It is the coefficient $D_E$ after removing the momentum delta, not the full coincident-momentum expectation, that equals the inverse kernel. Applying $-\partial_E^2+m^2$ to the corresponding position-space covariance gives $\delta^{(D)}(x-y)$.

At $m\ne0$, the mass introduces a scale and the theory is not conformal. In two dimensions the $m=0$ theory has a zero-mode and infrared subtlety: the scalar itself has a logarithmic correlator rather than an ordinary power-law primary correlator. The derivative sector is nevertheless conformal, and its simplest fields obey

$$\langle \partial\phi(z)\partial\phi(0)\rangle\propto {1\over z^2}, \qquad \partial\phi\text{ has weights }(1,0).$$

This small caveat is worth remembering. Not every field appearing in a Lagrangian is automatically a primary field in the CFT sense. The primary operators are the fields with definite transformation laws under conformal maps.

Summary

Finite conformal maps in two dimensions act on primary fields through local holomorphic and antiholomorphic Jacobians. The weights $(h,\bar h)$ encode scaling dimension $\Delta=h+\bar h$ and spin $s=h-\bar h$. Local holomorphic maps are infinite-dimensional, but globally regular maps on the sphere reduce to Möbius transformations.

The covariance formula for correlators is both powerful and delicate. It is exact for global conformal transformations preserving the vacuum, and it relates correlators in conformally equivalent domains. Similar conceptual care is required for local symmetries in gauge theory and gravity: coordinate-labeled or charged fields are useful, but physical observables must be invariant or properly dressed.

The free scalar field reappears as a collection of harmonic oscillators in $D-1$ spatial momentum dimensions. The positive-frequency mode $e^{-i\omega t}/\sqrt{2\omega}$ is normalized by a Wronskian, and that Wronskian is precisely what makes the equal-time canonical commutator work. This oscillator picture will now be used to derive Wightman functions, commutators, time ordering, and the $i\epsilon$ prescription.

Common pitfalls

Mixing active and passive conventions. If one writes

$$U_f O(z)U_f^{-1}=(f')^hO(f(z)),$$

then the corresponding vacuum Ward identity has positive powers of $f'$. If instead one asks for the field components in the new coordinate $w=f(z)$, inverse powers appear; neither convention is wrong, but combining them is.

Treating every holomorphic map as a global symmetry. On the sphere, only Möbius transformations are globally regular. More general holomorphic maps organize local Ward identities and Virasoro symmetry, but they need not leave the vacuum, domain, or boundary conditions unchanged.

Using an undressed charged correlator as an observable. Before gauge fixing, a separated charged two-point function is not gauge invariant. A Wilson line is not decoration; its endpoint transformation supplies exactly the missing phase.

Reusing $d$ for two different dimensions. Conformal formulas count spacetime dimensions, whereas the oscillator integral counts spatial momentum components. Here $D$ is the Minkowski spacetime dimension and the momentum measure is therefore $d^{D-1}p$.

Exercises

Exercise 1: Möbius covariance of a two-point function

Show that the two-point function

$$G(z_1,z_2)={1\over z_{12}^{2h}}$$

is covariant under the holomorphic part of the Möbius transformation

$$f(z)={az+b\over cz+d}, \qquad ad-bc=1,$$

with primary weight $h$. Work on a connected configuration chart away from poles and coincident points, with all derivative and separation powers transported on compatible branches. For $h=1/2$, use a consistent spin-frame lift rather than separate principal square roots.

Solution

On the branches specified in the question, first compute

$$f'(z)={ad-bc\over(cz+d)^2}={1\over(cz+d)^2}.$$

Next,

$$\begin{aligned} f(z_1)-f(z_2) &={az_1+b\over cz_1+d}-{az_2+b\over cz_2+d} \\ &={ (az_1+b)(cz_2+d)-(az_2+b)(cz_1+d) \over(cz_1+d)(cz_2+d)}. \end{aligned}$$

The numerator is

$$a d(z_1-z_2)-bc(z_1-z_2)=(ad-bc)z_{12}=z_{12}.$$

Therefore

$$f(z_1)-f(z_2)={z_{12}\over(cz_1+d)(cz_2+d)}.$$

The transformed two-point function with the primary factors is

$$(f'(z_1))^h(f'(z_2))^h{1\over(f(z_1)-f(z_2))^{2h}}.$$

Substituting the formulas gives

$${1\over(cz_1+d)^{2h}(cz_2+d)^{2h}} {(cz_1+d)^{2h}(cz_2+d)^{2h}\over z_{12}^{2h}} ={1\over z_{12}^{2h}}.$$

So the correlator is Möbius covariant.

Exercise 2: Global holomorphic vector fields

Why are the globally regular infinitesimal holomorphic conformal transformations on the Riemann sphere generated only by

$$\epsilon(z)=\alpha+\beta z+\gamma z^2?$$

Solution

Near $z=0$, regularity allows a Taylor expansion

$$\epsilon(z)=\sum_{n\ge0}a_n z^n.$$

Now examine the same vector field near $z=\infty$ using $u=1/z$. Since

$$\epsilon(z)\partial_z =\epsilon(1/u){d u\over dz}\partial_u =-u^2\epsilon(1/u)\partial_u,$$

regularity at $u=0$ requires

$$-u^2\epsilon(1/u)$$

to have no negative powers of $u$. A term $a_n z^n$ becomes

$$-u^2 a_n u^{-n}=-a_n u^{2-n}.$$

This is regular at $u=0$ only if $2-n\ge0$, or $n\le2$. Hence only $n=0,1,2$ survive:

$$\epsilon(z)=\alpha+\beta z+\gamma z^2.$$

These are the infinitesimal generators of $PSL(2,\mathbb C)$.

Exercise 3: Wilson-line endpoint phases

Using the gauge transformations

$$\phi(x)\mapsto e^{+iq\alpha(x)}\phi(x), \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\alpha(x),$$

show that

$$\phi^\dagger(x_1)\exp\left(-iq\int_{x_1}^{x_2}A_\mu dx^\mu\right)\phi(x_2)$$

is gauge invariant.

Solution

The Wilson-line factor transforms as

$$\begin{aligned} \exp\left(-iq\int_{x_1}^{x_2}(A_\mu+\partial_\mu\alpha)dx^\mu\right) &= \exp\left(-iq\int_{x_1}^{x_2}A_\mu dx^\mu\right) \\ &\quad\times \exp\left(-iq[\alpha(x_2)-\alpha(x_1)]\right). \end{aligned}$$

The charged fields transform as

$$\phi^\dagger(x_1)\mapsto e^{-iq\alpha(x_1)}\phi^\dagger(x_1), \qquad \phi(x_2)\mapsto e^{+iq\alpha(x_2)}\phi(x_2).$$

Multiplying all factors gives the phase

$$e^{-iq\alpha(x_1)}e^{-iq[\alpha(x_2)-\alpha(x_1)]}e^{+iq\alpha(x_2)}=1.$$

Therefore the dressed bilocal operator is gauge invariant.

Exercise 4: Wronskian and canonical normalization

For $\omega>0$, let

$$f(t)={e^{-i\omega t}\over\sqrt{2\omega}}.$$

Verify the Wronskian condition

$$f\dot f^*-\dot f f^*=i,$$

and explain why this normalization is needed in the free-field expansion.

Solution

We have

$$\dot f=-i\omega f, \qquad \dot f^*=i\omega f^*.$$

Therefore

$$f\dot f^*-\dot f f^* =i\omega |f|^2+i\omega |f|^2 =2i\omega {1\over2\omega}=i.$$

In the field expansion, the equal-time commutator $[\phi,\pi]$ receives one contribution from the positive-frequency mode and one from the negative-frequency mode. The Wronskian is exactly the coefficient that remains after subtracting these two pieces. Setting it equal to $i$ ensures

$$[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta_V^{(D-1)}(\mathbf x-\mathbf y)$$

in the periodic box; $\delta_V$ becomes the ordinary spatial delta in the infinite-volume limit. A different normalization of $f$ would require a compensating change in the normalization of the creation and annihilation operators.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, 1997. DOI: 10.1007/978-1-4612-2256-9. Finite maps and primaries: §§5.1.1–5.1.5, pp.113–116; vacuum and zero mode: §§6.2.2–6.3.1, pp.157–160; boundaries and transported correlators: §§11.2.1 and 11.2.3, pp.413–414 and 419–420.