Global Conformal Generators and Descendant States

The previous lesson established the Virasoro algebra and the Schwarzian transformation law of the stress tensor. We now zoom in on the part of that structure that is already visible before using the full infinite-dimensional algebra: the global conformal generators. On the Riemann sphere these are $L_{-1}$, $L_0$, and $L_1$, corresponding to translations, dilatations/rotations, and special conformal transformations.

Global Ward identities fix the coordinate dependence of two- and three-point functions in the plane vacuum. Radial quantization translates these constraints into state language: primary states sit at the top of representation towers, and negative Virasoro modes generate descendants.

The final comparison concerns conserved internal currents. A charge that annihilates the vacuum gives selection rules; a nonzero local order-parameter contact forces a singular longitudinal correlator. Its physical Goldstone interpretation requires the additional vacuum, locality and spectral assumptions stated below.

Required background. Lesson 21 supplies the OPE and contour Ward argument; Lesson 22 distinguishes origin modes from local descendant contours. Lesson 23 and the previous lesson supply highest-weight modules, the identity vacuum and radial adjoints. The final current comparison uses the definitions in Lesson 19.

The three global generators

Use the plane identity vacuum with $\langle0|0\rangle=1$. Correlators are radially ordered at separated insertions. The vacuum is regular at the origin and infinity, with no extra insertion at infinity. Contours are counterclockwise; their deformations must meet no additional singularities or monodromy of the weighted stress-tensor correlator. Multivalued chiral expressions are evaluated on a fixed compatible branch domain. Where state norms are used, the radial adjoint is $L_n^\dagger=L_{-n}$; positivity is invoked only for a unitary theory.

We work in the holomorphic sector, with origin modes

$$L_n={1\over 2\pi i}\oint_0 dz\,z^{n+1}T(z), \qquad T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2},$$

and

$$[L_n,L_m]=(n-m)L_{n+m}+{c\over 12}n(n^2-1)\delta_{n+m,0}.$$

For a holomorphic primary field $\phi(z)$ of weight $h$,

$$T(z)\phi(w)\sim {h\phi(w)\over (z-w)^2}+{\partial\phi(w)\over z-w}.$$

The antiholomorphic sector has the corresponding barred relations.

The positive active pullback generated by a holomorphic vector field $\epsilon(z)\partial_z$ acts on a primary field by

$$\delta_\epsilon\phi(z) = {1\over 2\pi i}\oint_z dw\,\epsilon(w)T(w)\phi(z) = \epsilon(z)\partial\phi(z)+h\partial\epsilon(z)\phi(z).$$

Here the small contour encloses $z$. The inverse, passive field transformation has the opposite infinitesimal sign (Di Francesco et al. 1997, § 5.1.4, p. 116, Eqs. (5.22)–(5.23)). Choosing $\epsilon(z)=z^{n+1}$ gives the mode action

$$\boxed{ [L_n,\phi(z)] = \left(z^{n+1}\partial_z+(n+1)h z^n\right)\phi(z). }$$

For a displaced insertion $z\ne0$ and nonglobal $n$, this commutator comes from the difference of two origin contours enclosing the insertion between them. The resulting small contour about $z$ excludes the origin, where $w^{n+1}$ may be singular. A local descendant mode instead uses the weight $(w-z)^{n+1}$, as in Lesson 22’s mode comparison. The contour realization of the commutator is given in Di Francesco et al. 1997, § 6.1.2, pp. 154–155, Eqs. (6.13)–(6.18).

The modes $L_{-1}$, $L_0$, and $L_1$ correspond to

$$\begin{aligned} [L_{-1},\phi(z)]&=\partial\phi(z),\\ [L_0,\phi(z)]&=(z\partial+h)\phi(z),\\ [L_1,\phi(z)]&=(z^2\partial+2hz)\phi(z). \end{aligned}$$

They close among themselves:

$$[L_0,L_{-1}]=L_{-1}, \qquad [L_0,L_1]=-L_1, \qquad [L_1,L_{-1}]=2L_0.$$

This is the global conformal algebra $sl_2$. The central term vanishes on this subalgebra because $n(n^2-1)=0$ for $n=-1,0,1$. Möbius maps therefore have zero Schwarzian derivative. For transformations preserving the real plane, combine the two sectors: $L_0+\bar L_0$ generates dilations, while $i(L_0-\bar L_0)$ generates rotations. A real holomorphic parameter multiplying $\epsilon=z$ alone describes dilation, not a rotation (Di Francesco et al. 1997, § 5.1.3, p. 115, Eqs. (5.18)–(5.20)).

The word “global” is doing real work. A general holomorphic function $\epsilon(z)$ is locally allowed in two dimensions, but on the compact Riemann sphere a globally nonsingular holomorphic vector field can have at most a quadratic polynomial coefficient. The infinite Virasoro algebra is local; the $sl_2$ algebra is globally well-defined on the sphere.

Ward identities from vacuum invariance

In radial quantization the vacuum is invariant under the global conformal group:

$$L_{-1}|0\rangle=L_0|0\rangle=L_1|0\rangle=0,$$

and similarly for the bra vacuum. Therefore, for a product of primary fields,

$$0= \langle 0|[L_n,\phi_1(z_1)\cdots\phi_N(z_N)]|0\rangle, \qquad n=-1,0,1.$$

Using the commutator as a derivation gives the global conformal Ward identities

$$\boxed{ \sum_{i=1}^N \left(z_i^{n+1}\partial_{z_i}+(n+1)h_i z_i^n\right) \langle \phi_1(z_1)\cdots\phi_N(z_N)\rangle=0, \qquad n=-1,0,1. }$$

Equivalently, insert the charge

$$Q_\epsilon={1\over 2\pi i}\oint dz\,\epsilon(z)T(z)$$

on a contour surrounding all insertions. Here $\epsilon$ is one of $1,z,z^2$. The identity-vacuum condition at infinity makes the large contour vanish. Under the stated contour-domain conditions, deforming it inward produces the sum of residues above. Lesson 21’s contour construction shows this local-to-global step.

The case $n=-1$ is translation invariance:

$$\sum_i\partial_{z_i}G_N=0.$$

The case $n=0$ is scale and rotation covariance:

$$\sum_i(z_i\partial_{z_i}+h_i)G_N=0.$$

The case $n=1$ is special conformal covariance:

$$\sum_i(z_i^2\partial_{z_i}+2h_i z_i)G_N=0.$$

These three equations already fix the holomorphic dependence of two- and three-point functions. For example, translation invariance implies that a two-point function depends only on $z_{12}=z_1-z_2$. Scaling then gives

$$G_{12}(z_1,z_2)={C_{12}\over z_{12}^{h_1+h_2}},$$

and the special conformal Ward identity forces $h_1=h_2$ unless $C_{12}=0$. Thus, more precisely,

$$\boxed{ \langle \phi_1(z_1)\phi_2(z_2)\rangle ={C_{12}\over (z_1-z_2)^{2h_1}}, \qquad C_{12}=0\ \text{unless}\ h_1=h_2. }$$

Equal weights are necessary, not sufficient: internal quantum numbers and the choice of operator basis determine the matrix $C_{12}$.

For three insertions, write the holomorphic dependence as $C_{123}\prod_{i<j}z_{ij}^{-\alpha_{ij}}$. Global covariance at each insertion requires

$$\alpha_{12}+\alpha_{13}=2h_1,\qquad \alpha_{12}+\alpha_{23}=2h_2,\qquad \alpha_{13}+\alpha_{23}=2h_3.$$

Solving these three equations gives

$$\boxed{ \langle \phi_1(z_1)\phi_2(z_2)\phi_3(z_3)\rangle = {C_{123}\over z_{12}^{h_1+h_2-h_3} z_{13}^{h_1+h_3-h_2} z_{23}^{h_2+h_3-h_1}} }$$

in the holomorphic sector. The corresponding barred factors restore the full two-dimensional expression. These two- and three-point covariance formulas are Di Francesco et al. 1997, § 5.1.5, pp. 116–117, Eqs. (5.24)–(5.26). The constants $C_{123}$ are dynamical CFT data. Symmetry fixes the coordinate dependence, but it does not determine which fields exist or what their OPE coefficients are.

Primary states and descendants

The state–operator correspondence assigns to a local operator the state

$$|\phi\rangle=\lim_{z\to0}\phi(z)|0\rangle.$$

If $\phi$ is primary of holomorphic weight $h$, the commutator formula immediately implies

$$L_n|\phi\rangle=0\quad(n>0), \qquad L_0|\phi\rangle=h|\phi\rangle.$$

The primary state is therefore a highest-weight state for the Virasoro algebra. Negative modes generate descendant states:

$$L_{-n_1}L_{-n_2}\cdots L_{-n_k}|\phi\rangle, \qquad n_i>0.$$

The level of this descendant is

$$N=n_1+n_2+\cdots+n_k,$$

and its $L_0$ eigenvalue is $h+N$, because

$$[L_0,L_{-n}]=nL_{-n}.$$

The simplest descendant is the translation descendant:

$$L_{-1}|\phi\rangle = \partial\phi(0)|0\rangle.$$

This is the state-language version of the local commutator $[L_{-1},\phi(z)]=\partial\phi(z)$.

Ordering the negative modes with $n_1\ge n_2\ge\cdots$ gives the PBW basis of the abstract Verma module. Its image in a physical representation can satisfy additional relations; the irreducible representation is the appropriate quotient, as explained in Lesson 23. At level $2$, for example, the Verma module contains

$$L_{-2}|h\rangle, \qquad L_{-1}^2|h\rangle.$$

They have the same scaling dimension $h+2$, so they can mix under changes of basis. They need not both be physical independent states; null vectors may appear at special values of $h$ and $c$. That is the entry point to BPZ equations and minimal models on the next pages.

The identity module and the stress tensor

The vacuum corresponds to the identity operator. Since the identity has $h=0$ and is constant,

$$L_{-1}|0\rangle=0.$$

Global conformal invariance also gives

$$L_0|0\rangle=L_1|0\rangle=0.$$

More generally, regularity of the stress tensor at the origin when acting on the vacuum implies (Di Francesco et al. 1997, § 6.2.2, p. 157, Eqs. (6.26)–(6.27))

$$L_n|0\rangle=0, \qquad n\ge -1.$$

The first Virasoro descendant not forced to vanish by global invariance is therefore

$$L_{-2}|0\rangle.$$

This state is the stress tensor state. Indeed, from the mode expansion,

$$T(z)|0\rangle = \sum_n L_n z^{-n-2}|0\rangle =L_{-2}|0\rangle+zL_{-3}|0\rangle+z^2L_{-4}|0\rangle+\cdots,$$

so

$$T(0)|0\rangle=L_{-2}|0\rangle.$$

With the declared unit vacuum norm and radial adjoint, its norm is controlled by the central charge:

$$\langle 0|L_2L_{-2}|0\rangle = \langle 0|[L_2,L_{-2}]|0\rangle = {c\over2}.$$

This is the state version of the two-point function

$$\langle T(z)T(w)\rangle={c/2\over (z-w)^4}.$$

Thus $c$ fixes the normalization of the stress-tensor excitation above the vacuum; it should not in general be read as a literal count of fields. Reflection positivity in a unitary CFT implies $c\ge0$, and $c=0$ forces the stress-tensor state to be null. This is the $h=0$, $n=2$ case of the descendant norm formula in Di Francesco et al. 1997, § 7.2.1, p. 205, Eq. (7.19).

Lorentzian charges and radial generators

For a real Lorentzian symmetry parameter $a$, define a Hermitian physical charge $Q_{\rm phys}$ on an appropriate domain by

$$U(a)=e^{iaQ_{\rm phys}},\qquad \phi_a=U(a)\phi U(a)^{-1}.$$

If the chosen active transformation is $\delta\phi=a\,\mathcal D\phi$, differentiating at $a=0$ gives

$$i[Q_{\rm phys},\phi]=\mathcal D\phi, \qquad [Q_{\rm phys},\phi]=-i\mathcal D\phi.$$

For example, the physical translation convention is $i[P_\mu,\phi(x)]=\partial_\mu\phi(x)$. The factor of $i$ is required for a Hermitian charge acting on a Hermitian field. The Euclidean contour relation $[L_n,\phi]=\mathcal D_n\phi$ uses radial generators; they obey $L_n^\dagger=L_{-n}$, rather than all being individually Hermitian physical charges.

For a conserved symmetric traceless Lorentzian stress tensor $\Theta_{\mu\nu}$, the coordinates $x^\pm=t\pm x$ give $\partial_-\Theta_{++}=0$ and $\partial_+\Theta_{--}=0$ away from insertions. These equations express chirality. The normalization of physical charges requires the Lorentzian stress tensor and its integration surface. Lesson 19’s component continuation relates that tensor to the Euclidean metric tensor and the residue-normalized $T(z)$.

Momentum-space Ward identities and soft limits

For an ordinary internal symmetry with conserved current $J^\mu$, write $D$ for spacetime dimension. In the convention

$$\Delta\mathcal O_k=i e_k\mathcal O_k,$$

localize the variation as $\delta_\alpha\mathcal O_k=\alpha(x_k)\Delta\mathcal O_k$ and define the Euclidean current by $\delta_\alpha S_E=\int d^D x\,J^\mu\partial_\mu\alpha$. Assume an invariant measure and contact prescriptions preserving the symmetry. Then the Euclidean local Ward identity has the form

$$\partial_\mu\langle J^\mu(x)\mathcal O_1(x_1)\cdots\mathcal O_N(x_N)\rangle = -i\sum_{k=1}^N e_k\delta^{(D)}(x-x_k) \langle \mathcal O_1\cdots\mathcal O_N\rangle.$$

Define the current-inserted correlator $G_J^\mu(q;p_1,\ldots,p_N)$ with the site-wide forward Fourier phase $e^{+iq\cdot x+i\sum_k p_k\cdot x_k}$. Thus $\partial_\mu\mapsto-iq_\mu$, and Fourier transformation gives

$$\boxed{ q_\mu G_J^\mu(q;p_1,\ldots,p_N) = +\sum_{k=1}^N e_k\,G(p_1,\ldots,p_k+q,\ldots,p_N). }$$

A source written with the opposite Fourier phase is translated by reversing every momentum, $G_{\rm site}(q,p_k)=G_{-}(-q,-p_k)$. Changing the generator convention reverses the displayed charge sign, but neither change alters the momentum-shift structure. The momenta in this formula all use the forward phase; Lesson 18’s current-insertion calculation gives the routing and contact derivation.

For comparison, the proper photon vertex in Lesson 19’s Lorentzian QED convention is defined by its Feynman insertion $-i\Gamma_\mu$. The interaction is $\mathcal L_{\rm int}=-e\bar\psi\gamma^\mu\psi A_\mu$ with $e>0$ (site signed electron charge $-e$). Let $G_{\rm f}$ denote that lesson’s stripped fermion propagator:

$$S_F(p)=iG_{\rm f}(p),\qquad \Gamma_\mu^{(0)}=e\gamma_\mu,$$

so the proper identity is

$$q^\mu\Gamma_\mu(p+q,p) =e\left[G_{\rm f}^{-1}(p+q)-G_{\rm f}^{-1}(p)\right].$$

Its free check is $e q^\mu\gamma_\mu=e[(p+q)\!\!\!/-m-(p\!\!\!/-m)]$. Fermion-leg amputation of the full connected current insertion above need not produce this proper vertex: dynamical photons allow additional photon-reducible contributions. Lesson 19 derives the proper identity from the effective action and states its regular soft-limit hypotheses. Schwartz places the charge outside his vertex, so $\Gamma_\mu=e\Gamma_{\mu,\mathrm{Schwartz}}$ (Schwartz 2014, § 19.5.1, p. 352, Eqs. (19.74)–(19.80)).

Vacuum charges and Goldstone spectral weight

For a well-defined Lorentzian charge, integrating the local current identity over a spatial slice yields its commutator action. Write

$$Q=\int d^{D-1}x\,J^0(t,\mathbf x).$$

Then the commutator satisfies

$$\langle0|[Q,\mathcal O_1\cdots\mathcal O_N]|0\rangle = \sum_k\langle0|\mathcal O_1\cdots[Q,\mathcal O_k]\cdots\mathcal O_N|0\rangle.$$

If the chosen charge annihilates the vacuum,

$$Q|0\rangle=0,$$

the integrated identity gives a selection rule. If the vacuum is not invariant, one cannot move $Q$ through the correlator and kill the vacuum. In infinite volume the global charge may itself fail to exist as an operator on the vacuum. The precise argument uses compactly supported charges $Q_R$ and their limit in commutators with fixed local observables, with time smearing when required for the operator-valued distributions. The local current Ward identity gives a direct formulation.

To define its singular correlator without mixing Euclidean and Feynman conventions, let $\Phi$ be a local scalar order parameter and factor out the symmetry parameter:

$$\delta_\alpha\Phi=\alpha\,\Delta\Phi,\qquad B=\langle\Delta\Phi\rangle.$$

Choose the infinite-volume vacuum before removing its symmetry-breaking source. With the current and symmetry-preserving contact prescription just defined, the Euclidean identity is

$$\partial_\mu\langle J_\mu(x)\Phi(0)\rangle =-B\,\delta^{(D)}(x).$$

Define the mixed Euclidean correlator and the positive squared momentum by

$$C_\mu(q)=\int d^D x\,e^{iq\cdot x}\langle J_\mu(x)\Phi(0)\rangle, \qquad q_E^2=\sum_{\mu=1}^{D}q_\mu^2.$$

The transform gives $-iq_\mu C_\mu=-B$, hence, at $q_E^2>0$,

$$q_\mu C_\mu(q)=-iB,\qquad C_\mu^\parallel(q) ={q_\mu q_\nu C_\nu(q)\over q_E^2} =-iB\,{q_\mu\over q_E^2}.$$

For $B\ne0$, a correlator bounded at the origin cannot satisfy the contact identity. The longitudinal part scales as $1/\lvert q\rvert$, while its contraction is the nonzero constant $-iB$. Transverse terms remain undetermined. This is an exact statement about the defined Euclidean correlator; no raw time-ordered or Wightman object has been identified with it.

The physical Goldstone conclusion additionally assumes a translation- and Lorentz-invariant vacuum, a local conserved nonanomalous global current, a positive Hilbert-space inner product, an energy-momentum spectrum in the closed forward light cone, and the regulated local-charge commutator that implements the symmetry. A nonzero order-parameter variation then forces massless spectral content in the current–order-parameter correlator. When that content has a one-particle description, it gives the Goldstone pole. The Euclidean projection alone does not establish a particle scattering construction. Weinberg 2005, Volume II, § 19.2, pp. 169–172 develops the intermediate-state, locality and conservation argument. The corresponding Lesson 19 derivation keeps the same $B$ and current normalization.

This comparison is useful for conformal field theory. The global conformal generators annihilate the CFT vacuum, so their Ward identities are honest constraints on correlators. Negative Virasoro modes acting on a primary do not break the vacuum symmetry; they generate local descendant states inside a representation. Broken internal charges are different: the current has physical soft spectral weight. Similar-looking commutators therefore have different physical meanings depending on the state of the vacuum.

Under Coleman’s local scalar-order-parameter, local-current and positive-spectrum assumptions, the continuously broken vacuum just described is excluded in relativistic $1+1$ dimensions. The infrared argument does not exclude massless states of every kind (Coleman 1973, pp. 259–264, PDF). The Goldstone discussion here compares the conformal vacuum with a broken phase in a spacetime dimension and theory where these physical premises can hold.

Summary

The global conformal generators $L_{-1}$, $L_0$, and $L_1$ form an $sl_2$ subalgebra with no central extension. Their action on a primary field is

$$[L_n,\phi(z)]=\left(z^{n+1}\partial_z+(n+1)h z^n\right)\phi(z), \qquad n=-1,0,1.$$

Vacuum invariance turns this local transformation rule into global Ward identities for correlation functions. These identities fix the coordinate dependence of two- and three-point functions and are the first layer of conformal kinematics.

Radial quantization repackages the same information in representation language. A primary state obeys $L_n|h\rangle=0$ for $n>0$, while negative modes generate descendants. The vacuum module is special: global descendants vanish, and the first descendant not kinematically excluded is $L_{-2}|0\rangle=T(0)|0\rangle$, whose norm is $c/2$.

Finally, the same Ward-identity logic appears for ordinary conserved currents. If a well-defined charge annihilates the vacuum, integrated Ward identities become selection rules. In a phase with spontaneous breaking, the local current Ward identity instead requires soft massless spectral weight. This physical Goldstone mechanism is distinct from the representation-theoretic null relations studied next.

Common pitfalls

The most common sign trap is the relation between vector fields and mode labels. With the convention $L_n={1\over2\pi i}\oint z^{n+1}T$, the action on primaries is $[L_n,\phi]=(z^{n+1}\partial+(n+1)hz^n)\phi$, and the algebra is $[L_n,L_m]=(n-m)L_{n+m}+\cdots$. Some texts absorb a minus sign into the vector field basis.

A second pitfall is to confuse global and local conformal transformations. The modes $L_{-1},L_0,L_1$ are globally defined on the sphere and have no Schwarzian anomaly. Generic $L_n$ are local conformal generators in a coordinate patch; they are essential in radial quantization, but they are not all globally nonsingular transformations of the sphere.

Finally, descendants are not automatically independent. At special values of $h$ and $c$, linear combinations of descendants can be null. Those null states are not optional decoration; they are what make minimal models exactly solvable.

A final pitfall is to treat $Q|0\rangle\ne0$ literally in infinite volume. The global charge often has infrared-divergent norm in a broken phase; regulated charges and local Ward identities are the safe formulation of the Goldstone argument.

Exercises

Exercise 1: Global two-point function

Use the global Ward identities in the unit-normalized plane identity vacuum to derive the holomorphic two-point function of two primary fields at distinct points on a fixed compatible branch domain.

Solution

Let

$$G(z_1,z_2)=\langle \phi_1(z_1)\phi_2(z_2)\rangle.$$

Translation invariance gives

$$(\partial_{z_1}+\partial_{z_2})G=0,$$

so $G=f(z_{12})$, where $z_{12}=z_1-z_2$. Scale covariance gives

$$(z_1\partial_{z_1}+z_2\partial_{z_2}+h_1+h_2)G=0.$$

Since $z_1\partial_{z_1}+z_2\partial_{z_2}=z_{12}\partial_{z_{12}}$, this implies

$$f(z_{12})=Cz_{12}^{-(h_1+h_2)}.$$

The special conformal identity gives

$$\left(z_1^2\partial_{z_1}+z_2^2\partial_{z_2}+2h_1z_1+2h_2z_2\right)G=0.$$

Using the form above, this equation reduces to

$$(h_1-h_2)(z_1-z_2)G=0.$$

Thus either $G=0$ or $h_1=h_2$. Therefore

$$\langle \phi_1(z_1)\phi_2(z_2)\rangle ={C_{12}\over z_{12}^{2h_1}}, \qquad C_{12}=0\ \text{unless}\ h_1=h_2.$$

Exercise 2: Stress-tensor state norm

With $\langle0|0\rangle=1$, $L_n^\dagger=L_{-n}$ and the regular identity vacuum $L_n|0\rangle=0$ for $n\ge-1$, show that the norm of the stress-tensor state $L_{-2}|0\rangle$ is $c/2$.

Solution

Using radial quantization, $L_n^\dagger=L_{-n}$. Hence

$$\|L_{-2}|0\rangle\|^2 =\langle0|L_2L_{-2}|0\rangle.$$

Since $L_2|0\rangle=0$, we may replace $L_2L_{-2}$ by the commutator:

$$\langle0|L_2L_{-2}|0\rangle =\langle0|[L_2,L_{-2}]|0\rangle.$$

The Virasoro algebra gives

$$[L_2,L_{-2}]=4L_0+{c\over12}2(2^2-1)=4L_0+{c\over2}.$$

Because $L_0|0\rangle=0$,

$$\|L_{-2}|0\rangle\|^2={c\over2}.$$

This agrees with the normalization of $\langle T(z)T(w)\rangle$.

Exercise 3: Translation Ward identity

Let $G_N=\langle\phi_1(z_1)\cdots\phi_N(z_N)\rangle$ at separated insertions in the normalized plane identity vacuum. Assume $L_n|0\rangle=0$ for $n\ge-1$ and $L_n^\dagger=L_{-n}$, so that both $L_{-1}|0\rangle=0$ and $\langle0|L_{-1}=0$. Derive the translation Ward identity from these vacuum conditions and $[L_{-1},\phi_i]=\partial_i\phi_i$.

Solution

Since the vacuum is translation invariant,

$$\langle0|[L_{-1},\phi_1(z_1)\cdots\phi_N(z_N)]|0\rangle=0.$$

The commutator is a derivation:

$$[L_{-1},\phi_1\cdots\phi_N] = \sum_{i=1}^N \phi_1\cdots [L_{-1},\phi_i]\cdots\phi_N.$$

Using $[L_{-1},\phi_i]=\partial_{z_i}\phi_i$ gives

$$0= \sum_{i=1}^N \partial_{z_i}\langle\phi_1(z_1)\cdots\phi_N(z_N)\rangle.$$

Therefore

$$\boxed{\sum_i\partial_{z_i}G_N=0.}$$

This is the statement that a simultaneous translation of all insertion points does not change the correlator.

Exercise 4: Singular current correlator and finite divergence

Use the defined Euclidean identity $q_\mu C_\mu=-iB$ with $B\ne0$ in a setting where the order parameter is allowed. Find its longitudinal solution at $q_E^2>0$ and explain why it can have a finite nonzero contraction despite being singular at the origin. Compare it with the regular candidate $C_\mu^{\rm reg}=-iBq_\mu/(q_E^2+M^2)$, where $M^2>0$.

Solution

The longitudinal projector gives

$$C_\mu^\parallel ={q_\mu q_\nu C_\nu\over q_E^2} =-iB\,{q_\mu\over q_E^2}.$$

For $q=\lambda n$ with fixed nonzero Euclidean $n$ and $\lambda>0$, this becomes $C_\mu^\parallel=-iBn_\mu/(\lambda n_E^2)$. Its $1/\lambda$ singularity supplies exactly the finite contraction

$$q_\mu C_\mu^\parallel=-iB.$$

Multiplying by the derivative factor $-i$ gives $-iq_\mu C_\mu=-B$, the Fourier transform of the position-space contact $-B\delta^{(D)}(x)$. Conservation away from the insertion is therefore consistent with a nonzero distributional divergence at the insertion. One does not assign a finite value to $C_\mu(0)$.

The regular candidate instead has

$$q_\mu C_\mu^{\rm reg} =-iB\,{q_E^2\over q_E^2+M^2} \longrightarrow0 \qquad(q\to0).$$

It cannot reproduce the fixed nonzero contact. The local Ward identity has established the singular longitudinal part, while the physical spectral hypotheses in the text are needed to interpret it as Goldstone spectral weight.

References

• Coleman, Sidney. “There Are No Goldstone Bosons in Two Dimensions.” Communications in Mathematical Physics 31 (1973), 259–264. DOI. Open PDF.
• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Springer, 1997. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge University Press, 2005 paperback; 11th printing, 2012, consulted. ISBN 9780521670548. Chapter 19 DOI.