Schwinger Model and Gauge-Invariant Correlators

The previous page solved a massless two-dimensional fermion in a fixed Abelian background gauge field. The result was almost embarrassingly simple: a chiral propagator is the free chiral propagator multiplied by endpoint phases. That simplicity is deceptive in the best possible way. Once the gauge field is dynamical, those endpoint phases must be averaged over, and the answer depends crucially on gauge invariance.

This page studies the cleanest example: massless QED in two spacetime dimensions, usually called the Schwinger model. It is a small theory with a ridiculous amount of physics packed inside it. There are no transverse photons in $1+1$ dimensions, but the fermion determinant gives the gauge-invariant electric field a massive propagator. Charged fermion two-point functions are not gauge-invariant objects, but Wilson-line dressed correlators are. The same determinant and phase factors also lead naturally to the Cauchy determinants and vertex-operator formulas that become bosonization on the next page.

Required background. Gauge fields in two dimensions supplies the exact background-field propagator, the distinction between open lines and closed-loop determinants, and the light-cone decomposition used here. Helpful background. Ward identities and chiral symmetries supplies the vector Ward identity and axial anomaly, while current correlators and polarization tensors supplies the response-function viewpoint.

QED₂ and the absence of a transverse photon

Conventions and normalization. This page mostly uses Euclidean functional integrals. The gauge coupling is placed in the Maxwell term rather than in the covariant derivative:

$$S_E[A,\bar\psi,\psi] =\int d^2x\,\bar\psi\gamma_\mu(\partial_\mu-iA_\mu)\psi +{1\over4e^2}\int d^2x\,F_{\mu\nu}F_{\mu\nu}.$$

Thus a unit-charge field transforms as

$$\psi(x)\mapsto e^{+i\theta(x)}\psi(x), \qquad \bar\psi(x)\mapsto \bar\psi(x)e^{-i\theta(x)}, \qquad A_\mu(x)\mapsto A_\mu(x)+\partial_\mu\theta(x).$$

The Euclidean gamma matrices obey

$$\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu\nu}.$$

In two dimensions

$${1\over4e^2}F_{\mu\nu}F_{\mu\nu}={1\over2e^2}F_{12}^2.$$

With these conventions the Schwinger mass for one massless Dirac fermion is

$$\boxed{m_\gamma^2={e^2\over\pi}.}$$

If instead one writes $D_\mu=\partial_\mu-ie\mathcal A_\mu$ with a canonically normalized gauge field $\mathcal A_\mu$, then $A_\mu=e\mathcal A_\mu$ and the same physical mass is obtained.

Local formulas and global sectors. The determinant formulas below are for the Euclidean plane, or equivalently for the nonzero modes in the trivial flux sector. On a compact surface, harmonic gauge fields, spin structure, quantized flux, and fermion zero modes add global dependence. They do not change the local polarization tensor or the Schwinger mass $m_\gamma^2=e^2/\pi$.

The Euclidean Schwinger model is

$$Z=\int \mathcal D A_\mu\,\mathcal D\bar\psi\,\mathcal D\psi\, \exp\left[-S_E[A,\bar\psi,\psi]\right].$$

The important starting point is not the determinant, but the kinematics. In four spacetime dimensions a massless gauge field has two transverse polarizations. In $1+1$ dimensions there is no transverse spatial direction. The gauge field still has components $A_0,A_1$, but gauge redundancy and Gauss’ law remove local photon oscillators. The gauge-invariant field strength has one independent component, the electric field.

This is why the Schwinger model should not be described as “the photon eats a scalar” in the Higgs sense. There is no elementary scalar field with a vacuum expectation value. Instead, the fermion loop changes the correlator of the gauge-invariant electric field. A theory with no transverse photon becomes equivalent, in its gauge-invariant sector, to a massive scalar excitation.

The mass dimension also already hints at special behavior. In two dimensions,

$$[A_\mu]=1, \qquad [e]=1, \qquad [\psi]={1\over2}.$$

The gauge coupling is itself a mass scale. There is no need for dimensional transmutation here; the scale $e$ is present from the beginning. What is nontrivial is that this scale appears as a gauge-invariant pole in correlation functions.

An open charged fermion propagator is gauge covariant, not gauge invariant. A Wilson-line dressing supplies the missing endpoint phase and produces a gauge-invariant charged-pair operator.

Three correlators to keep separate

The Schwinger model is simple enough that one can compute many objects exactly, but the objects are not interchangeable.

• $G_A(x,y)$ is the fermion propagator in a fixed background $A$; it is gauge covariant.
• $\langle G_A(x,y)\rangle$ after a gauge-field average is zero unless a gauge is fixed or a dressing is supplied.
• $\langle \psi(x)\,\mathcal W_\Gamma(y,x)\,\bar\psi(y)\rangle$ is a gauge-invariant charged-pair correlator, but it depends on the chosen dressing path $\Gamma$.

This page repeatedly moves among these three objects. The physics is clearest when the endpoint phases are kept visible.

Integrating out the fermion

For a fixed background $A_\mu$, the fermion integral is Gaussian:

$$\int \mathcal D\bar\psi\,\mathcal D\psi\, \exp\left[-\int d^2x\,\bar\psi\gamma_\mu(\partial_\mu-iA_\mu)\psi\right] =\det(\gamma_\mu D_\mu),$$

where

$$D_\mu=\partial_\mu-iA_\mu.$$

Define the effective action $W_E[A]$ by

$$e^{-W_E[A]}=\det(\gamma_\mu D_\mu), \qquad W_E[A]=-\operatorname{Tr}\log(\gamma_\mu D_\mu).$$

For one massless Dirac fermion, gauge-invariant regularization gives the exact Abelian result

$$\boxed{ W_E[A] ={1\over2\pi}\int d^2x\, A_\mu \left(\delta_{\mu\nu}-{\partial_\mu\partial_\nu\over\partial^2}\right) A_\nu . }$$

Equivalently,

$$\boxed{ W_E[A] ={1\over2\pi}\int d^2x\, F_{12}{1\over-\partial^2}F_{12}. }$$

The second form is often the more physical one. It says that the determinant depends only on the gauge-invariant electric field, not on the pure-gauge part of $A_\mu$. The nonlocal operator $(-\partial^2)^{-1}$ is the two-dimensional Coulomb Green function. Thus a massless fermion loop turns a local Maxwell theory into a theory with a nonlocal electric-field term.

In momentum space, let

$$\int_p\equiv\int {d^2p\over(2\pi)^2}, \qquad P^T_{\mu\nu}(p)=\delta_{\mu\nu}-{p_\mu p_\nu\over p^2}.$$

Then

$$W_E[A]={1\over2}\int_p A_\mu(-p)\Pi_{\mu\nu}(p)A_\nu(p), \qquad \Pi_{\mu\nu}(p)={1\over\pi}P^T_{\mu\nu}(p).$$

The tensor is transverse:

$$p_\mu\Pi_{\mu\nu}(p)=0.$$

This transversality is the momentum-space form of vector gauge invariance. A regulator that spoils it has changed the theory.

Why the determinant stops at quadratic order

The word “exact” deserves an explanation. Expanding the logarithm of the determinant in powers of $A_\mu$ generates connected correlation functions of the free vector current:

$$W_E[A]-W_E[0] =-\sum_{n\geq1}{(+i)^n\over n!} \int d^2x_1\cdots d^2x_n\, A_{\mu_1}(x_1)\cdots A_{\mu_n}(x_n) \langle j_{\mu_1}(x_1)\cdots j_{\mu_n}(x_n)\rangle_{0,c}.$$

Here the sign follows from the Euclidean coupling $-iA_\mu j_\mu$ used above. For a massless Abelian Dirac fermion in two dimensions, the current algebra is Gaussian: the current can be represented as a derivative of a free scalar. Its connected correlators vanish for $n>2$. The one-point function vanishes, and the two-point function is the transverse polarization tensor $P^T_{\mu\nu}/\pi$. Consequently, all local nonzero-mode dependence of the determinant is already contained in the quadratic term.

This truncation is special. A fermion determinant is always a one-loop object, but in a generic theory that one loop has nonzero vertices with arbitrarily many external gauge fields. Non-Abelian currents, fermion masses, and global holonomy sectors invalidate the simple Gaussian argument.

The massless fermion loop produces a transverse quadratic term $\Pi_{\mu\nu}=P^T_{\mu\nu}/\pi$. Combined with the Maxwell action, the transverse gauge field has denominator $p^2+m_\gamma^2$ with $m_\gamma^2=e^2/\pi$.

The Schwinger mass

The total effective action for the gauge field is

$$S_{\mathrm{eff}}[A] ={1\over4e^2}\int d^2x\,F_{\mu\nu}F_{\mu\nu}+W_E[A].$$

Since only the transverse part of $A_\mu$ is physical, write $A_\mu=A^T_\mu+\partial_\mu\lambda$ with

$$p_\mu A^T_\mu(p)=0.$$

The Maxwell term is

$${1\over4e^2}\int F_{\mu\nu}F_{\mu\nu} ={1\over2e^2}\int_p p^2 A^T_\mu(-p)A^T_\mu(p),$$

and the determinant is

$$W_E[A]={1\over2\pi}\int_p A^T_\mu(-p)A^T_\mu(p).$$

Therefore

$$S_{\mathrm{eff}}[A] ={1\over2}\int_p A^T_\mu(-p) \left({p^2\over e^2}+{1\over\pi}\right) A^T_\mu(p).$$

The transverse gauge-field propagator is proportional to

$${e^2\over p^2+e^2/\pi}.$$

Thus the gauge-invariant mass scale is

$$\boxed{m_\gamma^2={e^2\over\pi}.}$$

The field strength has the Euclidean two-point function

$$\boxed{ \langle F_{12}(p)F_{12}(-p)\rangle ={e^2p^2\over p^2+m_\gamma^2}. }$$

The numerator $p^2$ is not a mistake. Since $F_{12}$ is a derivative of $A_\mu$, its two-point function contains a contact term. Indeed,

$${e^2p^2\over p^2+m_\gamma^2} =e^2-{e^2m_\gamma^2\over p^2+m_\gamma^2}.$$

At separated points, the contact term drops out and the remaining correlation decays with mass $m_\gamma$. This is the precise sense in which the Schwinger model has a massive gauge-invariant excitation.

There is a useful real-time derivation of the same mass. In Minkowski signature, the vector current is conserved, while the axial current has the anomaly

$$\partial_\mu j_5^\mu=-{1\over\pi}F_{01}$$

in the unit-charge convention above. With $\epsilon^{01}=+1$, the vector and axial currents are dual:

$$j_5^\mu=-\epsilon^{\mu\nu}j_\nu.$$

The minus sign is fixed here by $\epsilon^{01}=+1$, the mostly-minus metric, and $\gamma^5=\gamma^0\gamma^1$; changing any of those conventions requires translating both this identity and the anomaly together. Combining this relation with Maxwell’s equation gives

$$(\Box+m_\gamma^2)F_{01}=0, \qquad m_\gamma^2={e^2\over\pi}.$$

This derivation is less useful for computing all correlators, but it makes the physics vivid: the anomaly turns the electric field into a massive propagating scalar.

What is massive in the Schwinger model?

The phrase “the photon becomes massive” is useful but slightly dangerous in $1+1$ dimensions. There was no transverse photon oscillator to begin with. The gauge-invariant statement is instead that the electric-field correlator has a massive pole at

$$p^2=-m_\gamma^2$$

after analytic continuation to Minkowski signature. In Euclidean language the separated-point part of

$$\langle F_{12}(p)F_{12}(-p)\rangle =e^2-{e^2m_\gamma^2\over p^2+m_\gamma^2}$$

is governed by the denominator $p^2+m_\gamma^2$. The first term is a contact term. It matters for Ward identities and short-distance normalization, but it does not produce a long-distance force.

This distinction is helpful when comparing QED₂ with the Higgs mechanism. The Schwinger mass is generated by vacuum polarization and the anomaly; it is not a local gauge-noninvariant Proca term.

Screening of external charges

Pure electrodynamics in one spatial dimension confines external charges linearly. Put charges $+Q$ and $-Q$ a distance $R$ apart, with $Q$ measured in units of the dynamical fermion charge. Gauss’ law forces a constant electric field between them, so the energy grows like

$$V_{\mathrm{pure},Q}(R)={Q^2e^2\over2}|R|.$$

In the massless Schwinger model the fermion determinant modifies the static gauge propagator. In a convenient gauge, the static kernel is

$$D_{00}(k_0=0,k)={e^2\over k^2+m_\gamma^2}.$$

Therefore the interaction energy of the two opposite external charges is

$$V_Q(R)=Q^2\int_{-\infty}^{\infty}{dk\over2\pi}\, (1-\cos kR){e^2\over k^2+m_\gamma^2}.$$

Using

$$\int_{-\infty}^{\infty}{dk\over2\pi}\,{e^{ikR}\over k^2+m^2} ={1\over2m}e^{-m|R|},$$

we get

$$\boxed{ V_Q(R)={Q^2e^2\over2m_\gamma}\left(1-e^{-m_\gamma |R|}\right). }$$

At small separation, this reduces to the pure one-dimensional Coulomb result:

$$V_Q(R) ={Q^2e^2\over2}|R| -{Q^2e^2m_\gamma\over4}R^2 +O(|R|^3).$$

At large separation, it saturates:

$$V_Q(R)\longrightarrow {Q^2e^2\over2m_\gamma}.$$

The massless dynamical fermions screen external charges. This is why one must be careful with the word “confinement” in two-dimensional gauge theory. Pure QED₂ has a linear potential, whereas the massless Schwinger model screens. Massive matter adds pair-production thresholds and charge-lattice effects rather than a single universal interpolation; the later confinement-and-screening lesson treats those distinctions.

In pure QED₂, the static potential between opposite external charges grows linearly. In the massless Schwinger model, vacuum polarization gives the gauge field the Schwinger mass and the potential saturates at large separation.

Open fermion propagators are gauge covariant

Now consider the fermion Green function in a fixed background:

$$G_A(x,y)=\langle x|\left(\gamma_\mu D_\mu\right)^{-1}|y\rangle.$$

Under a gauge transformation,

$$\gamma_\mu D_\mu[A+\partial\theta] =e^{+i\theta(x)}\gamma_\mu D_\mu[A]e^{-i\theta(x)}$$

as an operator acting on charge-one fields. Therefore the inverse transforms as

$$\boxed{ G_{A+\partial\theta}(x,y) =e^{+i\theta(x)}G_A(x,y)e^{-i\theta(y)}. }$$

This is exactly the transformation law of the operator product $\psi(x)\bar\psi(y)$. It is covariant, not invariant.

Suppose we try to define the gauge-field averaged open propagator without gauge fixing:

$$\langle G(x,y)\rangle_A ={1\over Z}\int \mathcal D A\,e^{-S_{\mathrm{eff}}[A]}G_A(x,y).$$

Strictly speaking, both the numerator and denominator contain the infinite gauge-orbit volume. One can regulate the theory, divide by that common volume, and then use gauge invariance to project insertions onto the invariant sector. Changing variables $A\mapsto A+\partial\theta$ gives

$$\langle G(x,y)\rangle_A =e^{+i\theta(x)}e^{-i\theta(y)}\langle G(x,y)\rangle_A.$$

The function $\theta(x)$ is arbitrary. Unless $x=y$, the only gauge-invariant answer is

$$\boxed{\langle G(x,y)\rangle_A=0.}$$

This is the local gauge-theory version of a simple warning: a charged field is not an observable. One may compute a charged propagator after choosing a gauge, but then the answer is a gauge-dependent diagnostic, not a gauge-invariant correlation function. The same distinction appears in gravity: coordinate components may transform covariantly, while an observable must also specify how its insertion points are identified.

This statement is sometimes called Elitzur’s theorem in lattice language. In continuum perturbation theory it is often hidden because gauge fixing is introduced early. The Schwinger model is a good place to keep it visible.

Wilson-line dressed fermion correlators

To build a gauge-invariant charged-pair correlator, connect the two charged insertions by a Wilson line. For a path $\Gamma$ from $y$ to $x$, define

$$\mathcal W_\Gamma(y,x) =\exp\left(-i\int_\Gamma dz^\mu A_\mu(z)\right).$$

Under $A\mapsto A+\partial\theta$,

$$\mathcal W_\Gamma(y,x) \mapsto \exp\left(-i\theta(x)+i\theta(y)\right)\mathcal W_\Gamma(y,x).$$

Therefore

$$\psi(x)\mathcal W_\Gamma(y,x)\bar\psi(y)$$

is gauge invariant:

$$e^{+i\theta(x)} \left(e^{-i\theta(x)+i\theta(y)}\right) e^{-i\theta(y)}=1.$$

After integrating out the fermions, the dressed two-point function becomes

$$\boxed{ \mathcal G_\Gamma(x,y) ={1\over Z}\int\mathcal D A\, \exp[-S_{\mathrm{eff}}[A]]\, G_A(x,y)\, \exp\left(-i\int_\Gamma A\right). }$$

This formula separates three pieces of physics:

1. $G_A(x,y)$ is the propagation of a fermion in a fixed background.
2. $e^{-S_{\mathrm{eff}}[A]}$ includes the Maxwell action and all closed fermion loops.
3. The Wilson line supplies the electric dressing required by gauge invariance.

For a pure gauge field $A_\mu=\partial_\mu\alpha$, the fixed-background propagator has the endpoint form

$$G_A(x,y)=e^{+i\alpha(x)}G_0(x-y)e^{-i\alpha(y)},$$

while

$$\exp\left(-i\int_y^x dz^\mu\partial_\mu\alpha\right) =e^{-i\alpha(x)+i\alpha(y)}.$$

The phases cancel, and the dressed object reduces to the free propagator. For a field with nonzero curvature, the answer depends on the path $\Gamma$. Different paths differ by a Wilson loop around the area between them. In two-dimensional gauge theory that area is not a minor detail; it measures electric flux.

The path dependence has a simple physical interpretation. A gauge-invariant charged-pair operator does not create a naked fermion and antifermion. It creates them together with a chosen electric string connecting them. The Schwinger mass then determines how that string is screened by the dynamical fermions.

This bilocal operator is neutral as a whole; it does not establish the existence of an isolated charged asymptotic particle. The physical spectrum of the one-flavor massless Schwinger model contains a neutral massive boson. Different dressings can have different overlaps with that spectrum and different short-distance or perimeter contributions.

Gauge-fixed charged propagators and anomalous powers

Although open charged propagators are not gauge-invariant observables, they are still useful inside a fixed gauge or in closely related auxiliary-field problems such as the Thirring model. The endpoint phase formula makes their structure transparent.

For one chirality in light-cone coordinates, if

$$A_+=\partial_+\alpha,$$

then

$$G_-(x,y;A)=e^{+i\alpha(x)}G_-^{(0)}(x-y)e^{-i\alpha(y)}.$$

A Gaussian average of the phase gives

$$\left\langle e^{+i[\alpha(x)-\alpha(y)]}\right\rangle =\exp\left[-{1\over2}\left\langle(\alpha(x)-\alpha(y))^2\right\rangle\right].$$

If the phase field has a logarithmic two-point function,

$$\left\langle(\alpha(x)-\alpha(y))^2\right\rangle =\kappa\log{(x-y)^2\over a^2}+\text{constant},$$

then

$$\left\langle e^{-i[\alpha(x)-\alpha(y)]}\right\rangle \propto \left({a^2\over (x-y)^2}\right)^{\kappa/2}.$$

Thus the charged propagator acquires an anomalous power:

$$\langle G_-(x,y)\rangle_{\mathrm{fixed\ gauge}} \propto G_-^{(0)}(x-y) \left({a^2\over (x-y)^2}\right)^{\kappa/2}.$$

This is the mechanism behind the anomalous dimensions in two-dimensional fermion models. The exponent $\kappa$ of an open propagator is gauge- and prescription-dependent; it is not by itself a physical scaling dimension. In the Schwinger model, connected correlators of local gauge-invariant operators cluster exponentially because the gauge-invariant spectrum is gapped. The phase-factor calculation is still the bridge to bosonization: a fermion behaves like an exponential of a scalar whose two-point function is logarithmic.

Neutral correlators and the determinant structure

Gauge-invariant local operators must be neutral. Examples include the vector current

$$j_\mu=\bar\psi\gamma_\mu\psi,$$

and scalar or pseudoscalar bilinears such as

$$\bar\psi\psi, \qquad \bar\psi i\gamma^5\psi.$$

When charged fields are brought to the same point to define such bilinears, a short Wilson line should be included before taking the limit. This point-splitting prescription is the gauge-invariant way to define composite operators.

Even before the gauge field is integrated over, free chiral fermions already contain the algebraic pattern that will become bosonization. For a free complex chiral fermion with coordinate $z$, Wick’s theorem gives

$$\left\langle \psi(z_1)\cdots\psi(z_n) \psi^\dagger(w_1)\cdots\psi^\dagger(w_n) \right\rangle =\det_{1\le i,j\le n}\left({1\over z_i-w_j}\right).$$

For $n=2$,

$$\det \begin{pmatrix} {1\over z_1-w_1} & {1\over z_1-w_2} \\ {1\over z_2-w_1} & {1\over z_2-w_2} \end{pmatrix} = {1\over(z_1-w_1)(z_2-w_2)} -{1\over(z_1-w_2)(z_2-w_1)}.$$

The general Cauchy determinant is

$$\boxed{ \det\left({1\over z_i-w_j}\right) = {\prod_{i<j}(z_i-z_j)\prod_{i<j}(w_j-w_i) \over \prod_{i,j}(z_i-w_j)}. }$$

This formula is identical in structure to a correlator of vertex operators of a free scalar $\varphi$ with

$$\langle \varphi(z)\varphi(w)\rangle=-\log(z-w).$$

Indeed,

$$\left\langle e^{i\varphi(z_1)}\cdots e^{i\varphi(z_n)} e^{-i\varphi(w_1)}\cdots e^{-i\varphi(w_n)}\right\rangle \propto {\prod_{i<j}(z_i-z_j)\prod_{i<j}(w_i-w_j) \over \prod_{i,j}(z_i-w_j)}.$$

The slight sign convention in the $w$-product is a fermion-ordering convention. The important point is conceptual: neutral fermion correlators have the same coordinate dependence as neutral products of scalar exponentials. The next page turns this observation into the bosonization dictionary.

The word neutral is an actual selection rule, not merely a convenient choice of examples. For normal-ordered vertex operators

$$V_{q_a}(z_a)=:e^{iq_a\varphi(z_a)}:,$$

split the scalar into its constant mode and its fluctuating part,

$$\varphi(z)=\varphi_0+\widetilde\varphi(z).$$

The constant-mode integral contains

$$\int d\varphi_0\, \exp\left(i\varphi_0\sum_a q_a\right),$$

so a noncompact boson gives a delta function imposing $\sum_a q_a=0$; a compact boson gives the corresponding discrete charge-selection rule. Once neutrality holds, the fluctuating Gaussian integral gives

$$\boxed{ \left\langle\prod_a V_{q_a}(z_a)\right\rangle \propto \prod_{a<b}(z_a-z_b)^{q_aq_b}, \qquad \sum_a q_a=0. }$$

Without neutrality the massless scalar zero mode makes the correlator vanish or leaves it infrared-regulator dependent. Equal numbers of $\psi$ and $\psi^\dagger$ in the fermion determinant are the same selection rule in fermionic language.

For two identical chiral fermions, Wick’s theorem gives a determinant: the direct pairing minus the exchange pairing. The Cauchy determinant form is the first visible hint of the vertex-operator representation of fermions.

Summary

The Schwinger model is exactly solvable because several special two-dimensional facts meet at once. A gauge field has no transverse photon polarization. A massless Dirac fermion splits into chiral components. On the plane, the nonzero-mode part of the Abelian fermion determinant is exactly quadratic and gauge invariant. Combining that determinant with the Maxwell term gives a massive gauge-invariant field-strength correlator with

$$m_\gamma^2={e^2\over\pi}.$$

The model also teaches a sharp lesson about observables. The open fermion propagator $G_A(x,y)$ is gauge covariant, so its gauge-field average vanishes unless a gauge is fixed. A physical charged-pair correlator must include a Wilson-line dressing. Local fermion bilinears and currents are neutral and can be defined gauge invariantly, often by point splitting with a short Wilson line.

Finally, neutral chiral fermion correlators are determinants, and Cauchy’s determinant has exactly the product structure of free-boson vertex operators. The scalar zero mode enforces the same neutrality condition as equal fermion and antifermion number. That observation is the doorway to bosonization, the sine-Gordon/Thirring relation, and the two-dimensional confinement/screening comparisons that follow.

Common pitfalls

Calling the Schwinger mass a Proca term. The mass is not inserted by hand: $A_\mu A_\mu$ would violate gauge invariance. The exact determinant produces the transverse structure $A_\mu P^T_{\mu\nu}A_\nu$, or equivalently $F(-\partial^2)^{-1}F$.

Treating an open propagator as an observable. The ordinary fermion two-point function is not gauge invariant. It can be useful after gauge fixing, but a gauge-invariant charged-pair correlator needs a Wilson line or another dressing, and the result depends on that dressing.

Reading “no photon” as “no gauge dynamics.” The statement means no transverse photon oscillator. The electric field, holonomies, Wilson loops, and response to sources remain meaningful, and the anomaly produces a neutral massive excitation.

Dropping the field-strength contact term. The massive long-distance physics is read from the pole or from separated-point correlators, but the full momentum-space numerator also contains a contact term needed by short-distance identities.

Applying the plane determinant to every global sector. On a compact surface, flux, holonomy, spin structure, and zero modes add information beyond $A P^T A/(2\pi)$. The local polarization tensor and Schwinger mass remain valid.

Ignoring the scalar neutrality condition. A massless scalar has a constant mode. Vertex-operator correlators are nonzero and infrared finite only when the total vertex charge vanishes.

Exercises

Exercise 1: Prove Wilson-line dressing is gauge invariant

Show that the Wilson-line dressed operator

$$\psi(x)\exp\left(-i\int_y^x dz^\mu A_\mu(z)\right)\bar\psi(y)$$

is gauge invariant under

$$\psi\mapsto e^{+i\theta}\psi, \qquad \bar\psi\mapsto\bar\psi e^{-i\theta}, \qquad A_\mu\mapsto A_\mu+\partial_\mu\theta.$$

Solution

The Wilson line transforms as

$$\exp\left(-i\int_y^x A\right) \mapsto \exp\left(-i\int_y^x A-i\int_y^x d\theta\right) = \exp\left(-i\theta(x)+i\theta(y)\right) \exp\left(-i\int_y^x A\right).$$

The fermion factors transform as

$$\psi(x)\mapsto e^{+i\theta(x)}\psi(x), \qquad \bar\psi(y)\mapsto \bar\psi(y)e^{-i\theta(y)}.$$

Multiplying the phases gives

$$e^{+i\theta(x)}e^{-i\theta(x)+i\theta(y)}e^{-i\theta(y)}=1.$$

So the dressed operator is gauge invariant.

Exercise 2: Derive the Schwinger mass and field-strength correlator

Starting from

$$S_{\mathrm{eff}}[A] ={1\over2}\int_p A^T_\mu(-p) \left({p^2\over e^2}+{1\over\pi}\right)A^T_\mu(p),$$

derive the Schwinger mass and the field-strength two-point function

$$\langle F_{12}(p)F_{12}(-p)\rangle ={e^2p^2\over p^2+e^2/\pi}.$$

Solution

The transverse gauge-field kernel is

$$K(p)={p^2\over e^2}+{1\over\pi} ={1\over e^2}\left(p^2+{e^2\over\pi}\right).$$

Therefore the transverse propagator is

$$\langle A^T_\mu(p)A^T_\nu(-p)\rangle =e^2{P^T_{\mu\nu}(p)\over p^2+e^2/\pi}.$$

The pole occurs at

$$p^2+{e^2\over\pi}=0$$

in Euclidean momentum, so the physical mass is

$$m_\gamma^2={e^2\over\pi}.$$

In two dimensions, the field strength is one derivative of the transverse gauge field. In momentum space,

$$F_{12}(p)F_{12}(-p)=p^2 A^T_\mu(p)A^T_\mu(-p)$$

inside the transverse subspace. Thus

$$\langle F_{12}(p)F_{12}(-p)\rangle =p^2{e^2\over p^2+e^2/\pi} ={e^2p^2\over p^2+e^2/\pi}.$$

Exercise 3: Derive the screened static potential

Evaluate

$$V(R)=\int_{-\infty}^{\infty}{dk\over2\pi}\,(1-\cos kR){e^2\over k^2+m^2}$$

and show that it tends to a constant as $R\to\infty$.

Solution

Use the standard integral

$$\int_{-\infty}^{\infty}{dk\over2\pi}\,{e^{ikR}\over k^2+m^2} ={1\over2m}e^{-m|R|}.$$

The term with $1$ gives

$$\int {dk\over2\pi}{1\over k^2+m^2}={1\over2m}.$$

The cosine term is the real part of the exponential integral:

$$\int {dk\over2\pi}{\cos kR\over k^2+m^2}={1\over2m}e^{-m|R|}.$$

Therefore

$$V(R)={e^2\over2m}\left(1-e^{-m|R|}\right).$$

As $R\to\infty$, the exponential vanishes and

$$V(R)\to {e^2\over2m}.$$

Thus the potential is screened.

Exercise 4: Recover the two-fermion Cauchy determinant

Use Wick’s theorem to show that

$$\left\langle\psi(z_1)\psi(z_2)\psi^\dagger(w_1)\psi^\dagger(w_2)\right\rangle ={1\over(z_1-w_1)(z_2-w_2)} -{1\over(z_1-w_2)(z_2-w_1)}.$$

Assume the free chiral propagator is

$$\langle\psi(z)\psi^\dagger(w)\rangle={1\over z-w}.$$

Solution

There are two nonzero contractions. The direct contraction pairs

$$z_1\leftrightarrow w_1, \qquad z_2\leftrightarrow w_2,$$

and gives

$${1\over z_1-w_1}{1\over z_2-w_2}.$$

The exchange contraction pairs

$$z_1\leftrightarrow w_2, \qquad z_2\leftrightarrow w_1.$$

To put the fermion operators into the order required for this contraction, one exchanges two fermionic operators, producing a minus sign. Hence the exchange contribution is

$$-{1\over z_1-w_2}{1\over z_2-w_1}.$$

Adding the two terms gives the stated expression. Equivalently, it is the determinant

$$\det \begin{pmatrix} {1\over z_1-w_1} & {1\over z_1-w_2} \\ {1\over z_2-w_1} & {1\over z_2-w_2} \end{pmatrix}.$$

Exercise 5: Enforce vertex-operator neutrality

Let $\varphi(z)=\varphi_0+\widetilde\varphi(z)$ be a free chiral scalar with

$$\langle\widetilde\varphi(z)\widetilde\varphi(w)\rangle =-\log(z-w).$$

For $V_{q_a}(z_a)=:e^{iq_a\varphi(z_a)}:$, show that integration over $\varphi_0$ enforces $\sum_aq_a=0$ and that, when this condition holds,

$$\left\langle\prod_aV_{q_a}(z_a)\right\rangle \propto\prod_{a<b}(z_a-z_b)^{q_aq_b}.$$

Use $q=+1$ at each $z_i$ and $q=-1$ at each $w_j$ to recover the product structure of the Cauchy determinant.

Solution

The constant mode factors out:

$$\prod_aV_{q_a}(z_a) \propto \exp\left(i\varphi_0\sum_aq_a\right) \prod_a:e^{iq_a\widetilde\varphi(z_a)}:.$$

Therefore

$$\int d\varphi_0\, \exp\left(i\varphi_0\sum_aq_a\right) =2\pi\delta\left(\sum_aq_a\right)$$

for a noncompact scalar. Thus the correlator vanishes unless the total charge is zero.

For the fluctuating field, the Gaussian identity and normal ordering remove the self-contractions:

$$\left\langle\prod_a:e^{iq_a\widetilde\varphi(z_a)}:\right\rangle =\exp\left[ -\sum_{a<b}q_aq_b \langle\widetilde\varphi(z_a)\widetilde\varphi(z_b)\rangle \right].$$

Since the two-point function is $-\log(z_a-z_b)$,

$$\left\langle\prod_aV_{q_a}(z_a)\right\rangle \propto \prod_{a<b}(z_a-z_b)^{q_aq_b}.$$

For charges $+1$ at $z_i$ and $-1$ at $w_j$, the $z$–$z$ and $w$–$w$ pairs appear in the numerator, while every $z$–$w$ pair appears in the denominator:

$$\left\langle \prod_iV_{+1}(z_i)\prod_jV_{-1}(w_j) \right\rangle \propto {\prod_{i<j}(z_i-z_j)\prod_{i<j}(w_i-w_j) \over \prod_{i,j}(z_i-w_j)}.$$

This agrees with the Cauchy determinant up to the overall sign fixed by the ordering of the fermionic operators.

References

• S. Coleman, “More About the Massive Schwinger Model,” Annals of Physics 101 (1976), 239–267, doi:10.1016/0003-4916(76)90280-3.
• J. Schwinger, “Gauge Invariance and Mass. II,” Physical Review 128 (1962), 2425–2429, doi:10.1103/PhysRev.128.2425.

Further reading

• S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, edited by B.-G. Chen et al. (World Scientific, 2018).
• A. M. Polyakov, Gauge Fields and Strings (Harwood Academic Publishers, 1987).
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed. (Princeton University Press, 2010).
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed. (Oxford University Press, 2021).