Gauge Fields in Two Dimensions

The previous two pages studied one-dimensional fermions from the many-body side. We found that the Fermi surface collapses to two Fermi points, that low-energy fields split into right- and left-movers, and that logarithms appear because the kinematics are effectively two-dimensional: one time direction and one space direction.

We now reorganize the same physics in the language of relativistic two-dimensional field theory. The great simplification is that a massless fermion in $1+1$ dimensions separates into two chiral components. Each component is governed by a first-order operator in one light-cone direction. As a result, the fermion Green function in an arbitrary Abelian background gauge field can be written almost exactly by inspection. This is the technical seed of the Schwinger model, the Thirring model, bosonization, and many of the two-dimensional examples that follow.

There is another simplification, just as important: a gauge field in $1+1$ dimensions has no transverse photon polarization. This does not mean that gauge fields are irrelevant. It means that their dynamics are mostly constraint, topology, holonomy, and vacuum polarization rather than propagating waves. Two dimensions are small enough to solve many things and still large enough for anomalies, confinement, screening, and nontrivial operator dimensions to show up. Tiny arena, dramatic cast.

Required background. Fermi-surface instabilities and one-dimensional fermions supplies the right-/left-moving decomposition and the logarithms that become anomalous powers below. Helpful background. Current correlators and polarization tensors supplies the transverse response kernel used for the fermion determinant, while Ward identities and chiral symmetries supplies the gauge-covariance logic. Three related objects remain distinct throughout: the fixed-background propagator $G(x,y;A)$ is gauge covariant and carries endpoint phases; the closed-loop determinant $e^{-W[A]}$ controls vacuum polarization; and a gauge-invariant observable is neutral or includes a Wilson-line dressing. A charged propagator may be useful in a chosen gauge, but only neutral or dressed operators are physical observables.

Gauge fields without transverse photons

Light-cone conventions. On this page we use real-time light-cone coordinates

$$x^+=x^0+x^1, \qquad x^-=x^0-x^1,$$

with

$$\partial_+={1\over2}(\partial_0+\partial_1), \qquad \partial_-={1\over2}(\partial_0-\partial_1).$$

The line element is

$$ds^2=dx^+dx^-.$$

The gauge field one-form is written as

$$A=A_+dx^+ + A_-dx^-,$$

so the gauge transformation is

$$A_\pm\mapsto A_\pm+\partial_\pm\theta.$$

For a unit-charge field, the covariant derivatives are

$$D_\pm=\partial_\pm-iA_\pm, \qquad \psi\mapsto e^{+i\theta}\psi.$$

Overall factors of $2$ in light-cone actions depend on whether one writes the measure as $d^2x=dx^0dx^1$ or as $dx^+dx^-/2$. To keep the inverse equations unambiguous, define

$$\delta^{(2)}_{\mathrm{LC}}(x-y) =\delta(x^+-y^+)\delta(x^--y^-),$$

the delta function normalized with the measure $dx^+dx^-$. The Cartesian delta function normalized with $dx^0dx^1$ is $2\delta^{(2)}_{\mathrm{LC}}$. The light-cone Green functions below satisfy $D_\pm G_\pm=\delta^{(2)}_{\mathrm{LC}}$.

In four spacetime dimensions, a massless gauge boson has two physical transverse polarizations. In $1+1$ dimensions there is no transverse direction. The gauge potential has two components, $A_0$ and $A_1$; gauge redundancy removes one combination, and Gauss’ law removes the other as an independent local oscillator. The gauge-invariant field strength has only one component,

$$F_{01}=\partial_0A_1-\partial_1A_0.$$

It is an electric field, not a wave with transverse polarizations.

In $1+1$ dimensions there is no transverse direction for a photon. The electric field $F_{01}$ is still physical, but it is fixed by constraints and sources rather than by an independent transverse wave mode.

This statement is clearest in temporal gauge, $A_0=0$. The Maxwell action is

$$S_M={1\over2e^2}\int d^2x\,E^2, \qquad E=\partial_0A_1.$$

The equation obtained by varying $A_0$ before imposing the gauge is Gauss’ law,

$$\partial_1E=-e^2\rho.$$

For external static charges, Gauss’ law determines $E$ algebraically from the charge distribution. Between a charge and an anticharge, the electric field is constant; the energy grows linearly with separation. This is the elementary reason why two-dimensional electrodynamics is a useful toy model for confinement, even though the detailed story depends strongly on whether light dynamical fermions are present.

In the absence of matter, the equation of motion

$$\partial_\mu F^{\mu\nu}=0$$

says in two dimensions that $F_{01}$ is constant locally. There is no local photon Hilbert space. But once fermions are integrated out, the gauge field can acquire a nonlocal effective action and, in the Schwinger model, a gauge-invariant mass scale. That will be one of the next lessons.

Massless fermions and chiral splitting

A massless Dirac fermion in two dimensions decomposes into two independent chiral components. With projectors $P_\pm$ onto the two eigenvalues of the two-dimensional chirality matrix, write

$$\psi=\psi_+ + \psi_-, \qquad \psi_\pm=P_\pm\psi.$$

The massless action separates into two first-order pieces. Up to the harmless overall light-cone normalization mentioned above,

$$S_0=i\int dx^+dx^-\, \left( \psi_-^\dagger\partial_+\psi_-+ \psi_+^\dagger\partial_-\psi_+ \right).$$

Thus the free equations are

$$\partial_+\psi_-=0, \qquad \partial_-\psi_+=0.$$

The field $\psi_-$ depends only on $x^-$, while $\psi_+$ depends only on $x^+$. In ordinary coordinates these are right- and left-moving fields. Coupling to a background Abelian gauge field gives

$$S[A]=i\int dx^+dx^-\, \left[ \psi_-^\dagger(\partial_+ - iA_+)\psi_- + \psi_+^\dagger(\partial_- - iA_-)\psi_+ \right].$$

The two chiralities couple to different light-cone components of the gauge field. This is why two-dimensional fermion problems often reduce to one-dimensional inverse operators.

For the $\psi_-$ component, define the background Green function by

$$(\partial_+ - iA_+(x))G_-(x,y;A)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

The $\psi_+$ equation is obtained by exchanging $+\leftrightarrow-$:

$$(\partial_- - iA_-(x))G_+(x,y;A)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

Because each equation contains only one derivative, the background field enters as a phase or Wilson line.

The free chiral propagator

The free inverse of $\partial_+$ is not simply $1/x^+$. The operator $\partial_+$ differentiates along $x^+$, but the chiral singularity sits at fixed $x^-$. With ordinary time ordering, the useful distribution is

$$\boxed{ G_-^{(0)}(x-y)= {1\over2\pi i}\, {1\over x^- - y^- - i0\,\operatorname{sgn}(x^+ - y^+)}. }$$

It satisfies

$$\partial_{x^+}G_-^{(0)}(x-y)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

The reason the $i0$ prescription involves $\operatorname{sgn}(x^+-y^+)$ is subtle the first time one sees it, but perfectly logical. The pole is at $x^-=y^-$. On that singular support,

$$x^0-y^0={1\over2}(x^+-y^+),$$

so ordinary time ordering is equivalent to ordering in $x^+$ at the pole. The discontinuity of the pole as $x^+-y^+$ crosses zero is exactly what produces the two-dimensional delta function.

Similarly,

$$\boxed{ G_+^{(0)}(x-y)= {1\over2\pi i}\, {1\over x^+ - y^+ - i0\,\operatorname{sgn}(x^- - y^-)} }$$

satisfies

$$\partial_{x^-}G_+^{(0)}(x-y)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

For the $\psi_-$ chirality, the operator is $D_+=\partial_+ - iA_+$. The free propagator is singular on $x^-=y^-$, and the integrating factor accumulates $A_+$ along each fixed-$x^-$ ray. A general bilocal Green function still requires both endpoint factors.

The propagator formula also gives the equal-time canonical anticommutator. The difference between the two time orderings is

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

so the jump of $G_-^{(0)}$ across $x^0=y^0$ reproduces the delta function in the spatial coordinate.

Euclidean bridge to complex coordinates

The real-time light-cone formulas have a Euclidean version that will be used heavily in the bosonization page. After Wick rotation, introduce

$$z=x^1+ix^2, \qquad \bar z=x^1-ix^2,$$

with

$$\partial_z={1\over2}(\partial_1-i\partial_2), \qquad \partial_{\bar z}={1\over2}(\partial_1+i\partial_2).$$

A right-moving chiral fermion has the holomorphic propagator

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

while a left-moving chiral fermion has the antiholomorphic propagator

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

The distributional $i0$ prescriptions above are the Lorentzian ancestors of these holomorphic singularities. This is why the determinant of many chiral propagators becomes a Cauchy determinant, and why the same functions can later be reproduced by free-boson vertex operators.

Exact Abelian Green functions in a background field

Now suppose first that the background is Abelian. Locally, for any fixed $x^-$, we can write

$$A_+(x^+,x^-)=\partial_+\alpha(x^+,x^-),$$

where, more precisely,

$$\alpha(x^+,x^-) =\alpha_0(x^-)+\int_{x_0^+}^{x^+}ds\,A_+(s,x^-)$$

for some reference point $x_0^+$ and boundary datum $\alpha_0(x^-)$. This is not a gauge-fixing assumption and does not imply that the full field strength vanishes. It is only the statement that a one-dimensional connection along the $x^+$ direction can be integrated. The freedom in $\alpha_0$ is part of the boundary prescription for the first-order inverse.

With $D_+=\partial_+ - iA_+$, the exact Green function is

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

Indeed,

$$(\partial_+ - i\partial_+\alpha) \left(e^{+i\alpha(x)}G_-^{(0)}(x-y)e^{-i\alpha(y)}\right) =e^{+i\alpha(x)}\partial_+G_-^{(0)}(x-y)e^{-i\alpha(y)}.$$

Using $\partial_+G_-^{(0)}=\delta^{(2)}_{\mathrm{LC}}$, the phase factors cancel at $x=y$, so

$$D_+G_-(x,y;A)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

The same construction for the other chirality gives

$$A_- = \partial_-\beta,$$

and

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

Under a gauge transformation, choose the boundary datum to transform as $\alpha_0(x^-)\mapsto\alpha_0(x^-)+\theta(x_0^+,x^-)$. Then $\alpha\mapsto\alpha+\theta$ and

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

The essential point is that the background field changes a chiral propagator by endpoint phases. This is the two-dimensional ancestor of several statements that will appear later: Wilson-line dressing, anomalous power laws, bosonization, and the exact solvability of the Schwinger model.

Gauge decomposition and field strength

Since $A_+$ and $A_-$ can each be integrated along their own light-cone direction, write

$$A_+=\partial_+\alpha, \qquad A_-=\partial_-\beta.$$

The field strength is

$$F_{+-}=\partial_+A_- - \partial_-A_+ =\partial_+\partial_-(\beta-\alpha).$$

Thus the gauge-invariant information is not $\alpha$ or $\beta$ separately, but their mismatch

$$\varphi=\beta-\alpha.$$

Under a gauge transformation,

$$A_\pm\mapsto A_\pm+\partial_\pm\theta,$$

we have

$$\alpha\mapsto\alpha+\theta, \qquad \beta\mapsto\beta+\theta,$$

and hence

$$\varphi\mapsto\varphi.$$

Locally, $A_+=\partial_+\alpha$ and $A_-=\partial_-\beta$. Gauge transformations shift $\alpha$ and $\beta$ together, so the difference $\beta-\alpha$ is gauge invariant and determines the curvature $F_{+-}$.

This decomposition is one of the small miracles of two-dimensional gauge theory. It separates the gauge redundancy from the curvature almost by eye. If $\alpha=\beta$ globally, the field is pure gauge and $F_{+-}=0$. If $\beta-\alpha$ is nonzero, then the two one-dimensional integrations cannot be glued into a single gauge function; their mismatch is the electric field.

On a space with nontrivial topology, such as a spatial circle or a thermal cylinder, there can also be zero modes and holonomies that are not captured by a single-valued local $\alpha$ or $\beta$. Those global variables are important in precise treatments, but the local formulas above are the workhorse for perturbative and operator calculations.

Fermion determinant and open-line propagator

There are two different objects that one must not confuse. The first is the open-line Green function in a fixed background,

$$G(x,y;A)=\langle x|(\gamma^\mu D_\mu)^{-1}|y\rangle.$$

The second is the fermion determinant, obtained by integrating out closed fermion loops:

$$e^{-W_E[A]} =\int \mathcal D\bar\psi\mathcal D\psi\, \exp\left[-\int d^2x\,\bar\psi\gamma^\mu D_\mu\psi\right] =\det(\gamma^\mu D_\mu).$$

Equivalently,

$$W_E[A]=-\operatorname{Tr}\log(\gamma^\mu D_\mu).$$

In a theory where the gauge field is also integrated over, the full fermion two-point function has the schematic form

$$\boxed{ \langle \psi(x)\bar\psi(y)\rangle ={1\over Z}\int \mathcal D A\, e^{-S_{g,E}[A]-W_E[A]}G(x,y;A). }$$

Here

$$Z=\int \mathcal D A\,e^{-S_{g,E}[A]-W_E[A]}.$$

The open propagator has an expansion in insertions of the background field,

$$G[A] =G_0+G_0(i\gamma^\mu A_\mu)G_0 +G_0(i\gamma^\mu A_\mu)G_0(i\gamma^\mu A_\mu)G_0+\cdots,$$

while $W[A]$ is made of closed fermion loops.

For fixed $A$, the open line gives $G[A]=(\gamma^\mu D_\mu)^{-1}$. Integrating out fermions also produces closed loops collected into $W[A]$. The full correlator averages the open line with the determinant included in the weight.

This distinction matters. If one averages $G(x,y;A)$ over $A$ but forgets $W[A]$, one has neglected vacuum polarization. That omission can define a controlled quenched approximation, but it is not the original dynamical-fermion theory. In two dimensions it misses precisely the closed-loop response responsible for the anomaly-related infrared physics.

For a massless unit-charge Dirac fermion on the Euclidean plane, preserving vector gauge invariance gives the universal nonlocal part of the Abelian determinant,

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

Equivalently,

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

This formula is finite and nonlocal. A renormalization prescription may separately add local gauge-invariant counterterms, such as a Maxwell term; it may not add an arbitrary longitudinal mass term while claiming to preserve the vector Ward identity. On a compact surface, flux sectors, harmonic gauge fields, spin structures, and fermion zero modes also contribute global factors not contained in this plane-wave kernel.

The connection to the light-cone mismatch becomes especially transparent in the Euclidean Hodge decomposition. On a simply connected patch write

$$A_\mu=\partial_\mu\lambda+\epsilon_{\mu\nu}\partial_\nu\chi, \qquad \epsilon_{12}=+1.$$

The transverse projector removes $\partial_\mu\lambda$, while

$$F_{12}=-\partial^2\chi.$$

After an integration by parts,

$$\boxed{ W_E[A]={1\over2\pi}\int d^2x\,(\partial_\mu\chi)^2. }$$

Thus the determinant sees only the transverse scalar $\chi$. This is the Euclidean counterpart of the Lorentzian statement that it depends on the mismatch $\beta-\alpha$, not on the common gauge shift of $\alpha$ and $\beta$. When the gauge field has its own Maxwell action, this same term produces the mass scale of the Schwinger model.

A useful normalization check is the following. In momentum space this same term is

$$W_E[A]={1\over2}\int_p A_\mu(-p)\,{1\over\pi}P^T_{\mu\nu}(p)\,A_\nu(p), \qquad P^T_{\mu\nu}=\delta_{\mu\nu}-{p_\mu p_\nu\over p^2}.$$

When it is added to the Maxwell term $(2e^2)^{-1}\int_p p^2 A^T_\mu(-p)A^T_\mu(p)$, the transverse denominator is $p^2/e^2+1/\pi$. Thus the mass in the next page is $m_\gamma^2=e^2/\pi$, not $2e^2/\pi$. The factor of $1/2$ in the quadratic-action convention is doing real work here.

Auxiliary gauge fields and current interactions

The same background-field method also organizes current-current interactions. Consider the Euclidean Thirring-type interaction

$$\mathcal L_E =\bar\psi\gamma_\mu\partial_\mu\psi +{g\over2}j_\mu j_\mu, \qquad j_\mu=\bar\psi\gamma_\mu\psi.$$

It can be represented by an auxiliary vector field:

$$\exp\left[-{g\over2}\int d^2x\,j_\mu j_\mu\right] \propto \int \mathcal D A_\mu\, \exp\left[-\int d^2x\, \left({1\over2g}A_\mu A_\mu-iA_\mu j_\mu\right)\right].$$

Thus the interacting fermion problem becomes a free fermion in a fluctuating background $A_\mu$, followed by a Gaussian average over $A_\mu$ modified by the determinant. For a chiral component, the exact background-field answer gives the schematic structure

$$\langle T\psi_-(x)\psi_-^\dagger(y)\rangle =G_-^{(0)}(x-y) \left\langle e^{+i[\alpha(x)-\alpha(y)]}\right\rangle_A.$$

If the $A$-average is Gaussian and

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

then

$$\left\langle e^{+i[\alpha(x)-\alpha(y)]}\right\rangle_A =\exp\left[-{1\over2}\left\langle [\alpha(x)-\alpha(y)]^2\right\rangle_A\right] \propto \left({a^2\over (x-y)^2}\right)^{\kappa/2}.$$

Therefore the fermion two-point function becomes an anomalous power law,

$$\langle T\psi_-(x)\psi_-^\dagger(y)\rangle \propto {1\over x^- - y^- - i0\operatorname{sgn}(x^+-y^+)} \left({a^2\over (x-y)^2}\right)^{\kappa/2}.$$

This is one of the most important lessons of two-dimensional field theory. A fluctuating gauge or auxiliary field does not merely shift a mass or coupling; it can change the scaling dimension of the fermion itself. In later language, the fermion behaves like an exponential of a free boson.

Abelian and non-Abelian backgrounds

For Abelian backgrounds, a single chiral equation is solved by an ordinary phase. For a non-Abelian background, $A_+$ is matrix-valued. Introduce a group-valued integrating factor $g(x)$ satisfying

$$(\partial_+ - iA_+)g=0.$$

At fixed $x^-$ it can be written as

$$g(x^+,x^-) =\mathcal P\exp\left[+i\int_{x_0^+}^{x^+}ds\,A_+(s,x^-)\right]g_0(x^-),$$

where $g_0(x^-)$ supplies the boundary data. The exact local inverse is then

$$\boxed{ G_-(x,y;A)=g(x)G_-^{(0)}(x-y)g^{-1}(y). }$$

Indeed, $D_+^xg(x)=0$, and on the support of $\partial_+G_-^{(0)}=\delta^{(2)}_{\mathrm{LC}}$ the matrix factors become $g(x)g^{-1}(x)=1$. If a gauge transformation acts as $\psi\mapsto h\psi$, the compatible choice $g\mapsto hg$ gives

$$G_-(x,y;A^h)=h(x)G_-(x,y;A)h^{-1}(y).$$

This is the non-Abelian version of the Abelian endpoint phases. Path ordering is essential because the matrices $A_+(s,x^-)$ at different $s$ need not commute.

The non-Abelian determinant is much richer than the Abelian one. The analogue of $W[A]$ contains the two-dimensional current algebra structure and is closely related to the Wess–Zumino–Witten functional. That is why the innocent-looking replacement

$$\bar\psi\gamma_\mu\psi \quad\longrightarrow\quad \bar\psi\gamma_\mu T^a\psi$$

changes the theory qualitatively. Closed loops now carry group generators, path ordering matters, and the effective action is no longer Gaussian in a single scalar phase.

Boundary data and Wilson-line shorthand

There is a small but consequential geometric point hidden by the usual Wilson-line shorthand. The segment

$$U(x^+,y^+;x^-) =\mathcal P\exp\left[+i\int_{y^+}^{x^+}ds\,A_+(s,x^-)\right].$$

obeys

$$(\partial_+ - iA_+(x))U(x^+,y^+;x^-)=0, \qquad U(y^+,y^+;x^-)=1.$$

But its lower endpoint is $(y^+,x^-)$, not the spacetime point $y=(y^+,y^-)$. Unless $x^-=y^-$, it therefore transforms with $\theta(y^+,x^-)$ rather than $\theta(y)$. The tempting expression

$$U(x^+,y^+;x^-)G_-^{(0)}(x-y)$$

solves the differential equation in its first argument but is not, by itself, the generally gauge-covariant bilocal Green function.

The endpoint-factor formula avoids this mistake. It transports $x$ and $y$ to boundary data on a reference slice and keeps both endpoint transformations:

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

Equivalently, one may connect $y$ to $x$ by a piecewise Wilson line whose transverse segment lies on the reference slice. The choice of that connector is boundary data; it is not fixed by the local equation $D_+G=\delta^{(2)}_{\mathrm{LC}}$. Retarded, advanced, Feynman, finite-temperature, and finite-volume inverses make different choices of this data.

On a circle or thermal cylinder, this boundary information includes holonomy and zero-mode sectors. The local integrating-factor solution remains correct, but no single-valued $\alpha$ or $g$ captures the entire answer globally.

Summary

Two-dimensional gauge theory is special for two independent reasons. First, the gauge field has no local transverse photon polarization. Its physical content is encoded in the electric field, constraints, holonomies, and the response of matter. Second, massless fermions split into chiral components, and each chiral component couples to only one light-cone component of the gauge field.

The central formula on this page is the exact chiral background propagator

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

with the analogous formula for $G_+$. This formula turns the problem of fermions in a fluctuating gauge or auxiliary field into the problem of averaging endpoint phases. In two dimensions such averages often produce logarithms, and exponentiated logarithms become anomalous powers. That is the bridge from Fermi-point logarithms to bosonization and exactly solvable models.

Common pitfalls

Confusing a directional primitive with a pure gauge. The statement $A_+=\partial_+\alpha$ is local and one-dimensional. It does not mean that the full gauge field is pure gauge. The field strength depends on the mismatch between the two potentials $\alpha$ and $\beta$.

Differentiating the wrong coordinate. The free chiral propagator for $\partial_+$ is singular in $x^-$, not in $x^+$. The derivative with respect to $x^+$ acts on the time-ordering prescription. This is a classic source of confused signs and missing delta functions.

Replacing endpoint phases by the wrong Wilson segment. A light-cone segment at fixed $x^-$ ends at $(y^+,x^-)$, not at a general point $(y^+,y^-)$. Use the two endpoint factors, or specify the reference-slice connector and boundary condition explicitly.

Equating the determinant with the open propagator. The determinant $W[A]$ is not the same object as the open propagator $G[A]$. Dropping $W[A]$ means dropping closed fermion loops. In two dimensions those loops can change the infrared physics completely.

Treating the absence of transverse photons as the absence of gauge physics. A two-dimensional gauge field has no local photon polarization, but it can mediate a linear Coulomb potential, carry holonomies, impose constraints, and acquire a gauge-invariant mass scale through matter polarization.

Exercises

Exercise 1: Verify the light-cone contact term

Verify the distribution identity

$$\partial_+\left[{1\over2\pi i}\,{1\over x^- - i0\operatorname{sgn}x^+}\right] =\delta(x^+)\delta(x^-).$$

Solution

For $x^+\neq0$, the sign is constant and the expression has no explicit smooth dependence on $x^+$, so the derivative vanishes away from $x^+=0$. The whole derivative is a contact term at $x^+=0$.

Across $x^+=0$, the denominator changes from $x^-+i0$ to $x^- - i0$. More precisely,

$$\partial_+{1\over x^- - i0\operatorname{sgn}x^+} =\delta(x^+)\left({1\over x^- - i0}-{1\over x^- + i0}\right).$$

Using

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

we obtain

$$\partial_+{1\over x^- - i0\operatorname{sgn}x^+} =2\pi i\delta(x^+)\delta(x^-).$$

Multiplying by $1/(2\pi i)$ gives the desired result.

Exercise 2: Verify the exact Abelian background propagator

Let $D_+=\partial_+ - iA_+$ and suppose $A_+=\partial_+\alpha$. Show that

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

satisfies

$$D_+^xG_-(x,y;A)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

Solution

Act with $D_+^x$:

$$(\partial_+^x-i\partial_+\alpha(x)) \left[e^{+i\alpha(x)}G_-^{(0)}(x-y)e^{-i\alpha(y)}\right].$$

The derivative of the phase is

$$\partial_+e^{+i\alpha(x)}=+i(\partial_+\alpha(x))e^{+i\alpha(x)}.$$

This cancels the $-i\partial_+\alpha$ term in $D_+$. Therefore

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

Since $\partial_+^xG_-^{(0)}(x-y)=\delta^{(2)}_{\mathrm{LC}}(x-y)$,

$$D_+^xG_-(x,y;A) =e^{+i\alpha(x)}\delta^{(2)}_{\mathrm{LC}}(x-y)e^{-i\alpha(y)}.$$

The delta function sets $x=y$, so the phases cancel. Hence

$$D_+^xG_-(x,y;A)=\delta^{(2)}_{\mathrm{LC}}(x-y).$$

Exercise 3: Isolate the gauge-invariant light-cone mismatch

Show that if

$$A_+=\partial_+\alpha, \qquad A_-=\partial_-\beta,$$

then the field strength is

$$F_{+-}=\partial_+\partial_-(\beta-\alpha),$$

and that $\beta-\alpha$ is invariant under $A_\pm\mapsto A_\pm+\partial_\pm\theta$.

Solution

By definition,

$$F_{+-}=\partial_+A_- - \partial_-A_+.$$

Substituting the decomposition gives

$$F_{+-}=\partial_+\partial_-\beta-\partial_-\partial_+\alpha.$$

For ordinary smooth fields, the derivatives commute, so

$$F_{+-}=\partial_+\partial_-(\beta-\alpha).$$

Under a gauge transformation,

$$A_+\mapsto A_+ + \partial_+\theta, \qquad A_-\mapsto A_- + \partial_-\theta.$$

This is implemented by

$$\alpha\mapsto\alpha+\theta, \qquad \beta\mapsto\beta+\theta.$$

Therefore

$$\beta-\alpha\mapsto(\beta+\theta)-(\alpha+\theta)=\beta-\alpha.$$

So $\beta-\alpha$ is gauge invariant.

Exercise 4: Perform the Euclidean auxiliary-field transformation

Use the Gaussian identity

$$\int \mathcal D A_\mu\, \exp\left[-\int d^2x\left({1\over2g}A_\mu A_\mu-iA_\mu j_\mu\right)\right] \propto \exp\left[-{g\over2}\int d^2x\,j_\mu j_\mu\right]$$

to derive the auxiliary-field representation of the Euclidean current-current interaction.

Solution

Complete the square:

$${1\over2g}A_\mu A_\mu-iA_\mu j_\mu ={1\over2g}(A_\mu-igj_\mu)(A_\mu-igj_\mu)+{g\over2}j_\mu j_\mu.$$

Therefore

$$\int \mathcal D A_\mu\, \exp\left[-\int d^2x\left({1\over2g}A_\mu A_\mu-iA_\mu j_\mu\right)\right]$$

is equal to

$$\exp\left[-{g\over2}\int d^2x\,j_\mu j_\mu\right] \int \mathcal D A_\mu\, \exp\left[-\int d^2x\,{1\over2g}(A_\mu-igj_\mu)^2\right].$$

The remaining Gaussian integral is independent of $j_\mu$ after shifting the integration variable. Thus it contributes only an overall normalization, giving the stated identity.

Exercise 5: Derive an anomalous power from a Gaussian phase

Assume a Gaussian average of a phase field gives

$$\left\langle [\alpha(x)-\alpha(0)]^2\right\rangle= \kappa\log{x^2\over a^2}$$

at large separation. Show that

$$\left\langle e^{-i[\alpha(x)-\alpha(0)]}\right\rangle =\left({a^2\over x^2}\right)^{\kappa/2}$$

up to a constant normalization.

Solution

For a Gaussian variable $X$ with zero mean,

$$\langle e^{iX}\rangle=e^{-{1\over2}\langle X^2\rangle}.$$

Here take

$$X=-[\alpha(x)-\alpha(0)].$$

Since $X^2=[\alpha(x)-\alpha(0)]^2$,

$$\left\langle e^{-i[\alpha(x)-\alpha(0)]}\right\rangle =\exp\left[-{1\over2}\kappa\log{x^2\over a^2}\right].$$

Thus

$$\left\langle e^{-i[\alpha(x)-\alpha(0)]}\right\rangle =\left({x^2\over a^2}\right)^{-\kappa/2} =\left({a^2\over x^2}\right)^{\kappa/2}.$$

A regulator-dependent additive constant in $\langle X^2\rangle$ would multiply this by a constant normalization.

References

• S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model,” Physical Review D 11 (1975), 2088–2097, doi:10.1103/PhysRevD.11.2088.
• 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).