Bosonization and Sine-Gordon–Thirring Duality

The previous page ended with a clue: neutral fermion correlators in two dimensions have the same algebraic structure as correlators of exponentials of a free scalar field. This is not an accident, and it is not merely a trick for computing determinants. In two spacetime dimensions, the operator algebra of a massless Dirac fermion can be represented by the operator algebra of a compact scalar. This is bosonization.

Bosonization is one of the places where quantum field theory behaves like a magician who actually explains the trick afterward. A fermion field, which anticommutes and has spin $1/2$, can be written as an exponential of a bosonic field, provided one treats zero modes, branch cuts, and Klein factors correctly. A four-fermion current-current interaction becomes a change of the scalar radius. A fermion mass term becomes a cosine potential. The massive Thirring model is therefore equivalent, in a precise sense, to the sine-Gordon model.

This page develops the dictionary in the normalization closest to the preceding two-dimensional notes. The goal is not to prove every global subtlety of bosonization on an arbitrary Riemann surface. The goal is to make the local operator identities and the sine-Gordon–Thirring map usable.

Required background. The Schwinger model and gauge-invariant correlators supplies the neutral fermion determinant that becomes a bosonic vertex-operator correlator. Helpful background. Gauge fields in two dimensions supplies the chiral propagators and complex-coordinate conventions.

Fermion determinants as boson correlators

Two-dimensional bosonization conventions. We use Euclidean complex coordinates

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

After analytic continuation, $z$ and $\bar z$ become the two light-cone directions $x^+$ and $x^-$. We use two independent chiral bosons and define the relative, sine-Gordon combination by

$$\Phi(z,\bar z)=\phi_R(z)-\phi_L(\bar z),$$

with two-point functions

$$\langle \phi_R(z)\phi_R(w)\rangle=-\log{z-w\over a}, \qquad \langle \phi_L(\bar z)\phi_L(\bar w)\rangle=-\log{\bar z-\bar w\over a},$$

and

$$\langle \phi_R(z)\phi_L(\bar w)\rangle=0.$$

The short-distance length $a$ is a UV cutoff. With this convention,

$$\langle \Phi(z,\bar z)\Phi(w,\bar w)\rangle =-\log{|z-w|^2\over a^2}.$$

The sum $\widetilde\Phi=\phi_R+\phi_L$ is the dual combination. Which of $\Phi$ and $\widetilde\Phi$ is called “the boson” varies across the literature. Keeping the relative combination explicit prevents a sign mismatch between the fermion mass operator and the topological current.

Normal ordering $: \cdots :$ is defined with respect to this free scalar. Overall constants in vertex operators depend on the cutoff convention and are absorbed into normalization constants.

Relation to canonical scalar normalization. The chiral convention above corresponds to the nonchiral free-boson action

$$S={1\over8\pi}\int d^2x\,(\partial_\mu\Phi)^2,$$

for which $:e^{i\phi_R}:$ has holomorphic weight $1/2$ and can represent a chiral fermion. If instead one uses the canonical normalization

$$S={1\over2}\int d^2x\,(\partial_\mu\varphi)^2,$$

then

$$\Phi=\sqrt{4\pi}\,\varphi.$$

In that convention the free-fermion mass operator corresponds to a sine-Gordon cosine with $\beta_{\mathrm{SG}}^2=4\pi$. Many apparent disagreements in bosonization formulas are only this rescaling.

The right-moving free fermion has propagator

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

Wick’s theorem gives the $n$-particle neutral correlator

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

The Cauchy determinant identity rewrites this as

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

The signs in the numerator depend on the ordering convention for the $w$ operators, but the determinant expression is unambiguous.

Now compare this with a Gaussian scalar. For vertex operators

$$V_\alpha(z)=:e^{i\alpha\phi_R(z)}:,$$

Wick’s theorem gives

$$\left\langle \prod_k V_{\alpha_k}(z_k)\right\rangle =\begin{cases} \displaystyle \prod_{i<j}\left({z_i-z_j\over a}\right)^{\alpha_i\alpha_j}, & \sum_k\alpha_k=0,\\[1.3em] 0, & \sum_k\alpha_k\ne 0, \end{cases}$$

where the second line is the zero-mode selection rule on the plane. Taking charges $+1$ at the $z_i$ and charges $-1$ at the $w_j$ gives

$$\left\langle \prod_{i=1}^n :e^{i\phi_R(z_i)}: \prod_{j=1}^n :e^{-i\phi_R(w_j)}: \right\rangle =a^n{ \prod_{i<j}(z_i-z_j) \prod_{i<j}(w_i-w_j) \over \prod_{i,j}(z_i-w_j) }$$

up to a sign fixed by the ordering of fermions. The factor $a^n$ is required by dimensional analysis and is canceled by the normalization of the $2n$ fermion operators. After that cancellation, this is exactly the same rational function as the fermion determinant.

A neutral right-moving fermion correlator is a Cauchy determinant. The same function is produced by a free chiral scalar with vertex charges $+1$ at fermion insertions and $-1$ at conjugate insertions.

Thus the local identification

$$\boxed{ \psi_R(z)\sim {F_R\over\sqrt a}:e^{i\phi_R(z)}:, \qquad \psi_R^\dagger(z)\sim {F_R^\dagger\over\sqrt a}:e^{-i\phi_R(z)}: }$$

reproduces all neutral right-moving correlators. Similarly,

$$\boxed{ \psi_L(\bar z)\sim {F_L\over\sqrt a}:e^{i\phi_L(\bar z)}:, \qquad \psi_L^\dagger(\bar z)\sim {F_L^\dagger\over\sqrt a}:e^{-i\phi_L(\bar z)}:. }$$

These prefactors match the preceding convention $\langle\psi_R(z)\psi_R^\dagger(w)\rangle=1/(z-w)$. If the fermion propagator is normalized instead as $1/[2\pi(z-w)]$, each prefactor acquires the familiar additional factor $1/\sqrt{2\pi}$.

The factors $F_R,F_L$ are Klein factors. They commute with the bosonic oscillators but anticommute among themselves, ensuring

$$\{\psi_R,\psi_L\}=0.$$

For many local correlation functions with equal numbers of each species, the Klein factors only provide the overall fermionic sign. For operator algebra, Hilbert-space construction, boundary conditions, or finite-size systems, they are not optional.

The determinant formula assumes a complex chiral fermion. For a real Majorana fermion, Wick’s theorem instead gives a Pfaffian of the antisymmetric matrix of two-point functions. Two Majorana fields combine into one Dirac field, which is why the charged Cauchy-determinant form is the most direct doorway to bosonization.

What the symbol “$\sim$” means

The formula $\psi_R\sim F_R:e^{i\phi_R}:$ is an operator dictionary inside correlation functions with a specified normalization and spin structure. It is not an equality of ordinary commuting classical fields. The exponential carries the correct local singularities; the Klein factor carries the global anticommutation data.

Scaling dimensions and the neutrality condition

The same Gaussian calculation immediately gives the scaling dimension of a vertex operator. For a chiral operator,

$$\langle V_\alpha(z)V_{-\alpha}(w)\rangle =\left({a\over z-w}\right)^{\alpha^2}.$$

A holomorphic primary field of weight $h$ has two-point function proportional to $(z-w)^{-2h}$, so

$$\boxed{h[V_\alpha]={\alpha^2\over2}.}$$

The fermion operator corresponds to $\alpha=1$ and therefore has $h=1/2$, as it should.

For the nonchiral vertex operator

$$V_{\alpha,\bar\alpha}(z,\bar z) =:e^{i\alpha\phi_R(z)+i\bar\alpha\phi_L(\bar z)}:,$$

we have

$$h={\alpha^2\over2}, \qquad \bar h={\bar\alpha^2\over2},$$

so the scaling dimension and Euclidean spin are

$$\Delta=h+\bar h={\alpha^2+\bar\alpha^2\over2}, \qquad s=h-\bar h={\alpha^2-\bar\alpha^2\over2}.$$

A right-moving spinor has spin $+1/2$, so any bosonic representation of an interacting right-moving fermion must satisfy

$$\boxed{\alpha^2-\bar\alpha^2=1.}$$

This simple equation is the seed of anomalous fermion dimensions in the massless Thirring model.

The neutrality condition is just as important. The nonchiral correlator

$$\left\langle\prod_k :e^{i\alpha_k\Phi(x_k)}:\right\rangle$$

contains the zero mode of $\Phi$. Since the free boson action depends only on derivatives, the constant mode is integrated over. The integral

$$\int d\Phi_0\,\exp\left(i\Phi_0\sum_k\alpha_k\right)$$

vanishes unless

$$\boxed{\sum_k\alpha_k=0.}$$

Equivalently, if one regulates the theory in a box of size $R$, nonneutral correlators carry powers of $R$ and disappear in the infinite-volume limit after normalizing the vacuum. This is the infrared counterpart of charge conservation.

The vertex-operator OPE

The operator product expansion of vertex operators follows from separating the contraction between nearby points from the normal-ordered product:

$$:e^{i\alpha\phi_R(z)}::e^{i\beta\phi_R(w)}: =\exp\left[-\alpha\beta\langle\phi_R(z)\phi_R(w)\rangle\right] :e^{i\alpha\phi_R(z)+i\beta\phi_R(w)}:.$$

Using

$$\langle\phi_R(z)\phi_R(w)\rangle=-\log{z-w\over a},$$

and expanding the slowly varying field $\phi_R(z)$ around $w$, we get

$$\boxed{ :e^{i\alpha\phi_R(z)}::e^{i\beta\phi_R(w)}: \sim \left({z-w\over a}\right)^{\alpha\beta} :e^{i(\alpha+\beta)\phi_R(w)}: \left[1+i\alpha(z-w)\partial\phi_R(w)+\cdots\right]. }$$

For $\alpha=1$ and $\beta=-1$,

$$:e^{i\phi_R(z)}::e^{-i\phi_R(w)}: \sim {a\over z-w}\left[1+i(z-w)\partial\phi_R(w)+\cdots\right],$$

which reproduces the fermion short-distance singularity. For $\alpha=\beta=1$,

$$:e^{i\phi_R(z)}::e^{i\phi_R(w)}: \sim {z-w\over a}:e^{2i\phi_R(w)}: +\cdots.$$

The zero as $z\to w$ is the bosonic representation of Pauli exclusion for two identical chiral fermions. The anticommutation sign is encoded by the branch choice and Klein factors; the vanishing of the local product is already visible in the OPE.

When two vertex operators approach each other, their charges add. The singular or vanishing prefactor is determined by the product of charges, $(z-w)^{\alpha\beta}$.

This OPE is the local engine behind bosonization. The global theory contains more data: compactification radius, allowed charges, spin structures, and Klein factors. But the short-distance algebra already knows why exponentials of a free scalar can behave like fermions.

Current dictionary and charge normalization

The vector current is the derivative of the relative boson. In the chiral normalization used above, and up to the overall sign fixed by the orientation convention for $\epsilon^{\mu\nu}$,

$$j^\mu={1\over2\pi}\epsilon^{\mu\nu}\partial_\nu\Phi.$$

Equivalently, in the canonical free-fermion normalization $\Phi=\sqrt{4\pi}\,\varphi$,

$$j^\mu={1\over\sqrt\pi}\epsilon^{\mu\nu}\partial_\nu\varphi.$$

The charge is therefore a winding number:

$$Q=\int dx\,j^0={1\over2\pi}\left[\Phi(+\infty)-\Phi(-\infty)\right] ={1\over\sqrt\pi}\left[\varphi(+\infty)-\varphi(-\infty)\right].$$

At nonzero Thirring coupling the canonically normalized sine-Gordon field has a coupling-dependent radius, and the corresponding dictionary becomes $j^\mu=(\beta_{\mathrm{SG}}/2\pi)\epsilon^{\mu\nu}\partial_\nu\varphi$. The free point $\beta_{\mathrm{SG}}^2=4\pi$ reproduces the coefficient $1/\sqrt\pi$.

This formula is the bridge between the local vertex-operator dictionary and the soliton interpretation below. A fermion is local in fermion variables but becomes a kink in the bosonic field.

The massless Thirring model as a free boson with a shifted radius

The massless Thirring model is a Dirac fermion in two dimensions with a current-current interaction:

$$\mathcal L_{\mathrm{Th},0} =\bar\psi i\gamma^\mu\partial_\mu\psi -{g_{\mathrm{Th}}\over2}j_\mu j^\mu, \qquad j^\mu=\bar\psi\gamma^\mu\psi.$$

In two dimensions this interaction is special. It is classically marginal: since $[\psi]=1/2$, the current has dimension $1$, so $j_\mu j^\mu$ has dimension $2$, the same as the Lagrangian density. Quantum mechanically, the model remains exactly solvable. The interaction does not create an arbitrary complicated non-Gaussian bosonic theory; it changes the normalization, or radius, of the boson.

A convenient way to encode this is to represent the interacting right- and left-moving fermions by

$$\psi_R\sim F_R:e^{i a\phi_R+i b\phi_L}:, \qquad \psi_L\sim F_L:e^{i b\phi_R+i a\phi_L}:,$$

with

$$a^2-b^2=1.$$

Then

$$\langle\psi_R(z,\bar z)\psi_R^\dagger(0,0)\rangle \propto {1\over z^{a^2}\bar z^{b^2}} ={1\over z}\,{1\over |z|^{2b^2}}.$$

The first factor gives the spinor transformation law; the second is an anomalous power. In Lorentzian notation this has the form

$$\langle T\psi_R(x)\psi_R^\dagger(0)\rangle \propto {1\over x^+}\,f(x^+x^-),$$

where the function $f$ is a power law in the conformal massless theory. Thus the Thirring interaction changes the scaling dimension of the fermion without changing its spin.

There are several equivalent normalizations in the literature. In the canonical Coleman normalization one writes a scalar $\varphi$ with kinetic term

$${1\over2}(\partial_\mu\varphi)^2,$$

and the Thirring coupling is related to the sine-Gordon coupling by

$$\boxed{ {4\pi\over \beta_{\mathrm{SG}}^2}=1+{g_{\mathrm{Th}}\over\pi}. }$$

In the chiral normalization above, the corresponding parameter is

$$\beta_{\mathrm{SG}}^2=4\pi\lambda^2, \qquad \lambda=a-b.$$

At $g_{\mathrm{Th}}=0$, one has $b=0$, $a=1$, and $\beta_{\mathrm{SG}}^2=4\pi$, the free-fermion point.

Adding a mass: the cosine perturbation

The massive Thirring model adds

$$\Delta\mathcal L=m\bar\psi\psi.$$

In chiral components,

$$\bar\psi\psi=\psi_L^\dagger\psi_R+\psi_R^\dagger\psi_L.$$

Using the interacting-fermion representation,

$$\psi_L^\dagger\psi_R \sim :e^{i(a-b)(\phi_R-\phi_L)}:.$$

Using the relative field already introduced above, define only the remaining exponent

$$\lambda=a-b.$$

Then the mass operator becomes

$$\boxed{ \bar\psi\psi\ \longleftrightarrow\ C_m\cos(\lambda\Phi), }$$

where $C_m$ is a cutoff-dependent normalization constant. Therefore the massive Thirring model maps to a sine-Gordon theory,

$$\boxed{ S_{\mathrm{SG},E} =\int d^2x\left[ {1\over2}(\partial_\mu\varphi)^2 +{\mu\over\beta_{\mathrm{SG}}^2}\left(1-\cos(\beta_{\mathrm{SG}}\varphi)\right) \right] }$$

in canonical normalization, or equivalently to

$$S_E={1\over8\pi}\int d^2x\,(\partial_\mu\Phi)^2 +\widetilde\mu\int d^2x\,[1-\cos(\lambda\Phi)]$$

in the chiral normalization of the discussion, with $\beta_{\mathrm{SG}}=\sqrt{4\pi}\lambda$. The precise relation between $m$, $\mu$, $\widetilde\mu$, and the cutoff is nonuniversal; the relation between the operator algebras and the dimensionless coupling $\beta_{\mathrm{SG}}$ is universal once the renormalization convention is fixed.

The current-current interaction of the massless Thirring model changes the compact-boson radius, represented by $K$ in the diagram and by the exponents $a,b$ in the operator dictionary. The fermion mass operator then becomes the sine-Gordon cosine.

This is the massive Thirring/sine-Gordon equivalence. It is strong–weak in an important sense. From

$${4\pi\over \beta_{\mathrm{SG}}^2}=1+{g_{\mathrm{Th}}\over\pi},$$

large positive Thirring coupling corresponds to small sine-Gordon coupling. In that regime the sine-Gordon semiclassical soliton is a useful description of the fermionic excitation.

Scaling dimension of the cosine

The mass perturbation also provides a direct consistency check. Because

$$\left\langle :e^{i\lambda\Phi(x)}: :e^{-i\lambda\Phi(0)}: \right\rangle \propto {1\over |x|^{2\lambda^2}},$$

the cosine has scaling dimension

$$\boxed{ \Delta_{\cos}=\lambda^2 ={\beta_{\mathrm{SG}}^2\over4\pi} ={1\over1+g_{\mathrm{Th}}/\pi}. }$$

At the free-fermion point, $\Delta_{\cos}=1$, exactly the dimension of $\bar\psi\psi$. The cosine is relevant when $\beta_{\mathrm{SG}}^2<8\pi$, marginal at $8\pi$, and irrelevant on the other side of that boundary. This is the renormalization-group content hidden inside the apparently simple replacement of a fermion mass by a periodic potential.

Solitons as fermions

The sine-Gordon potential is periodic. Its classical vacua are

$$\beta_{\mathrm{SG}}\varphi=2\pi n, \qquad n\in\mathbb Z.$$

A finite-energy static configuration must approach vacua at spatial infinity:

$$\varphi(x\to-\infty)={2\pi n_-\over\beta_{\mathrm{SG}}}, \qquad \varphi(x\to+\infty)={2\pi n_+\over\beta_{\mathrm{SG}}}.$$

The integer

$$Q_{\mathrm{top}}=n_+-n_- ={\beta_{\mathrm{SG}}\over2\pi}\left[\varphi(+\infty)-\varphi(-\infty)\right]$$

is the soliton number. The one-soliton solution for the potential

$$V(\varphi)={m_{\mathrm{SG}}^2\over\beta_{\mathrm{SG}}^2} \left(1-\cos\beta_{\mathrm{SG}}\varphi\right)$$

is

$$\boxed{ \varphi_{\mathrm{sol}}(x-X) ={4\over\beta_{\mathrm{SG}}}\arctan e^{m_{\mathrm{SG}}(x-X)}. }$$

It interpolates from $0$ to $2\pi/\beta_{\mathrm{SG}}$. For a static finite-energy solution, the first integral of the field equation is

$${1\over2}\left({d\varphi\over dx}\right)^2 =V(\varphi),$$

where the integration constant vanishes because both terms go to zero in a vacuum. The energy can therefore be evaluated without inserting the detailed profile:

$$M_{\mathrm{sol}} =\int_0^{2\pi/\beta_{\mathrm{SG}}} d\varphi\,\sqrt{2V(\varphi)}.$$

Evaluating the elementary integral gives

$$\boxed{ M_{\mathrm{sol}}={8m_{\mathrm{SG}}\over\beta_{\mathrm{SG}}^2}. }$$

The interacting bosonized current,

$$j^\mu={\beta_{\mathrm{SG}}\over2\pi} \epsilon^{\mu\nu}\partial_\nu\varphi,$$

identifies fermion number with topological charge:

$$Q_F=\int dx\,j^0 ={\beta_{\mathrm{SG}}\over2\pi}\left[\varphi(+\infty)-\varphi(-\infty)\right] =Q_{\mathrm{top}}.$$

Thus the elementary fermion of the Thirring description is represented, in the sine-Gordon description, by a soliton. The antifermion is the antisoliton.

Topological charge alone does not prove fermionic statistics. The nonlocal soliton operator and its branch prescription supply that extra information; in the full Coleman–Mandelstam dictionary, those operators obey the massive Thirring anticommutation relations.

The sine-Gordon field has infinitely many equivalent vacua. A soliton interpolates between neighboring vacua and carries one unit of fermion number. A background electric field or theta-like source tilts the periodic potential, making adjacent vacua energetically inequivalent and foreshadowing pair creation.

This identification is one of the sharpest lessons of two-dimensional field theory: what is elementary in one set of variables may be extended and topological in another. The word “duality” should be taken literally here. The two descriptions organize the same Hilbert space in different coordinates.

Tilted vacua and the electric-field interpretation

The sine-Gordon potential also gives a compact way to visualize electric backgrounds in two-dimensional gauge theory. A background electric field, or equivalently a theta-like source, can add a term linear in the boson,

$$V(\varphi)\to {m_{\mathrm{SG}}^2\over\beta_{\mathrm{SG}}^2} \left(1-\cos\beta_{\mathrm{SG}}\varphi\right) -h_E\varphi.$$

The coefficient $h_E$ is proportional to the applied electric field; its exact normalization depends on the gauge-field and bosonization conventions.

The shift

$$\varphi\to\varphi+{2\pi\over\beta_{\mathrm{SG}}}$$

no longer relates exactly degenerate vacua. Neighboring minima are tilted relative to each other. In the fermionic language, producing a soliton–antisoliton pair changes the electric flux between them. If the electric field is strong enough, pair production is energetically favored.

This is not yet the full confinement/screening story of QED₂, but it is already the right picture. The bosonic field keeps track of charge through its winding. A region between two solitons can sit in a different branch of the electric field. The next page uses this idea to discuss linear potentials, screening by massless fermions, and confinement in one spatial dimension.

Summary

Bosonization begins from a concrete identity: the Cauchy determinant of free chiral fermion correlators equals a neutral correlator of scalar vertex operators. With the chiral scalar normalized by

$$\langle\phi_R(z)\phi_R(w)\rangle=-\log(z-w),$$

the operator $:e^{i\phi_R}:$ has weight $1/2$ and reproduces a right-moving fermion. The zero mode enforces charge neutrality, while the vertex-operator OPE encodes both the fermion pole and the short-distance exclusion of identical chiral fermions.

The massless Thirring interaction changes the boson radius and gives the fermion an anomalous dimension while preserving its spin. The fermion mass term becomes a cosine perturbation. This turns the massive Thirring model into the sine-Gordon model, with the Coleman relation

$${4\pi\over\beta_{\mathrm{SG}}^2}=1+{g_{\mathrm{Th}}\over\pi}$$

in a standard normalization. In the sine-Gordon variables, fermion number is the topological winding number of the bosonic field, and the elementary fermion is a soliton.

Common pitfalls

Confusing the two beta symbols. Here $\beta_{\mathrm{SG}}$ is the sine-Gordon coupling, not an RG beta function. The chiral exponents were called $a,b$ to avoid the worst collision.

Dropping Klein factors from the full operator algebra. They often disappear from neutral local correlators, but they are needed so that right- and left-moving fermions anticommute.

Identifying the sum and relative bosons. With the fermion phases used here, the mass operator and vector current involve $\Phi=\phi_R-\phi_L$; the sum $\widetilde\Phi=\phi_R+\phi_L$ is the dual field. Changing the sign convention for the left-moving vertex exchanges these names, but the dictionary must change everywhere at once.

Treating the mass coefficient as universal. The relation between the Thirring mass parameter $m$ and the sine-Gordon cosine coefficient $\mu$ depends on the UV normalization of the composite operator $\bar\psi\psi$. The relation between scaling dimensions and the dimensionless coupling is the universal part.

Reading bosonization as a classical field identity. The massive Thirring/sine-Gordon relation is a quantum operator equivalence. A classical Dirac field does not literally equal a classical exponential of a scalar.

Exercises

Exercise 1: Vertex-operator weight

Using

$$\langle\phi_R(z)\phi_R(w)\rangle=-\log{z-w\over a},$$

show that

$$\left\langle :e^{i\alpha\phi_R(z)}::e^{-i\alpha\phi_R(w)}:\right\rangle =\left({a\over z-w}\right)^{\alpha^2}.$$

What is the conformal weight of $:e^{i\alpha\phi_R}:$?

Solution

For normal-ordered exponentials of a Gaussian field,

$$\langle :e^A::e^B:\rangle=e^{\langle AB\rangle}.$$

Here

$$A=i\alpha\phi_R(z), \qquad B=-i\alpha\phi_R(w),$$

so

$$\langle AB\rangle =\alpha^2\langle\phi_R(z)\phi_R(w)\rangle =-\alpha^2\log{z-w\over a}.$$

Therefore

$$\left\langle :e^{i\alpha\phi_R(z)}::e^{-i\alpha\phi_R(w)}:\right\rangle =\exp\left[-\alpha^2\log{z-w\over a}\right] =\left({a\over z-w}\right)^{\alpha^2}.$$

A holomorphic primary of weight $h$ has two-point function proportional to $(z-w)^{-2h}$, so

$$h={\alpha^2\over2}.$$

Exercise 2: The two-by-two Cauchy identity

Prove the $n=2$ Cauchy identity

$$\det \begin{pmatrix} {1\over z_1-w_1} & {1\over z_1-w_2}\\[0.5em] {1\over z_2-w_1} & {1\over z_2-w_2} \end{pmatrix} ={ (z_1-z_2)(w_2-w_1) \over (z_1-w_1)(z_1-w_2)(z_2-w_1)(z_2-w_2)}.$$

Solution

Compute the determinant directly:

$$D={1\over(z_1-w_1)(z_2-w_2)} -{1\over(z_1-w_2)(z_2-w_1)}.$$

Putting the two terms over a common denominator gives

$$D={ (z_1-w_2)(z_2-w_1)-(z_1-w_1)(z_2-w_2) \over (z_1-w_1)(z_1-w_2)(z_2-w_1)(z_2-w_2)}.$$

The numerator is

$$(z_1z_2-z_1w_1-w_2z_2+w_2w_1) -(z_1z_2-z_1w_2-w_1z_2+w_1w_2)$$

or

$$z_1(w_2-w_1)+z_2(w_1-w_2)=(z_1-z_2)(w_2-w_1).$$

This proves the formula.

Exercise 3: Spin from chiral charges

Let

$$V_{a,b}(z,\bar z)=:e^{ia\phi_R(z)+ib\phi_L(\bar z)}:.$$

Show that its Euclidean spin is

$$s={a^2-b^2\over2}.$$

Use this to derive the condition for $V_{a,b}$ to represent a right-moving spinor.

Solution

The right-moving and left-moving weights are

$$h={a^2\over2}, \qquad \bar h={b^2\over2}.$$

The Euclidean spin is the difference

$$s=h-\bar h={a^2-b^2\over2}.$$

A right-moving spinor has spin $+1/2$, so

$${a^2-b^2\over2}={1\over2}.$$

Therefore

$$a^2-b^2=1.$$

Exercise 4: Fermion mass as a cosine

Using

$$\psi_R\sim F_R:e^{ia\phi_R+ib\phi_L}:, \qquad \psi_L\sim F_L:e^{ib\phi_R+ia\phi_L}:,$$

show that the mass operator $\psi_L^\dagger\psi_R+\psi_R^\dagger\psi_L$ is proportional to $\cos[(a-b)(\phi_R-\phi_L)]$.

Solution

Ignoring Klein factors and normalization constants, the first term is

$$\psi_L^\dagger\psi_R \sim :e^{-ib\phi_R-ia\phi_L}::e^{ia\phi_R+ib\phi_L}:.$$

At coincident points the product must be renormalized, but the exponent is simply

$$i(a-b)\phi_R+i(b-a)\phi_L =i(a-b)(\phi_R-\phi_L).$$

Thus

$$\psi_L^\dagger\psi_R\sim :e^{i(a-b)(\phi_R-\phi_L)}:.$$

Similarly,

$$\psi_R^\dagger\psi_L\sim :e^{-i(a-b)(\phi_R-\phi_L)}:.$$

Adding them gives

$$\psi_L^\dagger\psi_R+\psi_R^\dagger\psi_L \propto \cos[(a-b)(\phi_R-\phi_L)].$$

Exercise 5: The classical sine-Gordon soliton

For the sine-Gordon energy functional

$$E=\int_{-\infty}^{\infty}dx\left[ {1\over2}\left({d\varphi\over dx}\right)^2 +{m^2\over\beta^2}(1-\cos\beta\varphi) \right],$$

show that the one-soliton solution

$$\varphi_{\mathrm{sol}}(x)={4\over\beta}\arctan e^{mx}$$

interpolates from $0$ to $2\pi/\beta$. Then verify that it satisfies the first-order equation

$${d\varphi\over dx}={2m\over\beta}\sin{\beta\varphi\over2}.$$

Solution

As $x\to-\infty$, $e^{mx}\to0$, so

$$\varphi_{\mathrm{sol}}(-\infty)=0.$$

As $x\to+\infty$, $\arctan e^{mx}\to\pi/2$, so

$$\varphi_{\mathrm{sol}}(+\infty)={4\over\beta}{\pi\over2}={2\pi\over\beta}.$$

Now differentiate. Let $u=e^{mx}$. Then

$${d\varphi\over dx} ={4\over\beta}{1\over1+u^2}mu ={4m\over\beta}{u\over1+u^2}.$$

Since

$${\beta\varphi\over2}=2\arctan u,$$

we use

$$\sin(2\arctan u)={2u\over1+u^2}.$$

Therefore

$${2m\over\beta}\sin{\beta\varphi\over2} ={2m\over\beta}{2u\over1+u^2} ={4m\over\beta}{u\over1+u^2} ={d\varphi\over dx}.$$

This proves the first-order equation.

References

• S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model,” Physical Review D 11 (1975), 2088–2097.
• S. Mandelstam, “Soliton operators for the quantized sine-Gordon equation,” Physical Review D 11 (1975), 3026–3030.
• D. C. Mattis and E. H. Lieb, “Exact solution of a many-fermion system and its associated boson field,” Journal of Mathematical Physics 6 (1965), 304–312.

Further reading

• S. Coleman, Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press, 1985).
• T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, 2004).
• A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, 1998).
• 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).