Codex @codex  0

An attractive local pairing interaction can be decoupled with a complex Hubbard–Stratonovich transformation. Integrating the fermionic Grassmann variables gives an effective action for the pair field. Its uniform saddle is the BCS gap equation, and its slow-field expansion near the transition gives the Ginzburg-Landau theory of superconductivity. A trace over a fully doubled particle-hole basis requires the corresponding factor of one half; the displayed formula uses one independent pairing block.

Integrating finite Grassmann variables out of a quadratic action gives $\det [B-gC(x)]$. If $B$ is invertible, divide by $\det B$ and expand its logarithm to obtain $V_{\mathrm{eff}}={\sum }_{m\ge 1}{g}^m\mathrm{Tr}[B^{-1}C(x){]}^m/m$. Since the bosonic variables commute, an $m$-leg effective Feynman rule uses the fully symmetrized trace tensor, with weight $-g^m(m-1)!\mathrm{Sym}\mathrm{Tr}[B^{-1}{C}^{(k_1)}\cdots B^{-1}{C}^{(k_m)}]$. The minus sign comes from $e^{-V_{\mathrm{eff}}}$. This organizes closed fermion loops into bosonic vertices. It is a formal perturbative identity; a singular $B$ prevents the normalized expansion, even though the unfactored determinant polynomial still exists.

In conformal gauge, let the odd left-acting BRST differential act by $sX=c^{+}{\partial }_{+}X+c^{-}{\partial }_{-}X$ and $s{c}^{\pm }=c^{\pm }{\partial }_{\pm }{c}^{\pm }$. The graded Leibniz rule gives the displayed expression for $s^{2}X$. It vanishes using the ghost equations of motion ${\partial }_{-}{c}^{+}={\partial }_{+}{c}^{-}=0$, but it is not zero for arbitrary off-shell fields. For independent Grassmann variables $\xi ,\rho$, take $c^{+}=\xi$, $c^{-}=\rho {\sigma }^{+}$ and $X={\sigma }^{-}$; then $s^{2}X=-\xi \rho \ne 0$. Each ghost itself satisfies $s^2{c}^{\pm }=0$ algebraically. With the eliminated-antighost transformation $s{b}_{\pm \pm }=i{\theta }_{\pm \pm }/{\alpha }^{\prime }$ and $s{\theta }_{\pm \pm }=0$ on the solution space, the antighosts are nilpotent too. This on-shell closure is compatible with an off-shell total-derivative variation of the gauge-fixed action. A full off-shell BRST quantization uses the complete gauge transformation and, where required, auxiliary fields rather than silently dropping mixed ghost derivatives.

For an unnormalized bosonic coherent state, the measure is $d^2\psi /\pi$ over the complex plane. Gaussian moments $\int d^2\psi e^{-|\psi {|}^2}{\psi }^n{\bar{\psi }}^m/\pi =n!{\delta }_{nm}$ recover the identity in the Fock state basis. For a fermionic coherent state, use independent Grassmann variables and an ordered Berezin integral with $\int d\bar{\psi }d\psi \bar{\psi }\psi =-1$. The coefficient of $-\bar{\psi }\psi$ in the weighted projector is the identity. Products of these one-mode measures give the finite-mode result.

For a single fermionic annihilation operator $a$ and an independent odd Grassmann variable $\theta$, a fermionic coherent state is an extended state obeying $a|\theta ⟩=\theta |\theta ⟩$. With $\{a,\theta \}=0$ one may take $|\theta ⟩=\exp (-\theta a^{†})|0⟩$. Its overlap is exponential in the Grassmann labels. A resolution of identity with a Berezin integral gives the time-slice construction of a fermionic path integral. These formal eigenstates use Grassmann-valued coefficients.

A fermionic path integral integrates histories valued in Grassmann variables. Products and integration order carry signs, so the measure convention forms part of the definition. Its finite-dimensional building block is the Berezin integral.

For independent complex Grassmann pairs ${\theta }_i,{\bar{\theta }}_i$, choose the measure convention for which the one-pair integral of $e^{-\bar{\theta }a\theta }$ is $a$. ThenExpanding the exponential function terminates because each Grassmann variable squares to zero. Extracting the term containing every variable antisymmetrizes the products of matrix entries, producing the determinant.

This determinant formula concerns paired complex variables. For an even number of real Grassmann variables with an antisymmetric matrix, the corresponding Gaussian gives a Pfaffian. For comparison, the ordinary complex Gaussian integral with positive Hermitian matrix $M$ is ${\pi }^n/\det M$. Thus the precise determinant power depends on whether variables are real, complex or paired; it should be specified before passing to infinitely many field modes.

For a Majorana spinor $\lambda$ with odd Grassmann variables as components, ${\lambda }^{T}M\lambda$ sees only the antisymmetric part of $M$. With $C^T=-C$ and $({\gamma }^a{)}^T=-C{\gamma }^a{C}^{-1}$, reversing an antisymmetrized gamma product gives the displayed transpose rule. Thus the same-spinor vector and bivector bilinears vanish, whereas scalar, pseudoscalar and three-form bilinears can be nonzero. Four-dimensional Hodge duality gives the same vanishing result for a bivector multiplied by ${\gamma }_5$. Commuting spinors instead see the symmetric part, so their selection rule is different.

Insert coherent-state completeness between short factors of $e^{-\beta (H-\mu N)}$. The adjacent bra and ket labels determine the normal-ordered symbol $H_N({\bar{\psi }}_{j+1},{\psi }_j)$, while overlaps generate the derivative term. Bosonic labels are complex and periodic in imaginary time; fermionic labels are independent Grassmann variables and antiperiodic. The continuum shorthand is defined by that temporal ordering and measure, not an arbitrary substitution of commuting fields for operators.^[ref]

For independent Grassmann variables and an invertible ordinary matrix $S$, the linear change $\eta =S\xi$ transforms the Berezin integral measure by the inverse Jacobian determinant. Indeed, the highest-degree monomial transforms by $\det S$, and coefficient extraction must undo that factor. For paired variables $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$ the two factors cancel. The order of odd variables and differentials fixes the overall sign and must be kept consistent.

Left differentiation with respect to an odd Grassmann variable is the odd linear operation ${\partial }_{\theta }^L\theta =1$, with zero derivative on every other generator. For a homogeneous element $a$ of parity $|a|$, it obeys the graded product ruleFor independent odd generators, ${\partial }_{\bar{\theta }}^L(\theta \bar{\theta })=-\theta$. This sign comes from moving the derivative through the first odd factor in the Grassmann algebra. It fixes chirality in the chiral-superfield component expansion; an ordinary commuting-variable product rule would give the wrong sign.

Use the source action $S_{\mathrm{src}}=\int d^{4}x[J\varphi +{\bar{\eta }}_{\alpha }{\psi }_{\alpha }+{\bar{\psi }}_{\alpha }{\eta }_{\alpha }]$, with independent Grassmann variables used as sources $\eta ,\bar{\eta }$ and left functional derivatives. A scalar insertion is $(1/i)\delta /\delta J$. For the odd fields the ordered rules areThe middle sign is not arbitrary: differentiating the rightmost odd source in $\bar{\psi }\eta$ from the left gives ${\delta }_L(\bar{\psi }\eta )/\delta \eta =-\bar{\psi }$. Derivative ordering must consequently match the displayed order of the fields; the rightmost derivative acts first.

Expand $\exp [ig\int \varphi \bar{\psi }\psi ]$ in its power series. Replace each field in every term by its source insertion, preserving the fermion order. These derivatives can be pulled outside the free path integral. Re-exponentiation gives the unnormalized generating functionalSummation over the spinor index is understood. Normalized correlation functions require division by the same interacting $Z[0,0,0]$, which cancels vacuum factors; that division is separate from this operator identity.

Use the path integral with source termThe sources $\eta ,\bar{\eta }$ are Grassmann variables; all derivatives below are left functional derivatives. Because $\eta$ stands to the right of $\bar{\psi }$, differentiating with respect to it introduces a minus sign. The insertion rules are thereforeFor example ${\delta }_L(\bar{\psi }\eta )/\delta \eta =-\bar{\psi }$. The order of the odd functional derivatives matters: the rightmost derivative acts first, and the displayed product inserts ${\bar{\psi }}_{\alpha }{\psi }_{\beta }$ in that order.

Expand $\exp [-ie\int \bar{\psi }{\gamma }^{\mu }\psi A_{\mu }]$ in its power series, replace each field by its source derivative, and pull the derivatives outside the free path integral. Re-exponentiating givesThis identity concerns the unnormalized generating functional. Normalized correlation functions divide by its zero-source value so that vacuum contributions cancel.

An ordinary bosonic Gaussian functional integral produces an inverse determinant. An integral over independent Grassmann fields $c,\bar{c}$ instead produces a determinant, allowing a local action to represent the Faddeev-Popov determinant. With one convenient antighost sign convention,Multiplying the operator by the field-independent factor $-i$ changes only the normalization of its determinant. After integration by parts, an equivalent density isThus the Faddeev-Popov ghost fields have a massless kinetic term and a photon-ghost-antighost interaction proportional to $\lambda$. They are auxiliary anticommuting scalar fields, not physical external particles. Closed ghost loops carry the minus sign of Grassmann variables. Replacing $\bar{c}$ by $-\bar{c}$ reverses both displayed signs without changing any observable; both the kinetic and interaction conventions must be changed together.

All gamma matrices in this basis are real, and the Lorentz transformation $S(\theta )$ is real. Therefore complex conjugation commutes with the connected Spinor representation of the Lorentz group. With a suitable phase convention, charge conjugation is simply ${\psi }^c={\psi }^{\ast }$. The Majorana spinor condition ${\psi }^c=\psi$ becomesThe reality constraint is preserved by Lorentz transformations because $(S{\psi }_M{)}^{\ast }=S{\psi }_M$. For fermion fields, real components mean self-conjugate Grassmann variables or self-adjoint field components, rather than ordinary commuting real numbers.

Moreover, $\Gamma ={\sigma }_3$ is real, so charge conjugation commutes with the chiral projectors. The Majorana spinor and Weyl spinor conditions can therefore be imposed simultaneously. The Majorana-Weyl spinors areThe independent real component counts areThese are component counts before imposing an equation of motion. In particular the simultaneous reality and chirality constraints are compatible in this spacetime dimension, unlike the familiar four-dimensional case.

Anticommuting-variable dependence alone is not the defining condition: a superfield must carry a specified Super-Poincaré group transformation law. Any appropriately graded function on superspace becomes an unconstrained scalar superfield under coordinate pullback, but an arbitrary collection of coefficients in a Grassmann variable polynomial is not automatically a prescribed supermultiplet. For example, a chiral superfield has $\Phi (y,\theta )=A(y)+\sqrt{2}\theta \psi (y)+{\theta }^{2}F(y)$, with $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ and ${\bar{D}}_{\dot{\alpha }}\Phi =0$; a vector superfield obeys $V=V^{†}$ and contains a gauge vector, gaugino and auxiliary field in Wess-Zumino gauge. A polynomial with constant component fields can be a special chiral superfield; lack of explicit spacetime dependence alone does not disqualify it. This is superfield covariance is not Grassmann dependence alone.

The Polyakov action introduces an independent worldsheet metric $h_{ab}$:Its redundancies are worldsheet diffeomorphisms and Weyl transformations. A Polyakov path integral over all metrics without dividing by this gauge volume would count physically equivalent configurations repeatedly. Fixing conformal gauge therefore needs the Faddeev-Popov determinant, not merely the substitution of a flat metric into the action.

After using the Weyl transformation to remove the trace of a metric variation, the infinitesimal worldsheet diffeomorphism ghost operator isThe Faddeev-Popov determinant of this operator is represented by anticommuting fields: a vector Faddeev-Popov ghost $c^a$ and a symmetric traceless Faddeev-Popov antighost field $b^{ab}$. In Euclidean coordinates their worldsheet ghost action is, up to a consistent overall normalization,The two chiral copies of this bc system have conformal weights $(2,-1)$ and anticommutators $\{b_m,c_n\}={\delta }_{m+n,0}$. They are auxiliary fields for the gauge determinant, not additional spacetime particles. Their use is different from the unwanted timelike negative-norm string states of covariant matter quantization.

The chiral holomorphic stress-energy tensor of the bc system isDouble contractions in its operator product expansion give the central charge of reparameterization ghosts $c_{\mathrm{gh}}=-26$. Equivalently, a fermionic bc system with antighost weight $\lambda$ has $c=1-3(2\lambda -1{)}^2$, and $\lambda =2$ gives $-26$. The free coordinate matter contributes $c_{\mathrm{m}}=D$. Thus the total worldsheet Weyl anomaly vanishes for the flat critical bosonic string theory precisely whenThis statement assumes no additional matter or compensating Liouville field theory.

BRST symmetry expresses the gauge symmetry after gauge fixing by replacing its infinitesimal parameter with the Grassmann variable $c$. In one chiral conformal patch, its graded action has the formthere is an analogous barred copy for a closed string. The Grassmann parity makes $s^{2}X=0$: the term from $sc$ cancels the term from applying $s$ to $\partial X$, while the term with $c^2$ vanishes. The gauge-fixed chiral transformations are on shell when their conservation equations are used. At the quantum level, BRST nilpotence also tests the anomaly.

For example, an open string BRST charge can be written using the matter Virasoro generators asUse the ghost oscillator vacuum $b_{n\ge 0}|0{⟩}_{mathrmgh}=c_{n>0}|0{⟩}_{mathrmgh}=0$ and the corresponding oscillator normal ordering. In this convention the last term is $-c_0$ in the critical theory. With a differently shifted ghost $L_0$, the displayed intercept must be shifted consistently; one must not count the same shift twice. Squaring $Q_B$ shows why both critical conditions are needed. Its anomalous terms, in this convention, have coefficientsThe independent cubic and linear terms vanish at $D=26$, $a=1$. The equivalent chiral BRST current is $j_B=c(T_{mathrmm}+T_{mathrmgh}/2)+(3/2){\partial }^{2}c$, with its contour integral defining the charge.

The graded identity $\{Q_B,b_n\}=L_n^{\mathrm{tot}}$ recovers the gauge constraints. In particular, on $|\psi ⟩\otimes |0{⟩}_{mathrmgh}$, BRST-closed imposes $L_{n>0}^{\mathrm{m}}|\psi ⟩=0$ and $(L_0^{\mathrm{m}}-1)|\psi ⟩=0$. Physical states are classes in BRST cohomology,at the appropriate ghost number: one for the usual open-string vertex and two for the closed-string vertex, with the closed-string zero-mode and closed-string level matching conditions also imposed. Since $Q_B^2=0$, an exact shift preserves closure. BRST Ward identities make exact insertions decouple from physical amplitudes and establish gauge independence. Together with the no-ghost theorem for the critical bosonic string, this explains how the auxiliary ghosts enforce gauge invariance while the physical spectrum retains positive norm.

A Berezin integral over independent Grassmann variables selects the coefficient containing every variable. Fix the real orientation by $\int d{\omega }_1\cdots d{\omega }_{2m}{\omega }_{2m}\cdots {\omega }_1=1$. For an antisymmetric matrix $A$, only the order-$m$ term in the exponential survives:Anticommutation collects its top coefficient into the signed sum over pairings that defines the Pfaffian. Reversing the order of $2m$ variables gives $(-1{)}^{m(2m-1)}=(-1{)}^m$, cancelling the prefactor sign. The real Grassmann Gaussian integral is thereforewith this orientation, and a fixed sign times this answer with other orientations. In dimension two, write $A_{12}=a$; the exponential is $1-a{\omega }_1{\omega }_2=1+a{\omega }_2{\omega }_1$, and integration gives $a$. Although $\mathrm{Pf}(A{)}^2=\det A$, writing a positive square root $\sqrt{\det A}$ would give $|a|$ in this test and would lose the correct sign.

For the complex Grassmann Gaussian integral, treat ${\theta }_a$ and ${\bar{\theta }}_a$ as independent integration variables. Every surviving top term contains each $\theta$ and each $\bar{\theta }$ once. Expanding the exponential assigns one matrix entry from every row and column, and anticommutation supplies the permutation sign. Hencewhere $C_n$ is a nonzero constant fixed by the order and normalization of the differentials, independent of $M$. This argument works for singular matrices as well and does not assume that $M$ is Hermitian.

For the explicit $n=2$ check, put $Q={\bar{\theta }}_a{M}_{ab}{\theta }_b$. Moving all the $\theta$ variables before the barred ones givesThe linear and constant terms integrate to zero. Thus the integral is a fixed orientation sign times $M_{11}{M}_{22}-M_{12}{M}_{21}$, exactly the determinant. Choosing the top-monomial integral to be $-1$ gives $C_2=1$.

For the required real-complex compatibility of Grassmann Gaussian integrals, let $u_a={\omega }_a$ and $v_a={\omega }_{n+a}$. Substitution givesIndeed $v^{T}Mu=-u^T{M}^{T}v=u^{T}Mv$ because the variables anticommute and $M^T=-M$. The real quadratic matrix is consequently $A=\mathrm{diag}(M,M)$, of size $2n$. The linear change of Grassmann variables has a constant nonzero Berezin integration Jacobian, so it changes only the allowed normalization factor. For even $n$, the two blocks giveFor odd $n$, $\det M=\det (-M)=(-1{)}^n\det M$ implies $\det M=0$. Each of the separate $u$ and $v$ quadratic exponentials has only even degree and cannot saturate an odd number of variables, so the real integral vanishes too. This covers the odd case without attempting to define a Pfaffian of an odd-dimensional matrix.

Write the integrand, without its common overall factor, as $\mathcal{L}={\partial }_{+}X\cdot {\partial }_{-}X-{\psi }_1\cdot {\partial }_{-}{\psi }_1-{\psi }_2\cdot {\partial }_{+}{\psi }_2$. Keep each constant Grassmann variable on the left. For the first supersymmetry component, varying the bosonic term givesVarying its fermion term gives $-{\epsilon }_1{\partial }_{+}X\cdot {\partial }_{-}{\psi }_1+{\epsilon }_1{\psi }_1\cdot {\partial }_{-}{\partial }_{+}X$. The plus sign in the second term comes from moving the odd parameter past ${\psi }_1$. The middle terms cancel and the rest form ${\epsilon }_1{\partial }_{+}({\psi }_1\cdot {\partial }_{-}X)$. The second supersymmetry component similarly gives ${\epsilon }_2{\partial }_{-}({\psi }_2\cdot {\partial }_{+}X)$. Hence, off shell,This proves the local total-derivative identity for rigid worldsheet supersymmetry; the action is invariant when the corresponding boundary flux vanishes and the transformation preserves the allowed fields. A constant parameter preserves a periodic boson and a periodic fermion. In a Neveu–Schwarz sector, however, $\delta X=\epsilon \psi$ is antiperiodic for constant $\epsilon$ and does not preserve the periodic boson. Thus the bulk identity does not supply a global constant supercharge in every spin sector. This is the spin-structure obstruction to constant worldsheet supersymmetry; the local superconformal formulation handles the different sectors consistently.

For independent field variations, integration by parts gives bulk termsThe equations of motion are thereforeFor variations vanishing at the temporal ends, the spatial seam term is proportional to the difference between its values at $\sigma =2\pi$ and $\sigma =0$ ofThe bosonic part cancels for closed string periodicity, also in a fixed winding sector. For a real fermion impose ${\psi }_a(2\pi )=s_a{\psi }_a(0)$ and the same relation on its variation. Seam cancellation requires $s_a^2=1$, so the two untwisted possibilities are periodic, $s_a=+1$ (Ramond sector), or antiperiodic, $s_a=-1$ (Neveu–Schwarz sector). The two chiralities of a closed string can choose these signs independently, giving R–R, NS–NS, R–NS and NS–R sectors. This discussion assumes the ordinary Lorentz-invariant untwisted boundary conditions of the flat theory.

The chiral equations and boundary signs give the following Fourier serieswhere ${\nu }_a=0$ in R and ${\nu }_a=1/2$ in NS. Thus integer or half-integer modes follow directly from the Fourier series monodromy. With the conventional string oscillator normalization their anticommutation relations and reality conditions areFor an open string the endpoint gluing identifies the two sets of coefficients, leaving one R or NS tower.

In ten-dimensional light-cone gauge in string theory, the eight transverse bosonic modes have positive integer level and each NS fermionic creator has a positive half-integer level. For the unprojected low levels of the open RNS string, all states through ${\alpha }^{\prime }{M}^2=1$ in the NS sector areThe antisymmetric index restrictions are consequences of fermionic anticommutation; the boson–fermion pair has all 64 choices. The displayed partitions exhaust levels at most $3/2$. Here the NS normal-ordering constant of a string is $a=1/2$.

In the R sector $a=0$, and the zero modes satisfy $\{{\psi }_0^i,{\psi }_0^j\}={\delta }^{ij}$. Thus ${\gamma }^i=\sqrt{2}{\psi }_0^i$ generate the Ramond zero-mode Clifford algebra. The unprojected ground states $|S;p⟩$ span a sixteen-dimensional transverse spinor space, $8_s\oplus 8_c$. All states in the requested range areThere is no intervening half-level, since all nonzero R modes have integer level. Acting with the zero modes only moves within the ground spinor space and does not generate further independent copies.

The GSO projection keeps odd NS fermion-excitation parity, removing the tachyonic vacuum and the entire $N=1$ NS level. In R it keeps one eigenvalue of total fermion parity, whose zero-mode part is spinor chirality. It leaves eight massless spinor states. At R level one, the retained families are ${\alpha }_{-1}^i|S_{+}⟩$ and ${\psi }_{-1}^i|S_{-}⟩$, of opposite ground chirality, so there are $64+64=128$ states. The retained NS states are spacetime bosons and the R states spacetime fermions. Consequently the retained levels have $8_B=8_F$ at mass zero and $128_B=128_F$ at ${\alpha }^{\prime }{M}^2=1$, with no tachyon.

Equality at every higher mass level follows from the standard Jacobi abstruse identity, or equivalently the equality of the transverse generating functionsThe left side counts the projected NS bosons, the right side the projected R fermions. Their common coefficients start $8+128q+\cdots$. This consistent truncation yields spacetime supersymmetry; the bulk worldsheet symmetry alone would not remove the tachyon or impose the projection.

The matrix $C{\gamma }^m$ is symmetric. Its contraction with two identical odd Grassmann variables is antisymmetric under their exchange, soThis is the vector case of the four-dimensional Majorana bilinear vanishing criterion.

The matrix $C$ is antisymmetric, so the scalar Majorana spinor bilinear is not identically zero. For example, choose two independent odd Grassmann variables ${\lambda }_{\alpha },{\lambda }_{\beta }$ at a pair of indices with $C_{\alpha \beta }\ne 0$. Their contribution is $2C_{\alpha \beta }{\lambda }_{\alpha }{\lambda }_{\beta }$, which is nonzero as a Grassmann polynomial. This is why a Majorana mass term is possible.