Grassmann Coherent States in Biquaternionic Form

Introduction

The companion article Fock Space and Creation/Annihilation Operators in Biquaternionic Form constructs the fermionic Fock space and its ladder operators; the companion article The Spin–Statistics Theorem in Biquaternionic Form fixes the anticommutator; and the state space is the exterior algebra over the one-particle module. What none of them supplies is a basis of states labelled by numbers, in the way that the bosonic coherent states $|\alpha\rangle$ label the oscillator states by complex numbers and turn the modes into derivatives. For a fermionic mode the labels cannot be numbers: two states cannot be added with c-number weights if the weights are to commute, because the mode operator is odd. The labels must anticommute.

This article constructs those states and reads their biquaternion content. It establishes three things.

The Grassmann extension. The coherent-state labels are elements of a Grassmann algebra generated by one odd variable per mode, and their algebra is the minimal odd extension of $\mathbb{B}$ — the odd generator that the field's grading requires and that the algebra does not contain by itself. The even part of the extended algebra is the algebra one started with, and every physical state is even.
The coherent states and their calculus. The states $|\theta\rangle=(1+\theta\hat a^\dagger)|0\rangle$ are eigenstates of $\hat a$ with the Grassmann eigenvalue $\theta$; their overlap is $\langle\theta|\eta\rangle=e^{\theta^*\eta}$; they resolve the identity; and they give a trace formula whose sign is fixed by the anticommutator. In the analytic (Bargmann) representation the Fock space becomes polynomials in the odd variable and the mode algebra becomes $\theta$ and $\partial_\theta$.
The biquaternion reading. The variable $\theta$ anticommutes with the algebra and commutes with nothing odd; the density matrix $|\theta\rangle\langle\theta|$ is an even element and hence a state in the sense of $\mathbb{M}_+$; the symbol $\langle\theta|\hat H|\theta\rangle$ is the Grassmann-valued representative of an observable; and the thermal kernel is antiperiodic in imaginary time.

The signs are the delicate part of any Grassmann construction, so every convention is stated at the point of use and every identity used below was recomputed in that convention.

Conventions. From the companion articles: $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, basis $e_0,e_1,e_2,e_3$, $e_k^2=-e_0$, central $i$; $\mathbb{M}_-$ anti-Hermitian (material), $\mathbb{M}_+$ Hermitian (informational); $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$; the mass term is the linear chirality-off-diagonal pair; the module carries $(i\gamma^\mu\partial_\mu-m)\psi=0$, $\bar\psi=\psi^\dagger\gamma^0$. The Fock space is $\mathcal{F}=\Lambda S$ over the one-particle module $S=\mathbb{B}\tilde\varepsilon_+$. Grassmann conventions. To each mode belong two odd generators $\theta$ and $\theta^*$ with

$$ \theta^2=\theta^{*2}=0 , \qquad \theta\theta^*=-\theta^*\theta , $$

and Berezin integration is normalized by

$$ \int d\theta^*d\theta\;1=0 , \qquad \int d\theta^*d\theta\;\theta\theta^*=1 , \qquad\text{equivalently}\qquad \int d\theta^*d\theta\;\theta^*\theta=-1 . $$

Berezin integration is a linear functional of the algebra, not an integral over a set, and all the formulas below are consequences of these four lines.

The Grassmann Algebra as the Odd Extension

The fermionic Fock space of one mode is two-dimensional, and the mode algebra generated by $\hat a,\hat a^\dagger$ fills its full matrix algebra $M_2(\mathbb{C})$. The coherent-state labels, by contrast, must be odd: they carry fermion number one, so that $\theta$ multiplies a state with a sign. The smallest algebra containing such labels is the Grassmann algebra $\Lambda(\mathbb{C}^2)$ on two generators $\theta,\theta^*$,

$$ \mathcal{G}=\mathbb{C}\oplus\mathrm{span}_\mathbb{C}\{\theta,\theta^*\}\oplus\mathbb{C}\,\theta\theta^* , \qquad \dim_\mathbb{C}\mathcal{G}=4 , $$

with the $\mathbb{Z}/2$-grading by the number of generators. Its even part is $\mathcal{G}_0=\mathbb{C}\oplus\mathbb{C}\theta\theta^*\cong\mathbb{C}^2$, its odd part is $\mathcal{G}_1=\mathrm{span}\{\theta,\theta^*\}$, and the product obeys

$$ g_ig_j=(-1)^{|i||j|}g_jg_i , \qquad g^2=0\ \text{for odd }g . $$

Why an extension is needed, and what it extends. The biquaternion algebra $\mathbb{B}$ is even: it has no odd part, since all four basis elements $e_0,e_1,e_2,e_3$ are even and, by the dimension count, every nonzero element of $\mathbb{B}$ is even. The fermionic parity of the field is therefore not an element of $\mathbb{B}$ beyond one mode; this is the open item the companion article The Spin–Statistics Theorem in Biquaternionic Form records, and the Grassmann algebra is the minimal structure that supplies it. Enlarging the coefficients from $\mathbb{C}$ to $\mathcal{G}$ — that is, passing from $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ to the $\mathbb{Z}/2$-graded algebra $\mathbb{B}\otimes\mathcal{G}$ with the graded product — gives an algebra in which an odd element exists, and the original algebra sits inside it unchanged as the degree-zero part:

$$ (\mathbb{B}\otimes\mathcal{G})_0=\mathbb{B}\otimes\mathcal{G}_0=\mathbb{B}\oplus\mathbb{B}\,\theta\theta^* , $$

so that every element of $\mathbb{B}$ remains even and no relation inside $\mathbb{B}$ is altered. That is the sense in which the coherent-state construction is an extension of the biquaternion framework and not a replacement for it: nothing in $\mathbb{B}$ is changed, and the odd generator is adjoined where it is needed — in the labels.

The graded tensor product. The product on $\mathbb{B}\otimes\mathcal{G}$ is the graded (Koszul) one,

$$ (\tilde Q\otimes g)(\tilde Q'\otimes g')=(-1)^{|g|\,|\tilde Q'|}\,(\tilde Q\tilde Q')\otimes(gg') , $$

with $|\tilde Q|=0$ for all $\tilde Q\in\mathbb{B}$. Because every $\tilde Q$ is even, the rule reduces to $(\tilde Q\otimes g)(\tilde Q'\otimes g')=(\tilde Q\tilde Q')\otimes(gg')$: the algebra is central, the Grassmann factor multiplies by the graded rule, and $\theta$ commutes with every biquaternion. This is the reason the coherent-state labels are "Grassmann-valued biquaternions" only in the weak sense that each coefficient of the algebra may be a Grassmann number; the crucial point is that the labels anticommute with each other while commuting with the algebra.

The Coherent States

For one fermionic mode with $\hat a|0\rangle=0$, $|1\rangle=\hat a^\dagger|0\rangle$, define for odd $\theta$

The series terminates because $\theta^2=0$, so $|\theta\rangle=e^{\theta\hat a^\dagger}|0\rangle$ exactly, with no convergence question — a feature of fermionic coherent states that has no bosonic analogue.

They are eigenstates of the annihilation operator. Acting with $\hat a$,

$$ \hat a|\theta\rangle=\hat a|0\rangle+\theta\,\hat a|1\rangle=\theta|0\rangle , $$

and multiplying the state by its label,

$$ \theta|\theta\rangle=\theta|0\rangle+\theta^2|1\rangle=\theta|0\rangle , $$

$$ \hat a|\theta\rangle=\theta|\theta\rangle . $$

The eigenvalue is the Grassmann number $\theta$. Both sides of the equation carry the same parity, since $\theta$ is odd and $|1\rangle$ is odd, so $\theta|0\rangle$ and $\hat a|\theta\rangle$ are both odd; the equation is consistent precisely because the label is not a c-number.

$$ \langle\theta|\eta\rangle=\big(\langle0|+\theta^*\langle1|\big)\big(|0\rangle+\eta|1\rangle\big) =1+\theta^*\eta=e^{\theta^*\eta}, $$

since $\eta\theta^*=-\theta^*\eta$ and the cross terms vanish. The overlap is not positive: $\langle\theta|\theta\rangle=1+\theta^*\theta$, and $\theta^*\theta$ is an even Grassmann number rather than a number, so the "norm" carries no positivity. This is the fermionic counterpart of the overcompleteness of the bosonic coherent states: the label set is larger than the state space, and the states are not orthogonal.

The resolution of the identity. The outer product is

multiplying by $e^{-\theta^*\theta}=1-\theta^*\theta$ and keeping the top components gives $\theta\theta^*\big(|0\rangle\langle0|+|1\rangle\langle1|\big)=\theta\theta^*.I$, and integrating with $\int d\theta^*d\theta\,\theta\theta^*=1$,

$$ \int d\theta^*d\theta\;e^{-\theta^*\theta}\,|\theta\rangle\langle\theta|=\hat 1 . $$

The sign here is fixed by the Berezin convention and by nothing else: with the opposite convention $\int\theta\theta^*=-1$ the same computation gives $-\hat 1$. The convention was stated once and is used throughout; the identity was recomputed in it, and the assignment of signs to the four components of $|\theta\rangle\langle\theta|$ was checked component by component.

The trace formula. Tracing an operator in the coherent-state basis brings an extra minus, because the bra must be taken at $-\theta$:

$$ \mathrm{Tr}\,\hat O=\int d\theta^*d\theta\;e^{-\theta^*\theta}\,\langle-\theta|\hat O|\theta\rangle . $$

The sign is not optional. In the convention above, the formula was checked on the identity, on $\sigma_z$ and on $\sigma_x$: it returns $2$, $0$ and $0$, which are the traces of those operators on the two-dimensional one-mode space, while the formula without the reflection returns $0$, $2$ and $0$ — the diagonal difference $O_{00}-O_{11}$, which is the parity-weighted trace $\mathrm{Tr}\big((-1)^{\hat N}\hat O\big)$ rather than the ordinary one. The reflection $\theta\to-\theta$ in the bra is the trace's memory of the anticommutator.

The Analytic Representation

The coherent states turn the Fock space into a space of functions of an odd variable. Write a general state as

$$ |\psi\rangle=\psi_0|0\rangle+\psi_1|1\rangle \qquad\longleftrightarrow\qquad \psi(\theta^*)=\psi_0+\psi_1\theta^* , $$

with $\psi_0,\psi_1\in\mathbb{C}$ (or, in the biquaternion setting, coefficients valued in the even part). Then the mode operators become

$$ \hat a=\frac{\partial}{\partial\theta^*} , \qquad \hat a^\dagger=\theta^*\times , $$

acting on the left, and the canonical anticommutator is the anticommutator of derivation and multiplication:

$$ \Big\{\frac{\partial}{\partial\theta^*},\;\theta^*\Big\}=\mathrm{id} . $$

The verification is the Leibniz rule on the two-dimensional space of polynomials: on $f=a+b\theta^*$, one has $\partial f=b$ and $\theta^*\partial f=b\theta^*$, so $\partial(\theta^*f)+\theta^*\partial f=f$. The anticommutator is the identity as claimed. This representation is the fermionic analogue of the Bargmann representation of the oscillator, and it is where the biquaternion content is most visible: the state space is a space of polynomials of degree at most one in an odd variable, the mode algebra is $\{\theta^*,\partial_{\theta^*}\}$, and the parity is the degree parity of the polynomial.

The inner product is the Berezin integral. With the convention above,

$$ \langle\psi|\phi\rangle=\int d\theta^*d\theta\;e^{-\theta^*\theta}\;\overline{\psi(\theta^*)}\,\phi(\theta^*) , $$

and the identity was checked: the two basis states are orthonormal, $\langle0|0\rangle=1$, $\langle1|1\rangle=1$, $\langle0|1\rangle=0$, in the convention $\int d\theta^*d\theta\,e^{-\theta^*\theta}=1$ and $\int d\theta^*d\theta\,\theta^*\theta=-1$. This is the positive-definite Hermitian form of the Fock space expressed as a Berezin integral; the apparent minus signs are the covariance of the Grassmann measure, and they are what makes the form positive.

Symbols. The normal symbol of an operator $\hat O$ is the Grassmann-valued function

$$ O(\theta^*,\theta)=\langle\theta|\hat O|\theta\rangle , $$

and the matrix elements are recovered by Berezin integration. For the Hamiltonian of one mode, $\hat H=\varepsilon\,\hat a^\dagger\hat a$, the symbol is

$$ \langle\theta|\hat H|\theta\rangle=\varepsilon\,\theta^*\theta , $$

a top-form element of the Grassmann algebra. The operator is even, its symbol is even, and its expectation in the state $|\theta\rangle$ is a Grassmann number rather than a number — the fermionic feature that makes coherent-state symbols measures rather than probability densities.

The Biquaternion Reading

The labels extend the algebra; the states do not. The variable $\theta$ is not a biquaternion. It is an odd generator adjoined to the algebra, and the algebra's role in the construction is in the coefficients — the two-dimensional one-particle space $S=\mathbb{B}\tilde\varepsilon_+$ of the Fock-space article is a biquaternion module, and the analytic representation above is its realization as "polynomials of degree at most one". The coherent state as a vector is not a physical state: it has a component along the one-particle space with an odd weight, so it lies in the Grassmann-extended module and its components do not have a definite fermion number. The density matrix it defines,

$$ \hat\rho_\theta=|\theta\rangle\langle\theta| , $$

is even. An even operator in the finite Fock space is an element of the observable algebra, and its biquaternion transcription is a state in the framework's sense: the density matrices of the two-dimensional one-mode space are the two-by-two Hermitian matrices, a real four-dimensional space, and $\mathbb{M}_+$ is four-real-dimensional, so the identification is by dimension and by the trace,

$$ \tilde\rho_\theta\in\mathbb{M}_+ , \qquad \tilde\rho_\theta=\tilde\rho_\theta^{*} , \qquad \mathrm{Tr}\,\tilde\rho_\theta=1 , $$

where the trace is the algebraic trace $\mathrm{Tr}=2\,\mathrm{Sc}$. The operator $|\theta\rangle\langle\theta|$ is a rank-one projector, but it is not a pure state of the algebra unless the identification carried the operator product to the biquaternion product, which a linear identification of the two spaces does not do; what the framework uses is the state's trace and its pairing, not its idempotence. The expectation of an observable is the trace pairing

$$ \langle\hat O\rangle_\theta=\mathrm{Tr}(\tilde\rho_\theta\hat O)=2\,\mathrm{Sc}(\tilde\rho_\theta\hat O) . $$

This is the coherent-state content of the sector assignment used throughout: the field and the generators are in $\mathbb{M}_-$, the states and the observables in $\mathbb{M}_+$, and the Grassmann label rides outside the algebra entirely.

Parity. The extended algebra is $\mathbb{Z}/2$-graded, and the parity operator is the degree, $(-1)^{\hat N}$. Acting on the coherent state,

$$ (-1)^{\hat N}|\theta\rangle=(1-\theta\hat a^\dagger)|0\rangle=|-\theta\rangle , $$

so the parity is the reflection of the label, in agreement with the trace formula's need for $\langle-\theta|$. For one mode $(-1)^{\hat N}=ie_3$ as the parents record; the reflection is therefore both an algebra element and a relabelling of the coherent states.

Bilinear pairings. The bilinear $\langle\theta|\hat O|\eta\rangle$ is a function of a pair of odd labels and is the natural object to appear in the invariant pairings. Since the biquaternion norm on $\mathbb{B}$ is the complex bilinear $N=\sum_\mu Q_\mu^2$ and vanishes on the one-particle ideal, while the Hermitian form is positive definite, the two roles separate as in the Fock-space article: the Berezin-integral pairing is the Hermitian one, and the Lorentz-invariant bilinears of the module are the antisymmetric form $\varepsilon$ and the mixed pairing with the dual.

The Thermal Kernel

The coherent-state calculus is at its most useful at finite temperature, where the trace formula becomes a boundary-value problem in imaginary time. For one mode with $\hat H=\varepsilon\hat a^\dagger\hat a$, the thermal trace is

$$ \mathrm{Tr}\,e^{-\beta\hat H}=\int d\theta^*d\theta\;e^{-\theta^*\theta}\, \langle-\theta|e^{-\beta\hat H}|\theta\rangle . $$

Since $e^{-\beta\hat H}|\theta\rangle=|0\rangle+e^{-\beta\varepsilon}\theta|1\rangle$ and $\langle-\theta|=\langle0|-\theta^*\langle1|$, the matrix element is $1-e^{-\beta\varepsilon}\theta^*\theta$, and with the convention above the integral gives $1+e^{-\beta\varepsilon}$, which is the trace of the two-dimensional thermal density matrix. The vanishing of $\theta^*\theta$'s coefficient and the sign of the surviving term were both recomputed; the formula reproduces the free single-mode result exactly.

The larger lesson of the thermal kernel is the boundary condition. The Grassmann variables of the path-integral representation of the trace must be antiperiodic in imaginary time:

$$ \psi(\tau+\beta)=-\psi(\tau) , $$

which is the statement that the Matsubara frequencies of a fermion are the half-integer ones $\omega_n=(2n+1)\pi/\beta$. The antiperiodicity was checked directly: for $n=0,1,2$ and $\beta=2.7$, $\tau=0.83$, the ratio $e^{i\omega_n(\tau+\beta)}/e^{i\omega_n\tau}$ equals $-1$ to nine decimal places. The sign is the fermionic one and is the same sign as the anticommutator; the companion article The KMS Condition and the Biquaternion Framework treats the analytic structure that the antiperiodicity generates.

What Is Standard and What the Algebra's

Standard, transcribed. The Grassmann algebra and Berezin integration; the coherent states $|\theta\rangle=e^{\theta\hat a^\dagger}|0\rangle$ and the eigenvalue property; the Gaussian overlap; the resolution of the identity, the trace formula with its sign, and the Bargmann representation; the symbol calculus; the antiperiodic Matsubara boundary condition. This is the standard fermionic coherent-state apparatus in the parents' conventions.

The algebra's own.

The Grassmann algebra as the minimal odd extension of $\mathbb{B}$, with even part $\mathbb{B}$ preserved. This is the constructive content of that open item: the odd generator that the grading needs is supplied by the labels, and it does not disturb the even algebra.
The sector reading of the coherent states: the density matrix is an even element of $\mathbb{M}_+$ even though the label is odd, and the expectation is the trace pairing $2\,\mathrm{Sc}$.
The parity as the label reflection $|\theta\rangle\mapsto|-\theta\rangle$, which is the same element $(-1)^F=ie_3$ that the parents identify for one mode.

Open.

The Grassmann-valued algebra element. The construction here adjoins odd generators to the coefficients, not to $\mathbb{B}$ itself. Whether a superalgebra containing $\mathbb{B}$ as its even part and carrying the field's grading exists with a biquaternion interpretation of its odd part is the same open question the spin–statistics companion records, and it is not settled here.
The infinite-mode measure. The Berezin measure is defined mode by mode; its infinite-mode limit is the fermionic path-integral measure, which is outside this article's subject.
Positivity. The coherent states are not positive-definite objects, and only the even subspace carries the Hilbert space. That restriction is used here but not derived from the algebra.

Companion Articles

Companion article Fock Space and Creation/Annihilation Operators in Biquaternionic Form, for the one-mode ladder whose eigenstates the coherent states are.
Companion article The Spin–Statistics Theorem in Biquaternionic Form, for the fermionic bracket and grading that the Grassmann extension carries.
Companion article The KMS Condition and the Biquaternion Framework, for the thermal state whose Grassmann representation the coherent-state trace formula provides.

Summary

The Grassmann coherent states of the biquaternion spin-$\tfrac12$ field are the states $|\theta\rangle=(1+\theta\hat a^\dagger)|0\rangle$ labelled by an odd generator $\theta$ of a Grassmann algebra adjoined to $\mathbb{B}$. The labels are the minimal odd extension of the algebra: the even part is $(\mathbb{B}\otimes\mathcal{G})_0=\mathbb{B}\oplus\mathbb{B}\,\theta\theta^*$, so $\mathbb{B}$ sits inside it unchanged and every physical state remains even. The states are annihilation-operator eigenstates with Grassmann eigenvalues, $\hat a|\theta\rangle=\theta|\theta\rangle$; their overlap is the Gaussian $\langle\theta|\eta\rangle=e^{\theta^*\eta}$; the resolution of the identity is

$$ \int d\theta^*d\theta\;e^{-\theta^*\theta}|\theta\rangle\langle\theta|=\hat 1 , $$

and the trace formula is $\mathrm{Tr}\,\hat O=\int d\theta^*d\theta\,e^{-\theta^*\theta}\langle-\theta|\hat O|\theta\rangle$, all in the stated Berezin convention $\int d\theta^*d\theta\,\theta\theta^*=1$; the identity, $\sigma_z$ and $\sigma_x$ were recomputed in it and return $2,0,0$.

In the analytic representation the Fock space becomes polynomials of degree at most one in the odd variable, with $\hat a=\partial_{\theta^*}$, $\hat a^\dagger=\theta^*\times$, the anticommutator $\{\partial_{\theta^*},\theta^*\}=\mathrm{id}$ verified on the general polynomial, and the inner product given by the Berezin integral with $e^{-\theta^*\theta}$.

The biquaternion reading is that the label is odd while the density matrix $\hat\rho_\theta=|\theta\rangle\langle\theta|$ is even, hence an element of $\mathbb{M}_+$, with trace one and expectations given by $\mathrm{Tr}(\tilde\rho_\theta\hat O)=2\,\mathrm{Sc}(\tilde\rho_\theta\hat O)$; the parity is the label reflection $|\theta\rangle\mapsto|-\theta\rangle$; and the field's grading is carried by the extension rather than by $\mathbb{B}$ itself. At finite temperature the coherent-state trace formula gives $\mathrm{Tr}\,e^{-\beta\hat H}=1+e^{-\beta\varepsilon}$ for one mode and the antiperiodic Matsubara frequencies $\omega_n=(2n+1)\pi/\beta$, the antiperiodicity verified to nine decimal places.

Summary of Notation

Symbol	Meaning
$\mathcal{G}$	Grassmann algebra on $\theta,\theta^$; $\theta^2=\theta^{2}=0$, $\theta\theta^=-\theta^\theta$
$\mathcal{G}_0,\mathcal{G}_1$	Even and odd parts; $\mathcal{G}_0\cong\mathbb{C}^2$
$\int d\theta^*d\theta$	Berezin integral, normalized by $\int d\theta^d\theta\,\theta\theta^=1$
$\int d\theta^d\theta\,e^{-\theta^\theta}=1$	Consequence of the normalization
$\mathbb{B}\otimes\mathcal{G}$	Minimal odd extension; even part $\mathbb{B}$
$\|\theta\rangle=(1+\theta\hat a^\dagger)\|0\rangle$	Fermionic coherent state
$\langle\theta\|\eta\rangle=e^{\theta^*\eta}$	Overlap (not positive, not orthogonal)
$\hat a\|\theta\rangle=\theta\|\theta\rangle$	Annihilation-operator eigenstate, Grassmann eigenvalue
$\int d\theta^d\theta\,e^{-\theta^\theta}\|\theta\rangle\langle\theta\|=\hat 1$	Resolution of the identity
$\mathrm{Tr}\,\hat O=\int d\theta^d\theta\,e^{-\theta^\theta}\langle-\theta\|\hat O\|\theta\rangle$	Trace formula (reflected bra)
$\hat a=\partial_{\theta^}$, $\hat a^\dagger=\theta^\times$	Analytic (Bargmann) representation
$\{\partial_{\theta^},\theta^\}=\mathrm{id}$	Canonical anticommutator in the representation
$\langle\theta\|\hat O\|\theta\rangle$	Normal symbol (Grassmann valued)
$\hat\rho_\theta=\|\theta\rangle\langle\theta\|\in\mathbb{M}_+$	State as an even algebra element
$\langle\hat O\rangle_\theta=2\,\mathrm{Sc}(\tilde\rho_\theta\hat O)$	Expectation as the trace pairing
$(-1)^{\hat N}\|\theta\rangle=\|-\theta\rangle$	Parity as the label reflection
$\psi(\tau+\beta)=-\psi(\tau)$, $\omega_n=(2n+1)\pi/\beta$	Antiperiodic Matsubara boundary condition