Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration

Let $\mathcal{A}={\mathcal{A}}_{\bar{0}}\oplus {\mathcal{A}}_{\bar{1}}$ be a finite-dimensional ${\mathbb{Z}}_2$-graded complex algebra whose homogeneous elements satisfy $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$, equipped with a conjugate-linear parity-preserving involutive anti-automorphism $a\mapsto a^{\ast }$ extending complex conjugation on $\mathbb{C}$. For the ordered index set $J=\{1,2,3,4\}$, let ${\mathrm{\Lambda }}_4$ be the complex vector space with specified basis $(e_S{)}_{S\subseteq J}$, including $e_{\mathrm{\varnothing }}=1$. Define $\nu (S,T)=\mathrm{\mid }\{(s,t)\in S\times T:s>t\}\mathrm{\mid },e_S{e}_T=\{\begin{array}{ll}0,& S\cap T\mathrm{\ne }\mathrm{\varnothing },\\ (-1{)}^{\nu (S,T)}{e}_{S\cup T},& S\cap T=\mathrm{\varnothing }\mathrm{.}\end{array}$ Put ${\xi }_1={\theta }^1=e_{\{1\}}$, ${\xi }_2={\theta }^2=e_{\{2\}}$, ${\xi }_3={\bar{\theta }}^{\dot{1}}=e_{\{3\}}$, and ${\xi }_4={\bar{\theta }}^{\dot{2}}=e_{\{4\}}$. Then each ${\xi }_i$ is odd, ${\xi }_i^2=0$, and ${\xi }_i{\xi }_j=-{\xi }_j{\xi }_i$ for $i\mathrm{\ne }j$. Give ${\mathrm{\Lambda }}_4$ the grading $\mathrm{\mid }{e}_S\mathrm{\mid }=\mathrm{\mid }S\mathrm{\mid }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. The superspace coefficient vector space is $\mathcal{A}{\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}_4$ with total parity and graded-tensor multiplication $(a\otimes e_S)(b\otimes e_T)=(-1{)}^{\mathrm{\mid }S\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}(ab)\otimes (e_S{e}_T)$ for homogeneous $a,b$. On the basis, the left derivative removes $i$ with sign $(-1{)}^{\mathrm{\mid }\{s\in S:s<i\}\mathrm{\mid }}$ and the right derivative removes $i$ with sign $(-1{)}^{\mathrm{\mid }\{s\in S:s>i\}\mathrm{\mid }}$ when $i\in S$, and both give zero when $i\mathrm{\notin }S$. On a coefficient-left homogeneous tensor $a\otimes e_S$, define ${\mathrm{\partial }}_{{\xi }_i}^L(a\otimes e_S)=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a\otimes {\mathrm{\partial }}_{{\xi }_i}^L{e}_S$ and ${\mathrm{\partial }}_{{\xi }_i}^R(a\otimes e_S)=a\otimes {\mathrm{\partial }}_{{\xi }_i}^R{e}_S$; in particular both vanish on $a\otimes 1$. The algebra axioms and derivative rules below are verified in the local well-definedness result. Complex conjugation on this coefficient algebra extends the given operation on $\mathcal{A}$ by $i^{\ast }=-i$, $({\theta }^{\alpha }{)}^{\ast }={\bar{\theta }}^{\dot{\alpha }}$, $({\bar{\theta }}^{\dot{\alpha }}{)}^{\ast }={\theta }^{\alpha }$, and $(uv{)}^{\ast }=v^{\ast }{u}^{\ast }$. In the coefficient-left tensor notation this means $(a\otimes e_S{)}^{\ast }=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }S\mathrm{\mid }}{a}^{\ast }\otimes e_S^{\ast }$, where $e_S^{\ast }$ is obtained by reversing the ordered generator word for $e_S$ and applying the stated generator conjugation.

For each $i\in J$, the left derivative ${\mathrm{\partial }}_{{\xi }_i}^L$ is the unique odd derivation of the graded tensor algebra that vanishes on $\mathcal{A}$ and has ${\mathrm{\partial }}_{{\xi }_i}^L{\xi }_j={\delta }_{ij}$; on homogeneous $u,v$ it obeys ${\mathrm{\partial }}_{{\xi }_i}^L(uv)=({\mathrm{\partial }}_{{\xi }_i}^{L}u)v+(-1{)}^{\mathrm{\mid }u\mathrm{\mid }}u({\mathrm{\partial }}_{{\xi }_i}^{L}v)\mathrm{.}$ The right derivative ${\mathrm{\partial }}_{{\xi }_i}^R$ is the unique right odd derivation vanishing on $\mathcal{A}$ with ${\mathrm{\partial }}_{{\xi }_i}^R{\xi }_j={\delta }_{ij}$ and ${\mathrm{\partial }}_{{\xi }_i}^R(uv)=(-1{)}^{\mathrm{\mid }v\mathrm{\mid }}({\mathrm{\partial }}_{{\xi }_i}^{R}u)v+u({\mathrm{\partial }}_{{\xi }_i}^{R}v)\mathrm{.}$ For one generator, write a polynomial uniquely with that generator on the left as $f=f_0+{\xi }_i{f}_1$, with $f_0,f_1$ independent of ${\xi }_i$, and define its Berezin integral by the complex-linear coefficient-extraction rule $\int d{\xi }_{i}f=f_1$. Thus $\int d{\xi }_{i}1=0$ and $\int d{\xi }_i{\xi }_i=1$. Fix the ordered measures $\int d^2\theta :=\int d{\theta }^{2}d{\theta }^1,\int d^2\bar{\theta }:=\int d{\bar{\theta }}^{\dot{2}}d{\bar{\theta }}^{\dot{1}},\int d^4\theta :=\int d{\bar{\theta }}^{\dot{2}}d{\bar{\theta }}^{\dot{1}}d{\theta }^{2}d{\theta }^1\mathrm{.}$ With this order, $\int d^4\theta$ extracts the coefficient of ${\theta }^1{\theta }^2{\bar{\theta }}^{\dot{1}}{\bar{\theta }}^{\dot{2}}$. For superspace coordinates $x=(x^0,x^1,x^2,x^3)\in {\mathbb{R}}^{1,3}$ in natural units $\mathrm{\hslash }=c=1$, assign mass dimensions $[x^{\mu }]=-1$, $[{\theta }^{\alpha }]=[{\bar{\theta }}^{\dot{\alpha }}]=-\frac{1}{2}$, $[d{\theta }^{\alpha }]=[d{\bar{\theta }}^{\dot{\alpha }}]=\frac{1}{2}$, and hence $[d^2\theta ]=[d^2\bar{\theta }]=1$ and $[d^4\theta ]=2$. These are formal engineering dimensions; no physical model or empirical unit convention is asserted.

Martin, §4.1.

Gates et al., §§3.1 and 3.7.

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$: The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$.

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$: $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$.

Real and imaginary parts, complex conjugation, and modulus: Real and imaginary parts, complex conjugation, and modulus.