Graßmann number

The Graßmann numbers (after Hermann Graßmann , often also in the English language adapted spelling Grassmann) are anti- commutating numbers that occur within the framework of the path integral formalism for fermions in the quantum field theories . A pioneer of their use in quantum field theory was Felix Berezin . Then they are mathematically the part of odd parity of a -graded algebra of commuting (parity ) and non-commuting (parity ) elements ( super algebra ). For the multiplication, the following applies to two elements : ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle P=0}$ ${\displaystyle P=1}$ ${\displaystyle A,B}$

${\displaystyle A\,B=(-)^{P_{A}\cdot P_{B}}B\,A}$.

properties

Let be Graßmann numbers and complex numbers . Then applies ${\displaystyle \zeta ,\eta ,\theta }$ ${\displaystyle a,b,c,d\in \mathbb {C} }$

Definitional properties

• Graßmann numbers are anti-commutative with regard to multiplication:
  ${\displaystyle \eta \theta =-\theta \eta }$
• Graßmann numbers are commutative with respect to addition :
  ${\displaystyle \eta +\theta =\theta +\eta }$
• Graßmann numbers are commutative with respect to multiplication with a complex number:
  ${\displaystyle a\eta =\eta a}$
• Graßmann numbers are associative in terms of both addition and multiplication
  ${\displaystyle (\eta +\theta )+\zeta =\eta +(\theta +\zeta )}$
  ${\displaystyle (\eta \theta )\zeta =\eta (\theta \zeta )}$
• All versions of the distributive law apply :
  ${\displaystyle a(\eta +\theta )=a\eta +a\theta }$
  ${\displaystyle \eta (\theta +\zeta )=\eta \theta +\eta \zeta }$
  ${\displaystyle \eta (a+b)=a\eta +b\eta }$

Inferences

• The sum of two Graßmann numbers is a Graßmann number:
  ${\displaystyle \zeta (\eta +\theta )=-(\eta +\theta )\zeta }$
• The product of a Graßmann number with a complex number is a Graßmann number:
  ${\displaystyle (a\eta )\theta =-\theta (a\eta )}$
• The product of two Graßmann numbers is not a Graßmann number:
  ${\displaystyle (\zeta \eta )\theta =-\zeta \theta \eta =\theta (\zeta \eta )}$
• In particular, the square of a Graßmann number is zero:
  ${\displaystyle \theta ^{2}=\theta \theta =-\theta \theta =0}$
• A function can be a maximum of the first order in a Graßmann variable: This is, for example, with the series representation of the exponential function .
  ${\displaystyle f(\eta ,\theta )=a+b\eta +c\theta +d\eta \theta }$
  ${\displaystyle \exp(\theta )=1+\theta }$

Integration and Differentiation

It is possible to define integral and differential calculus with regard to Graßmann numbers analogously to that with regard to functions of complex numbers:

• Differentiation of Graßmann numbers happens from the left. Be . Then: ${\displaystyle f(\eta ,\theta )=a+b\eta +c\theta +d\eta \theta }$
  ${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} \eta }}=b+d\theta }$
  ${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} \theta }}=c-d\eta }$
• As usual, the integration should represent a linear functional from the function space into the complex numbers, so it should apply:
  • ${\displaystyle \int f(\theta )\mathrm {\,} d\theta \in \mathbb {C} }$
  • ${\displaystyle \int (af(\theta )+bg(\theta )\,\mathrm {)} d\theta =a\int f(\theta )\,\mathrm {d} \theta +b\int g(\theta )\,\mathrm {d} \theta }$
• The integration rules for Graßmann numbers follow from this:
  ${\displaystyle \int \theta \,\mathrm {d} \theta =1}$
  ${\displaystyle \int 1\,\mathrm {d} \theta =0}$

application

Graßmann variables are required for the path integral formalism for fermions. To do this, you define the generating functional

${\displaystyle {\mathcal {Z}}[\eta ,{\bar {\eta }}]=\exp \left(-\mathrm {i} \int \mathrm {d} ^{4}x\left({\mathcal {L}}(\psi ,{\bar {\psi }})+\eta {\bar {\psi }}+\psi {\bar {\eta }}\right)\right)}$

with the Lagrangian for fermions , the fermionic Graßmann-valued fields and the Graßmann numbers . Then, for example, the 2-point correlation function (the fermionic propagator ) applies : ${\displaystyle {\mathcal {L}}}$ ${\displaystyle \psi ,{\bar {\psi }}}$ ${\displaystyle \eta ,{\bar {\eta }}}$

${\displaystyle \langle 0|T(\psi (x){\bar {\psi }}(y))|0\rangle ={\frac {\int {\mathcal {D}}\psi {\mathcal {D}}{\bar {\psi }}\,\psi (x){\bar {\psi }}(y){\mathcal {Z}}}{\int {\mathcal {D}}\psi {\mathcal {D}}{\bar {\psi }}{\mathcal {Z}}}}{\Bigg |}_{\eta ,{\bar {\eta }}=0}={\frac {1}{\int {\mathcal {D}}\psi {\mathcal {D}}{\bar {\psi }}{\mathcal {Z}}|_{\eta ,{\bar {\eta }}=0}}}\left({\frac {-\mathrm {i} \delta }{\delta {\bar {\eta }}(x)}}\right)\left({\frac {\mathrm {i} \delta }{\delta \eta (y)}}\right){\int {\mathcal {D}}\psi {\mathcal {D}}{\bar {\psi }}{\mathcal {Z}}}{\Bigg |}_{\eta ,{\bar {\eta }}=0}}$

Formal mathematical definition

Let be a -dimensional complex vector space with basis and ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle \theta _{i},i=1,\ldots ,n}$

${\displaystyle \Lambda (V)=\mathbb {C} \oplus V\oplus \left(V\wedge V\right)\oplus \left(V\wedge V\wedge V\right)\oplus \cdots \oplus \underbrace {\left(V\wedge V\wedge \cdots \wedge V\right)} _{n}\equiv \mathbb {C} \oplus \Lambda ^{1}V\oplus \Lambda ^{2}V\oplus \cdots \oplus \Lambda ^{n}V}$

the outer algebra (Graßmann algebra) over , where the outer product and the direct sum denotes. ${\displaystyle V}$ ${\displaystyle \wedge }$ ${\displaystyle \oplus }$

The Graßmann numbers are the elements of this algebra.

The symbol is usually left out in the notation for Graßmann numbers. ${\displaystyle \wedge }$

So Graßmann numbers are of the form

${\displaystyle z=\sum _{k=0}^{n}\sum _{i_{1},i_{2},\cdots ,i_{k}}c_{i_{1}i_{2}\cdots i_{k}}\theta _{i_{1}}\theta _{i_{2}}\cdots \theta _{i_{k}},}$

for strictly growing tuples with , and complex antisymmetric tensors of rank . ${\displaystyle k}$ ${\displaystyle (i_{1},i_{2},\ldots ,i_{k})}$ ${\displaystyle 1\leq i_{j}\leq n,1\leq j\leq k}$ ${\displaystyle c_{i_{1}i_{2}\cdots i_{k}}}$ ${\displaystyle k}$

The special case corresponds to the dual numbers introduced by William Clifford in 1873 . ${\displaystyle n=1}$

For infinite-dimensional vector spaces the series breaks ${\displaystyle V}$

${\displaystyle \Lambda _{\infty }(V)=\mathbb {C} \oplus \Lambda ^{1}V\oplus \Lambda ^{2}V\oplus \cdots }$

not off and the Graßmann numbers are of the form

${\displaystyle z=\sum _{k=0}^{\infty }\sum _{i_{1},i_{2},\cdots ,i_{k}}{\frac {1}{n!}}c_{i_{1}i_{2}\cdots i_{k}}\theta _{i_{1}}\theta _{i_{2}}\cdots \theta _{i_{k}}\equiv z_{B}+z_{S}=z_{B}+\sum _{k=1}^{\infty }\sum _{i_{1},i_{2},\cdots ,i_{k}}{\frac {1}{n!}}c_{i_{1}i_{2}\cdots i_{k}}\theta _{i_{1}}\theta _{i_{2}}\cdots \theta _{i_{k}},}$

where the super number is then referred to as the body and the soul . ${\displaystyle z_{B}}$ ${\displaystyle z_{S}}$ ${\displaystyle z}$

literature

• Michael D. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory, Perseus Books Publishing 1995, ISBN 0-201-50397-2 .

Web links

• Varilly: Graßmann Numbers in Quantum Mechanics