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 :
Z
2
{\ displaystyle \ mathbb {Z} _ {2}}
P
=
0
{\ displaystyle P = 0}
P
=
1
{\ displaystyle P = 1}
A.
,
B.
{\ displaystyle A, B}
A.
B.
=
(
-
)
P
A.
⋅
P
B.
B.
A.
{\ displaystyle A \, B = (-) ^ {P_ {A} \ cdot P_ {B}} B \, A}
.
properties
Let be Graßmann numbers and complex numbers . Then applies
ζ
,
η
,
θ
{\ displaystyle \ zeta, \ eta, \ theta}
a
,
b
,
c
,
d
∈
C.
{\ displaystyle a, b, c, d \ in \ mathbb {C}}
Definitional properties
Graßmann numbers are anti-commutative with regard to multiplication:
η
θ
=
-
θ
η
{\ displaystyle \ eta \ theta = - \ theta \ eta}
Graßmann numbers are commutative with respect to addition :
η
+
θ
=
θ
+
η
{\ displaystyle \ eta + \ theta = \ theta + \ eta}
Graßmann numbers are commutative with respect to multiplication with a complex number:
a
η
=
η
a
{\ displaystyle a \ eta = \ eta a}
Graßmann numbers are associative in terms of both addition and multiplication
(
η
+
θ
)
+
ζ
=
η
+
(
θ
+
ζ
)
{\ displaystyle (\ eta + \ theta) + \ zeta = \ eta + (\ theta + \ zeta)}
(
η
θ
)
ζ
=
η
(
θ
ζ
)
{\ displaystyle (\ eta \ theta) \ zeta = \ eta (\ theta \ zeta)}
All versions of the distributive law apply :
a
(
η
+
θ
)
=
a
η
+
a
θ
{\ displaystyle a (\ eta + \ theta) = a \ eta + a \ theta}
η
(
θ
+
ζ
)
=
η
θ
+
η
ζ
{\ displaystyle \ eta (\ theta + \ zeta) = \ eta \ theta + \ eta \ zeta}
η
(
a
+
b
)
=
a
η
+
b
η
{\ displaystyle \ eta (a + b) = a \ eta + b \ eta}
Inferences
The sum of two Graßmann numbers is a Graßmann number:
ζ
(
η
+
θ
)
=
-
(
η
+
θ
)
ζ
{\ displaystyle \ zeta (\ eta + \ theta) = - (\ eta + \ theta) \ zeta}
The product of a Graßmann number with a complex number is a Graßmann number:
(
a
η
)
θ
=
-
θ
(
a
η
)
{\ displaystyle (a \ eta) \ theta = - \ theta (a \ eta)}
The product of two Graßmann numbers is not a Graßmann number:
(
ζ
η
)
θ
=
-
ζ
θ
η
=
θ
(
ζ
η
)
{\ displaystyle (\ zeta \ eta) \ theta = - \ zeta \ theta \ eta = \ theta (\ zeta \ eta)}
In particular, the square of a Graßmann number is zero:
θ
2
=
θ
θ
=
-
θ
θ
=
0
{\ 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 .
f
(
η
,
θ
)
=
a
+
b
η
+
c
θ
+
d
η
θ
{\ displaystyle f (\ eta, \ theta) = a + b \ eta + c \ theta + d \ eta \ theta}
exp
(
θ
)
=
1
+
θ
{\ 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:
f
(
η
,
θ
)
=
a
+
b
η
+
c
θ
+
d
η
θ
{\ displaystyle f (\ eta, \ theta) = a + b \ eta + c \ theta + d \ eta \ theta}
d
f
d
η
=
b
+
d
θ
{\ displaystyle {\ frac {\ mathrm {d} f} {\ mathrm {d} \ eta}} = b + d \ theta}
d
f
d
θ
=
c
-
d
η
{\ displaystyle {\ frac {\ mathrm {d} f} {\ mathrm {d} \ theta}} = cd \ eta}
As usual, the integration should represent a linear functional from the function space into the complex numbers, so it should apply:
∫
f
(
θ
)
d
θ
∈
C.
{\ displaystyle \ int f (\ theta) \ mathrm {\,} d \ theta \ in \ mathbb {C}}
∫
(
a
f
(
θ
)
+
b
G
(
θ
)
)
d
θ
=
a
∫
f
(
θ
)
d
θ
+
b
∫
G
(
θ
)
d
θ
{\ 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:
∫
θ
d
θ
=
1
{\ displaystyle \ int \ theta \, \ mathrm {d} \ theta = 1}
∫
1
d
θ
=
0
{\ 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
Z
[
η
,
η
¯
]
=
exp
(
-
i
∫
d
4th
x
(
L.
(
ψ
,
ψ
¯
)
+
η
ψ
¯
+
ψ
η
¯
)
)
{\ 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 :
L.
{\ displaystyle {\ mathcal {L}}}
ψ
,
ψ
¯
{\ displaystyle \ psi, {\ bar {\ psi}}}
η
,
η
¯
{\ displaystyle \ eta, {\ bar {\ eta}}}
⟨
0
|
T
(
ψ
(
x
)
ψ
¯
(
y
)
)
|
0
⟩
=
∫
D.
ψ
D.
ψ
¯
ψ
(
x
)
ψ
¯
(
y
)
Z
∫
D.
ψ
D.
ψ
¯
Z
|
η
,
η
¯
=
0
=
1
∫
D.
ψ
D.
ψ
¯
Z
|
η
,
η
¯
=
0
(
-
i
δ
δ
η
¯
(
x
)
)
(
i
δ
δ
η
(
y
)
)
∫
D.
ψ
D.
ψ
¯
Z
|
η
,
η
¯
=
0
{\ 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
V
{\ displaystyle V}
n
{\ displaystyle n}
θ
i
,
i
=
1
,
...
,
n
{\ displaystyle \ theta _ {i}, i = 1, \ ldots, n}
Λ
(
V
)
=
C.
⊕
V
⊕
(
V
∧
V
)
⊕
(
V
∧
V
∧
V
)
⊕
⋯
⊕
(
V
∧
V
∧
⋯
∧
V
)
⏟
n
≡
C.
⊕
Λ
1
V
⊕
Λ
2
V
⊕
⋯
⊕
Λ
n
V
{\ 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.
V
{\ 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
z
=
∑
k
=
0
n
∑
i
1
,
i
2
,
⋯
,
i
k
c
i
1
i
2
⋯
i
k
θ
i
1
θ
i
2
⋯
θ
i
k
,
{\ 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 .
k
{\ displaystyle k}
(
i
1
,
i
2
,
...
,
i
k
)
{\ displaystyle (i_ {1}, i_ {2}, \ ldots, i_ {k})}
1
≤
i
j
≤
n
,
1
≤
j
≤
k
{\ displaystyle 1 \ leq i_ {j} \ leq n, 1 \ leq j \ leq k}
c
i
1
i
2
⋯
i
k
{\ displaystyle c_ {i_ {1} i_ {2} \ cdots i_ {k}}}
k
{\ displaystyle k}
The special case corresponds to the dual numbers introduced by William Clifford in 1873 .
n
=
1
{\ displaystyle n = 1}
For infinite-dimensional vector spaces the series breaks
V
{\ displaystyle V}
Λ
∞
(
V
)
=
C.
⊕
Λ
1
V
⊕
Λ
2
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
z
=
∑
k
=
0
∞
∑
i
1
,
i
2
,
⋯
,
i
k
1
n
!
c
i
1
i
2
⋯
i
k
θ
i
1
θ
i
2
⋯
θ
i
k
≡
z
B.
+
z
S.
=
z
B.
+
∑
k
=
1
∞
∑
i
1
,
i
2
,
⋯
,
i
k
1
n
!
c
i
1
i
2
⋯
i
k
θ
i
1
θ
i
2
⋯
θ
i
k
,
{\ 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 .
z
B.
{\ displaystyle z_ {B}}
z
S.
{\ displaystyle z_ {S}}
z
{\ 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
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">