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Units, Constants and Useful Formulas
Grassmann Numbers
-----------------
Grassmann numbers can always be represented by matrices.
Grassmann numbers are defined as ODD elements. Odd elements
are always ANTICOMMUTED:
{θ_{i},θ_{j}} = θ_{i}θ_{j} + θ_{j}θ_{i} = 0 ∴ θ_{i}θ_{j} = -θ_{j}θ_{i}
Ordinary numbers are defined as EVEN elements. Even elements
are always COMMUTED:
[n,m] = nm - mn = 0
A mix of Grassmann (odd elements) and ordinary numbers (even
elements) are always COMMUTED:
[θ,n] = θn - nθ = 0
- The product of 2 even numbers is an even number.
- The product of 2 odd numbers is an even number.
- The product of an even number and an odd number is an odd
number.
- The product of an odd number and an even number is an odd
number.
Functions
----------
Functions are expanded in terms of power series. Thus,
f(θ_{1}) = A + Bθ_{1}
and
f(θ_{1},θ_{2}) = A + B_{1}θ_{1} + B_{2}θ_{2} + Cθ_{2}θ_{1}
Where A and B can be Grassmann or ordinary numbers.
The number of terms can never go beyond those shown because
{θ_{i},θ_{i}} = 0
Functions can also either ODD or EVEN. In an odd function, all
terms are odd. In an even function all terms are even.
Thus, if f(θ_{1}) is odd, A must be odd and B must be even since
θ is odd. If f(θ_{1}) is even, A must be even and B must be odd.
When a Grassmann number passes another Grassmann number to
its left, it changes its sign. Thus, if f(θ_{1}) is even
f(θ_{1}) = A + Bθ_{1}
_{ } ≡ A - θ_{1}B
If f(θ_{1}) is odd:
f(θ_{1}) = A + Bθ_{1}
Differentiation
---------------
The general rule is to move the dependence adjacent to the partial
derivative while taking the above rules into account. Consider
the following examples:
Single variable:
f(θ_{1}) = A + Bθ_{1}
if f is odd => A must be odd and B must be even.
∂f/∂θ_{1} = ∂A/∂θ_{1} + (∂θ_{1}/∂θ_{1})B
_{ } = B
if f is even => A must be even and B must be odd.
∂f/∂θ_{1} = ∂A/∂θ_{1} - (∂θ_{1}/∂θ_{1})B
_{ } = -B
Two variables:
f(θ_{1},θ_{2}) = A + B_{1}θ_{1} + B_{2}θ_{2} + Cθ_{2}θ_{1}
if f is odd => A and C must be odd, B_{1} and B_{2} must be even.
∂f/∂θ_{1} = ∂A/∂θ_{1} + (∂θ_{1}/∂θ_{1})B_{1} + (∂θ_{2}/∂θ_{1})B_{2} + (∂θ_{1}/∂θ_{1})θ_{2}C
_{ } = 0 + B_{1} + 0 + Cθ_{2}
if f is even => A and C must be even, B_{1} and B_{2} must be odd.
∂f/∂θ_{1} = ∂A/∂θ_{1} - (∂θ_{1}/∂θ_{1})B_{1} - (∂θ_{2}/∂θ_{1})B_{2} - (∂θ_{1}/∂θ_{1})θ_{2}C
_{ } = 0 - B_{1} - 0 - Cθ_{2}
Integration
-----------
1. ∫dθ = 0
2. ∫θdθ = 1
3. ∫(af(θ) + bg(θ))dθ = a∫f(θ)dθ + b∫g(θ)dθ
4. ∫(∂f(θ)/∂θ)dθ = 0
Consider the following examples:
One variable:
f(θ) = A + Bθ
∫(A + Bθ)dθ = A∫dθ+ B∫dθ = 0 + B
Two variables,
f(θ_{1},θ_{2}) = A + B_{1}θ_{1} + B_{2}θ_{2} + Cθ_{2}θ_{1}
∫f(θ_{1},θ_{2})dθ_{1}dθ_{2} = ∫(A + B_{1}θ_{1} + B_{2}θ_{2} + Cθ_{2}θ_{1})dθ_{1}dθ_{2}
A∫dθ_{1}dθ_{2} = 0 since ∫dθ_{1} = 0
B_{1}∫θ_{1}dθ_{1}dθ_{2} = 0 since ∫θ_{1}dθ_{1} = 1
B_{2}∫θ_{2}dθ_{1}dθ_{2} = 0 since ∫θ_{2}dθ_{2} = 1
C∫θ_{2}θ_{1}dθ_{1}dθ_{2} = 1 since ∫θ_{1}dθ_{1} and ∫θ_{2}dθ_{2} = 1
By inspection we see that the result of the integration in
both cases is just equal to the coefficient of the last term
in the power series. THIS TURNS OUT TO BE THE GENERAL
RULE FOR INTEGRATING GRASSMANN FUNCTIONS.