com.andreimikhailov.minitheta.examples
Now introduce some bosonic (i.e. usual, commutative) variables
Super-monomial.
Super-monomial. As we already explained, a monomial of fermionic variables is just a set of Thetas And what is a monomial of bosonic variables? It is a Map, more precisely scala.collection.mutable.Map; Every bosonic variable is mapped to its power. For example
$X_1^5 X_2^8 X_5$
corresponds to Map(X(1) -> 5, X(2) -> 8, X(5) -> 1) (If the power is zero, i.e. the monomial does not contain a particular variable, then we simply drop this key.)
Linear combination of super-monomials
Fermionic variables.
This is the left derivative of the product M1 M2 with respect to Th(5).
This is the left derivative of the product M1 M2 with respect to Th(5). Notice that the sign is now plus, because we got another minus when carrying the derivative across Th(1)
This is a sample super-monomial
And their product.
And their product. Notice that there is a minus sign, because we had to exchange Th(2) and Th(5) to bring this to our defined ordering
And another one
Integer coefficients:
Integer coefficients:
integer
m times the unit of the ring
Rational coefficients:
Rational coefficients:
numerator
denominator
$m\over n$ times unit of the ring
Element of basis of the linear space:
Element of basis of the linear space:
is the name of the variable e.g. "x" or "y"
element of the basis
Sample system of linear equation Here x and y are variables, while a and b are constants.
Sample system of linear equation Here x and y are variables, while a and b are constants. The equations are:
x + 7y + a = 0 y + {1\over 13}b = 0
This is just a shortcut to treat an element of linear space as an element of free supercommutative algebra.
This is just a shortcut to treat an element of linear space as an element of free supercommutative algebra.
element of the linear space of super-polynomials
the same x, but as an element of free supercommutative algebra (a tautology)
This is a shortcut to contruct a base monomial
This is a shortcut to contruct a base monomial
set of fermions
map of bosons
corresponding element of the basis of the linear space of super-polynomials