#!/usr/bin/env cadabra2
\alpha_{n}^{\mu}::SelfNonCommuting;
\vac::LaTeXForm("|k\rangle");
def post_process(ex):
   substitute(ex, $\sqrt{2} \sqrt{2} -> 2$)
   collect_terms(ex)
{\alpha_{n}^{\mu}, \vac}::NonCommuting;
{\mu,\nu,\rho}::Integer(0..d-1).
{\mu,\nu,\rho}::Indices.
{m,n}::Integer.
{m,n}::Symbol.
\delta{#}::KroneckerDelta.
\delta{#}::Diagonal.
chi1:= \alpha_{-1}^{\mu} \alpha_{-1}^{\mu} \vac;
chi2:= \alpha_{0}^\mu \alpha_{-2}^\mu \vac;
chi3:= \alpha_{0}^{\mu} \alpha_{-1}^{\mu} \alpha_{0}^{\nu} \alpha_{-1}^{\nu})\vac;
rl:= { \alpha_{m}^{\mu} \alpha_{n}^{\nu} | m > n -> \alpha_{n}^{\nu} \alpha_{m}^{\mu} +  m\delta_{m}_{-n} \delta^{\mu\nu},
       \alpha_{m}^{\mu} \vac | m > 0 -> 0, 
       \alpha_{0}^{\mu} \vac -> \sqrt{2} k^{\mu} \vac, k^{\mu} k^{\mu} -> -1 };
def L(m):
	ins=''
	for n in range(-10,10):
		if n!=-10:
			ins+='+ '
		ins += r'1/2 \alpha_{'+str(m-n)+r'}^{\mu} \alpha_{'+str(n)+r'}^{\mu}'
	ex=Ex(ins)
	return ex
L0=L(0);
L1=L(1)
L2=L(2)
L1c1:= @(L1) @(chi1);
L1c2:= @(L1) @(chi2).
L1c3:= @(L1) @(chi3).
def act(ex):
	converge(ex):
		distribute(ex)
		substitute(ex, rl)
		distribute(ex)
		eliminate_kronecker(ex)
		rename_dummies(ex)

	return ex
act(L1c1);
act(L1c2);
act(L1c3);
L2c1:= @(L2) @(chi1).
L2c2:= @(L2) @(chi2).
L2c3:= @(L2) @(chi3).
act(L2c1);
act(L2c2);
act(L2c3);
tst1:=@(L1c1) + A @(L1c2) + B @(L1c3) = 0;
tst2:=@(L2c1) + A @(L2c2) + B @(L2c3) = 0;