mandelstam.txt
Sept. 12, 2018
Ted Woollett
http://web.csulb.edu/~woollett/
Maxima by Example
Ch. 12, Dirac Algebra and Quantum Electrodynamics

=========================
Examples of inputs to Maxima

The default list massL is created to be:
 massL =    [m,M] 
which means the symbols m and M are
recognised as mass symbols by the program.

You can check this list using 

display (massL)$

You can add other symbols to the list massL using:

Mmass (m1,m2,m3,m4)$

and then check the list again:

display (massL)$
=====================================

Examples of using the command set_invarR to populate the
list invarR with replacement rules for D(pa,pb) in terms
of Mandelstam variables s, t, and u.

You can copy and paste these if appropriate to your problem.

We have the basic convervation of 4-momentum in the scattering
process p1 + p2 -->  p3  + p4
    p1 + p2 = p3 + p4 for each Lorentz component,
	and then s, t, u are defined by:
	
    s = (p1+p2)^2 = (p3+p4)^2,
     t = (p1-p3)^2 = (p2-p4)^2, and  
     u = (p1-p4)^2 = (p2-p3)^2 

You can check the contents of this list using:

display (invarR)$
------------------------------------------------

      case all distinct masses  
	  
You should first use

Mmass (m1,m2,m3,m4)$     

so these symbols will be recognised as masses.	  

set_invarR (D(p1,p1) = m1^2,
         D(p1,p2) =  (s - m1^2 - m2^2)/2,
         D(p1,p3) = (m1^2 + m3^2 - t)/2,
         D(p1,p4) = (m1^2 + m4^2 - u)/2,
         D(p2,p2) = m2^2,
         D(p2,p3) = (m2^2 + m3^2 - u)/2,
         D(p2,p4) = (m2^2 + m4^2 - t)/2,
         D(p3,p3) = m3^2,
         D(p3,p4) = (s - m3^2 - m4^2)/2,
         D(p4,p4) = m4^2)$
		 
display (invarR)$
----------------------------------------------------

      case m1 = m3 = m, m2 = m4 = M  


set_invarR (D(p1,p1) = m^2,
         D(p1,p2) =  (s - m^2 - M^2)/2,
         D(p1,p3) = (2*m^2 - t)/2,
         D(p1,p4) = (m^2 + M^2 - u)/2,
         D(p2,p2) = M^2,
         D(p2,p3) = (M^2 + m^2 - u)/2,
         D(p2,p4) = (2*M^2 - t)/2,
         D(p3,p3) = m^2,
         D(p3,p4) = (s - m^2 - M^2)/2,
         D(p4,p4) = M^2)$
		 
---------------------------------------------------
case m1 = m2 = m,  m3 = m4 = M

set_invarR (D(p1,p1) = m^2,
         D(p1,p2) =  s/2 - m^2,
         D(p1,p3) = (m^2 + M^2)/2 - t/2,
         D(p1,p4) = (m^2 + M^2)/2 - u/2,
         D(p2,p2) = m^2,
         D(p2,p3) = (m^2 + M^2)/2 - u/2,
         D(p2,p4) = (m^2 + M^2)/2 - t/2,
         D(p3,p3) = M^2,
         D(p3,p4) = s/2 - M^2,
         D(p4,p4) = M^2)$
		 
------------------------------------------		 
 case all masses equal:
  case m1 = m2 = m3 = m4 = m 		 
		 
set_invarR (D(p1,p1) = m^2,
         D(p1,p2) = s/2 - m^2,
         D(p1,p3) = m^2 - t/2,
         D(p1,p4) = m^2 - u/2,
         D(p2,p2) = m^2,
         D(p2,p3) = m^2 - u/2,
         D(p2,p4) = m^2 - t/2,
         D(p3,p3) = m^2,
         D(p3,p4) = s/2 - m^2,
         D(p4,p4) = m^2)$		 
		 
---------------------------------------------

  case high energy: E >> any mass involved (HE): neglect masses

set_invarR (D(p1,p1) = 0,
         D(p1,p2) = s/2,
         D(p1,p3) =  - t/2,
         D(p1,p4) =  - u/2,
         D(p2,p2) = 0,
         D(p2,p3) =  - u/2,
         D(p2,p4) =  - t/2,
         D(p3,p3) = 0,
         D(p3,p4) = s/2 ,
         D(p4,p4) = 0)$

---------------------------------------------------
  
Possibly useful definition:

lam(a,b,c) := a^2 + b^2 + c^2 -2*(a*b + a*c + b*c)$