Lorentz Invariant Quantities
Total 4-Momentums
As was shown earlier the scalar product of a 4-Momentum vector with itself ,
P1⋅P1=E1E1−→p1⋅→p1=m21=s ,
and the length of a 4-Momentum vector composed of 4-Momentum vectors,
P2=(P1+P2)2=(E1+E2)2−(→p1+→p2)2=(m1+m2)2=s,
are invariant quantities.
It was further shown that
P∗2=P2
where P∗=(P∗1+P∗2)2 represents the 4-Momentum Vector in the CM frame
and P=(P1+P2)2 represents the 4-Momentum Vector in the initial Lab frame
which can be expanded to
{\mathbf P^*}^2={\mathbf P^{'*}}^2={\mathbf P}^2={\mathbf P^'}^2
where {\mathbf P^'}=({\mathbf P_1^'}+{\mathbf P_2^'})^2 represents the 4-Momentum Vector in the final Lab frame
and P′∗=(P′∗1+P′∗2)2 represents the 4-Momentum Vector in the final CM frame
New 4-Momentum Quantities
Working in just the CM frame, we can form new 4-Momentum Vectors comprised of 4-Momenta in this frame, with
P∗1−P′∗1=(E∗1−E′∗1p∗1(x)−p′∗1(x)p∗1(y)−p′∗1(y)p∗1(z)−p′∗1(z))=P∗a
P∗1−P′∗2=(E∗1−E′∗2p∗1(x)−p′∗2(x)p∗1(y)−p′∗2(y)p∗1(z)−p′∗2(z))=P∗b
P∗2−P′∗1=(E∗2−E1′∗p∗2(x)−p′∗1(x)p∗2(y)−p′∗1(y)p∗2(z)−p′∗1(z))=P∗c
P∗2−P′∗2=(E∗2−E′∗2p∗2(x)−p′∗2(x)p∗2(y)−p′∗2(y)p∗2(z)−p′∗2(z))=P∗d
Using the algebraic fact
(a−b)2=(b−a)2
and the fact that the length of these 4-Momentum Vectors are invariant,
(P∗1−P′∗1)2=(P∗12−2P∗1⋅P′∗1+P′∗1)=(E∗1−E′∗1p∗1(x)−p′∗1(x)p∗1(y)−p′∗1(y)p∗1(z)−p′∗1(z))2=(P∗a)2
(P∗1−P′∗2)2=(P∗12−2P∗1⋅P′∗2+P′∗2)=(E∗1−E′∗2p∗1(x)−p′∗2(x)p∗1(y)−p′∗2(y)p∗1(z)−p′∗2(z))2=(P∗b)2
(P∗2−P′∗1)2=(P∗22−2P∗2⋅P′∗1+P′∗1)=(E∗2−E′∗1p∗2(x)−p′∗1(x)p∗2(y)−p′∗1(y)p∗2(z)−p′∗1(z))2=(P∗c)2
(P∗2−P′∗2)2=(P∗22−2P∗2⋅P′∗2+P′∗2)=(E∗2−E′∗2p∗2(x)−p′∗2(x)p∗2(y)−p′∗2(y)p∗2(z)−p′∗2(z))2=(P∗d)2
Using the fact that the scalar product of a 4-momenta with itself is invariant,
P1⋅P1=E1E1−→p1⋅→p1=m21
We can simiplify the expressions
(P∗1−P′∗1)2=(m∗21−2P∗1⋅P′∗1+m′∗21)=(P∗a)2
(P∗1−P′∗2)2=(m∗21−2P∗1⋅P′∗2+m′∗22)=(P∗b)2
(P∗2−P′∗1)2=(m∗22−2P∗2⋅P′∗1+m′∗21)=(P∗c)2
(P∗2−P′∗2)2=(m∗22−2P∗2⋅P′∗2+m′∗22)=(P∗d)2
Finding the cross terms,
P∗1⋅P′∗=(E∗1p∗1(x)p∗1(y)p∗1(z))⋅(10000−10000−10000−1)⋅(E′∗p′∗1(x)p′∗1(y)p′∗1(z))=E∗1E′∗1−→p∗1⋅→p′∗1
(P∗1−P′∗1)2=(m∗21−2E∗1E′∗1−2→p∗1⋅→p′∗1+m′∗21)=(P∗a)2
(P∗1−P′∗2)2=(m∗21−2E∗1E′∗2−2→p∗1⋅→p′∗2+m′∗22)=(P∗b)2
(P∗2−P′∗1)2=(m∗22−2E∗2E′∗1−2→p∗2⋅→p′∗1+m′∗21)=(P∗c)2
(P∗2−P′∗2)2=(m∗22−2E∗2E′∗2−2→p∗2⋅→p′∗2+m′∗22)=(P∗d)2
Using the fact that in the CM frame,
→p∗1=−→p∗2
→p′∗1=−→p′∗2
Since this is an ellastic collision between identical particles, Energy is conserved,
E∗1=E′∗1
E∗2=E′∗2
Lastly as shown earlier, E∗1=E∗2
We can further simplify
(P∗1−P′∗1)2=(m∗21−2E∗2E′∗2−2→p∗2⋅→p′∗2+m′∗21)=(P∗a)2
(P∗2−P′∗2)2=(m∗22−2E∗2E′∗2−2→p∗2⋅→p′∗2+m′∗22)=(P∗d)2
⟹(P∗1−P′∗1)2=(P∗2−P′∗2)2
(P∗1−P′∗2)2=(m∗21−2E∗2E′∗1−2→p∗2⋅→p′∗1+m′∗22)=(P∗b)2
(P∗2−P′∗1)2=(m∗22−2E∗2E′∗1−2→p∗2⋅→p′∗1+m′∗21)=(P∗c)2
⟹(P∗1−P′∗2)2=(P∗2−P′∗1)2
Mandelstam Representation
⟹(P∗1+P∗2)2=(P′∗1+P′∗2)2≡s
⟹(P∗1−P′∗1)2=(P∗2−P′∗2)2≡t
⟹(P∗1−P′∗2)2=(P∗2−P′∗1)2≡s
