ORIGINAL ARTICLE: Ben Goren, Kamal Barley, Sergei K. Suslov, Matrix Approach to Helicity States of Dirac Free Particles. arXiv:2212.11994v2 [quant-ph]. https://doi.org/10.48550/arXiv.2212.11994.
Article Abstract: We derive the free wave solutions of the Dirac equation from the viewpoint of matrix algebra. The concept of spin and the corresponding helicity states are analyzed in detail. This consideration may help the readers to study mathematical methods of relativistic quantum mechanics.
We use elementary matrix algebra to derive the free wave solutions of the Dirac equation and examine the fundamental concepts of spin, polarization, and helicity states in details. This consideration can aid readers in studying the mathematical methods of relativistic quantum mechanics. We hope that our somewhat informal presentation may help beginners to enjoy the study of mathematics of relativistic quantum mechanics. To this end, we include important details of calculations which are usually omitted elsewhere. Computer algebra methods are useful for verification.
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(Last modified on June 12, 2023; 6:41 PM, Arizona time.)

3. Matrix algebra plane wave solutions

Here we verify our identities (3.1) and (3.2)

Identity (3.1):

This is the first matrix in the l.h.s. of (3.1) in 4x4 form, say E=R>0:
{{m*c^2-R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2-R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2-R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2-R}}//MatrixForm
2
c
m-R
0
c
p
3
c(
p
1
-
p
2
)
0
2
c
m-R
c(
p
1
+
p
2
)
-c
p
3
c
p
3
c(
p
1
-
p
2
)
-
2
c
m-R
0
c(
p
1
+
p
2
)
-c
p
3
0
-
2
c
m-R
This is the second matrix in the l.h.s. of (3.1), when R-R:
{{m*c^2+R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2+R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2+R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2+R}}//MatrixForm
2
c
m+R
0
c
p
3
c(
p
1
-
p
2
)
0
2
c
m+R
c(
p
1
+
p
2
)
-c
p
3
c
p
3
c(
p
1
-
p
2
)
-
2
c
m+R
0
c(
p
1
+
p
2
)
-c
p
3
0
-
2
c
m+R
Evaluating their product
{{m*c^2-R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2-R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2-R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2-R}}.{{m*c^2+R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2+R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2+R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2+R}};​​FullSimplify[%]//MatrixForm
4
c
2
m
-
2
R
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
4
c
2
m
-
2
R
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
4
c
2
m
-
2
R
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
4
c
2
m
-
2
R
+
2
c

2
p
1
+
2
p
2
+
2
p
3

This is the r.h.s. of (3.1) in 4x4 form.

Identity (3.2):

This is the matrices in the l.h.s. of (3.2):
{{m*c^2+R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2+R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2-R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2-R}}//MatrixForm
2
c
m+R
0
c
p
3
c(
p
1
-
p
2
)
0
2
c
m+R
c(
p
1
+
p
2
)
-c
p
3
c
p
3
c(
p
1
-
p
2
)
-
2
c
m-R
0
c(
p
1
+
p
2
)
-c
p
3
0
-
2
c
m-R
Evaluating the product in the l.h.s. of (3.2):
{{m*c^2+R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2+R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2-R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2-R}}.{{m*c^2+R,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2+R,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2-R,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2-R}};​​FullSimplify[%]//MatrixForm
2
(
2
c
m+R)
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
2
(
2
c
m+R)
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
2
(
2
c
m+R)
+
2
c

2
p
1
+
2
p
2
+
2
p
3

0
0
0
0
2
(
2
c
m+R)
+
2
c

2
p
1
+
2
p
2
+
2
p
3

This is the r.h.s. of (3.2).

4. Fermi’s bi-spinors

Verification of Fermi’s bi-spinors.

4.1 Equation (4.1):

This is the matrix in the l.h.s. of (4.1):
{{m*c^2,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2}}//MatrixForm
2
c
m
0
c
p
3
c(
p
1
-
p
2
)
0
2
c
m
c(
p
1
+
p
2
)
-c
p
3
c
p
3
c(
p
1
-
p
2
)
-
2
c
m
0
c(
p
1
+
p
2
)
-c
p
3
0
-
2
c
m
This is Fermi’s bi-spinor in the l.h.s. of (4.1); see also (34.27) in Fermi’s lectures:
{{(c*
p
3
)/(R-m*c^2)},{(c*(
p
1
+I*
p
2
))/(R-m*c^2)},{1},{0}}//MatrixForm
c
p
3
-
2
c
m+R
c(
p
1
+
p
2
)
-
2
c
m+R
1
0
Evaluating their product:
{{m*c^2,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2}}.{{(c*
p
3
)/(R-m*c^2)},{(c*(
p
1
+I*
p
2
))/(R-m*c^2)},{1},{0}};​​FullSimplify[%]//MatrixForm
-
cR
p
3
2
c
m-R
-
cR(
p
1
+
p
2
)
2
c
m-R
-
2
c

2
c
2
m
-mR+
2
p
1
+
2
p
2
+
2
p
3

2
c
m-R
0
Substitution from (3.4)
%/.
2
p
1
+
2
p
2
+
2
p
3
(R^2-m^2*c^4)/c^2;​​FullSimplify[%]//MatrixForm
-
cR
p
3
2
c
m-R
-
cR(
p
1
+
p
2
)
2
c
m-R
R
0
This is the r.h.s. of (4.1).

4.2 In a similar fashion for the fourth Fermi’s bi-spinor:

This is the matrix in the l.h.s. of (4.1), once again:
{{m*c^2,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2}}//MatrixForm
2
c
m
0
c
p
3
c(
p
1
-
p
2
)
0
2
c
m
c(
p
1
+
p
2
)
-c
p
3
c
p
3
c(
p
1
-
p
2
)
-
2
c
m
0
c(
p
1
+
p
2
)
-c
p
3
0
-
2
c
m
This is the last bi-spinor (34.27), up to normalization, in Fermi’s lectures:
{{(c*(
p
1
-I*
p
2
))/(R-m*c^2)},{-(c*
p
3
)/(R-m*c^2)},{0},{1}}//MatrixForm
c(
p
1
-
p
2
)
-
2
c
m+R
-
c
p
3
-
2
c
m+R
0
1
Evaluating their product:
{{m*c^2,0,c*
p
3
,c*(
p
1
-I*
p
2
)},{0,m*c^2,c*(
p
1
+I*
p
2
),-c*
p
3
},{c*
p
3
,c*(
p
1
-I*
p
2
),-m*c^2,0},{c*(
p
1
+I*
p
2
),-c*
p
3
,0,-m*c^2}}.{{(c*(
p
1
-I*
p
2
))/(R-m*c^2)},{-(c*
p
3
)/(R-m*c^2)},{0},{1}};​​FullSimplify[%]//MatrixForm
-
cR(
p
1
-
p
2
)
2
c
m-R
cR
p
3
2
c
m-R
0
-
2
c

2
c
2
m
-mR+
2
p
1
+
2
p
2
+
2
p
3

2
c
m-R
%/.
2
p
1
+
2
p
2
+
2
p
3
(R^2-m^2*c^4)/c^2;​​FullSimplify[%]//MatrixForm
-
cR(
p
1
-
p
2
)
2
c
m-R
cR
p
3
2
c
m-R
0
R
This is the last bi-spinor (34.27) in Fermi’s lectures multiplied by +R.

4.3 Fermi’s bi-spinors are linearly dependent:

The (transposed) matrix of all four Fermi’s bi-spinor (34.26-27) in Appendix B, in reverse order:
Thus the determinant of all four Fermi’s bi-spinors (34.26-27) is zero.

4.4 Verification of correct solutions:

This is the matrix in the l.h.s. of (4.1), once again:
These are two bi-spinors (3.8) for the positive energy eigenvalues:
Evaluating the product:
Those are our bi-spinors (3.8), up to normalization, or Fermi’s(34.26), multiplied by +R.
These are two bi-spinors (3.9) for the negative energy eigenvalues:
Evaluating the product:
Those are our bi-spinors (3.9) multiplied by -R.

5. Relativistic Helicity States

5.2 Nonrelativistic Helicity States

Equation (5.19): the first matrix.
Equation (5.19): the second matrix.
Their product:
This is the r.h.s of (5.19).
Checking (5.14) by matrix multiplication; both equations together as presented in the l.h.s. of (5.20):
This is the r.h.s of (5.20).
Verification of equations (5.15): the spinor (5.12) and its Hermitian conjugate.
For the first component of (5.15) with the first Pauli matrix:
For the second component of (5.15):
For the third component of (5.15):
Thus, all three components of the first (5.15), with plus, are given by:
In a similar fashion, for the spinor (5.13) and its Hermitian conjugate.
All three components of the second (5.15), with minus, are:
Thus both equations (5.15) are verified.

5.6 Relativistic Helicity States

◼
  • Helicity States
    • This implies that for the fist two bi-spinors, given by (5.78)-(5.79), the helicity signes are + and -. But for the last ones, given by (6.25)-(6.26), the helicity signes are - and +. Why? Hint: Compare two solutions, say, in (7.4), and/or in (5.36) and (6.8), for E>0 and E<0, respectively.
    ◼
  • Hamiltonian and bi-spinors in matrix form
    • Checking the first two bi-spinors in (5.78-79) by matrix multiplication:
    Compare with (5.78-79) - that is correct for the positive enegrgy eigenvalues E=R>0.
    Checking the last two bi-spinors in (6.25-26) by matrix multiplication, with p → -p or η → -η, in the Hamiltonian:
    Recall the matrix  once again:
    Thus, all four bi-spinors in (5.78-79), the first two columns, and in (6.25-26), with p → -p or η → -η, the last two ones, are linearly independent.
    Here, the first two columns correspond to positive energy eigenvalues and the last two - to the negative energy eigenvalues.
    ◼
  • Inverses
    • ◼
  • Decompositions of the Hamiltonian
    • Here, both matrix products are the same:
    This is Dirac’s Hamiltonian in the units (5.76).

    6. Matrix product identities

    Here we verify the projection operators, describing the polarization of spin one-half particles, introduced by Mitchel and Wightman (Figure 3, Ref. [35]), by direct matrix multiplications.

    Case E>0 and λ=1/2. Identities (6.19)-(6.24):

    Let us construct the tensor product in the l.h.s. of (6.17). This is our bi-spinor (5.78):
    This is its complex conjugate:
    This is the Dirac conjugate:
    This is their tensor product (6.24):
    This is the matrix in (6.24).
    This is the matrix in (6.19):
    This is the matrix in (6.23):
    This is multiplication of the last two matrices
    Now we compare this result with (6.24):
    Thus the matrix identity is verified.

    CITE THIS NOTEBOOK

    Matrix Approach to Helicity States of Dirac Free Particles​
    by Sergei Suslov​
    Wolfram Community, STAFF PICKS, June 9, 2023
    ​https://community.wolfram.com/groups/-/m/t/2933767