ISSN: 2689-7636
Annals of Mathematics and Physics
Short Communication       Open Access      Peer-Reviewed

Dirac spinor’s transformation under Lorentz mappings

J Yaljá Montiel-Pérez1, J López-Bonilla2* and VM Salazar del Moral2

1Centro de Investigación en Computación, Instituto Politécnico Nacional, México
2ESIME-Zacatenco, Instituto Politécnico Nacional, México
*Corresponding author: J López-Bonilla, ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX, México, E-mail: jlopezb@ipn.mx
Received: 30 June, 2020 | Accepted: 13 July, 2021 | Published: 15 July, 2021
Keywords: Dirac 4-spinor, Lorentz transformation, Dirac equation, Pauli and dirac matrices

Cite this as

Montiel-Pérez JY, López-Bonilla J, Salazar del Moral VM (2021) Dirac spinor’s transformation under Lorentz mappings. Ann Math Phys 4(1): 028-031. DOI: 10.17352/amp.000022

For a given Lorentz matrix, we deduce the Dirac spinor’s transformation in terms of four complex quantities.

Introduction

We have the Dirac equation for spin-1/2 particles [1-5] [ ( x μ )=( t, x, y, z ), =c=1 ]: MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qadaWadaGcpaqaaKqba+qadaqadaGcpaqaaKqzaeWdbiaadIhajuaGpaWaaWbaaSqabeaajugab8qacqaH8oqBaaaakiaawIcacaGLPaaajugabiabg2da9Kqbaoaabmaak8aabaqcLbqapeGaamiDaiaacYcacaGGGcGaamiEaiaacYcacaGGGcGaamyEaiaacYcacaGGGcGaamOEaaGccaGLOaGaayzkaaqcLbqacaGGSaGaaiiOaiabl+qiOjabg2da9iaadogacqGH9aqpcaaIXaaakiaawUfacaGLDbaajugabiaacQdaaaa@555F@

( i γ μ μ m 0 )ψ=0,            i= 1 ,             μ = x μ  ,     (1) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8317@

where ψ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqaqaaaaaaaaaWdbiabeI8a5baa@384F@ is a 4-spinor with the γ μ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqaqaaaaaaaaaWdbiabeo7aNLqba+aadaahaaWcbeqaaKqzaeWdbiabeY7aTbaaaaa@3B27@ matrices verifying the anticommutator [6-8]:

{ γ μ , γ ν }=2 g μν   I 4x4  ,            ( g μν )=Diag( 1,1,1,1 ).     (2) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7F47@

Here we shall use the Dirac-Pauli (or standard) representation [2,9]:

γ 0 =( I 0 0 I ),        γ j =( 0 σ j σ j 0 ),  j=1, 2, 3,      (3) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@73F5@

with the Cayley [10]-Sylvester [11]-Pauli [12] matrices:

σ 1 =( 0 1 1 0 ),                σ 2 =( 0 i i 0 ),                σ 3 =( 1 0 0 1 ),    (4) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqaqaaaaaaaaaWdbiabeo8aZLqba+aadaWgaaWcbaqcLbqapeGaaGymaaWcpaqabaqcLbqapeGaeyypa0tcfa4aaeWaaOWdaeaajugabuaabeqaciaaaOqaaKqzaeWdbiaaicdaaOWdaeaajugab8qacaaIXaaak8aabaqcLbqapeGaaGymaaGcpaqaaKqzaeWdbiaaicdaaaaakiaawIcacaGLPaaajugabiaacYcacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacqaHdpWCjuaGpaWaaSbaaSqaaKqzaeWdbiaaikdaaSWdaeqaaKqzaeWdbiabg2da9Kqbaoaabmaak8aabaqcLbqafaqabeGacaaakeaajugab8qacaaIWaaak8aabaqcLbqapeGaeyOeI0IaamyAaaGcpaqaaKqzaeWdbiaadMgaaOWdaeaajugab8qacaaIWaaaaaGccaGLOaGaayzkaaqcLbqacaGGSaGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaeq4Wdmxcfa4damaaBaaaleaajugab8qacaaIZaaal8aabeaajugab8qacqGH9aqpjuaGdaqadaGcpaqaaKqzaeqbaeqabiGaaaGcbaqcLbqapeGaaGymaaGcpaqaaKqzaeWdbiaaicdaaOWdaeaajugab8qacaaIWaaak8aabaqcLbqapeGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaaKqzaeGaaiilaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeinaiaabMcaaaa@8AFE@

to analyze the transformation law of under the orthochronic and proper Lorentz group [13-19]:

x ~ μ = L μ ν   x ν  ,    (5) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqaaaaaaaaaWdbiaadIhaaSWdaeqabaWdbiaac6haaaGcpaWaaWbaaSqabeaapeGaeqiVd0gaaOGaeyypa0Jaamita8aadaahaaWcbeqaa8qacqaH8oqBaaGcpaWaaSbaaSqaa8qacqaH9oGBa8aabeaak8qacaGGGcGaamiEa8aadaahaaWcbeqaa8qacqaH9oGBaaGccaGGGcGaaiilaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabMcaaaa@4B3C@

which implies the existence [2, 7, 20, 21] of a non-singular matrix S such that:

L μ ν  S  γ ν = γ μ  S ,    (6) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGmbWdamaaCaaaleqabaWdbiabeY7aTbaak8aadaWgaaWcbaWdbiabe27aUjaacckaa8aabeaak8qacaWGtbGaaiiOaiabeo7aN9aadaahaaWcbeqaa8qacqaH9oGBaaGccqGH9aqpcqaHZoWzpaWaaWbaaSqabeaapeGaeqiVd0MaaiiOaaaakiaadofacaGGGcGaaiilaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOnaiaabMcaaaa@4F15@

and we deduce the relativistic invariance of (1) if the Dirac 4-spinor obeys the transformation rule:

ψ ~ =S ψ .    (7) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqaaaaaaaaaWdbiabeI8a5bWcpaqabeaapeGaaiOFaaaakiabg2da9iaadofacaGGGcGaeqiYdKNaaiiOaiaac6cacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabEdacaqGPaaaaa@4497@

Here we exhibit a method to findS for a given Lorentz matrix.

Construction of the matrix S for a given Lorentz mapping.

The arbitrary complex quantities α, β, γ, δ verifying the constraint αδ – βγ = 1, generate a Lorentz matrix L=( L μ ν ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqaqaaaaaaaaaWdbiaadYeacqGH9aqpcaGGOaGaamitaKqba+aadaahaaWcbeqaaKqzaeWdbiabeY7aTbaajuaGpaWaaSbaaSqaaKqzaeWdbiabe27aUbWcpaqabaqcLbqapeGaaiykaaaa@411A@ via the relations [13-15, 17, 22-26]:

L 0 0 =  1 2 ( α α + β β + γ γ + δ δ ),     L 1 0 =  1 2 ( α  γ+ β δ )+ cc,         L 2 0 =  i 2 ( α γ β δ )+ cc, L 0 1 =  1 2 ( α β+ γ δ )+ cc,              L 1 1 =  1 2 ( α δ+ β γ )+ cc,          L 2 1 =  i 2 ( α δ + β γ )+ cc, L 0 2 =  i 2 ( α β+  γ δ )+ cc,           L 1 2 =  i 2 ( α δ+ β γ )+ cc,       L 2 2 =  1 2 ( α δ β γ )+ cc , L 0 3 =  1 2 ( α α β β +γ γ δ δ ),    L 1 3 =  1 2 ( α γ β δ )+ cc,          L 2 3 =  i 2 ( α γ + β δ )+ cc , L 3 0 =  1 2 ( α α +β β γ γ δ δ ),     L 3 1 =  1 2 ( α β γ δ )+ cc,          L 3 2 =  i 2 ( α β γ δ )+ cc , L 3 3 =  1 2 ( α α β β γ γ + δ δ ),       αδβγ=1,                                                                                                     (8) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@18EF@

where cc means the complex conjugate of all the previous terms.

The inverse problem is to obtain α, β, γ, δ if we know L, and the answer is [26-29]:

α= 1 D   Q 1 1 = 1 2D [ L 0 0 + L 0 3 + L 1 1 + L 2 2 + L 3 0 + L 3 3 i ( L 1 2 L 2 1 ) ], β= 1 D   Q 1 2 = 1 2D [ L 0 1 + L 1 0 L 1 3 + L 3 1 +i ( L 0 2 + L 2 0 L 2 3 + L 3 2 ) ], γ= 1 D   Q 2 1 = 1 2D [ L 0 1 + L 1 0 + L 1 3 L 3 1 i ( L 0 2 + L 2 0 + L 2 3 L 3 2 ) ], δ= 1 D   Q 2 2 = 1 2D [ L 0 0 L 0 3 + L 1 1 + L 2 2 L 3 0 + L 3 3 +i ( L 1 2 L 2 1 ) ],         (9) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@701D@

where D 2 = Q 1 1   Q 2 2 Q 1 2   Q 2 1 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@52E7@

From (6) are immediate the expressions [3, 30]:

L μ 0 = 1 4  tr ( γ 0   S 1  γ μ  S ),         L μ k = 1 4  tr ( γ k  S 1   γ μ  S ),     μ=0, , 3,    k=1, 2, 3,     (10) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AA10@

that is, if we know S then with (10) we can determine the Lorentz matrix; (10) generates the relations:

L 0 0 =2( b 0 2 b 1 2 b 2 2 b 3 2 )1,                         L 0 1 =2[  ( b 2   d 3 b 3   d 2 )+i ( b 0   d 1 b 1   d 0 )  ], L 0 2 =2[  ( b 3   d 1 b 1   d 3 )+i ( b 0   d 2 b 2   d 0 )  ],       L 0 3 =2[  ( b 1   d 2 b 2   d 1 )+i ( b 0   d 3 b 3   d 0 )  ], L 1 0 =2[  ( b 2   d 3 b 3   d 2 )+i ( b 0   d 1 b 1   d 0 )  ],     L 1 1 =2[  ( b 0 2 b 1 2 )+( d 2 2 + d 3 2 )  ]1, L 1 2 =2[ ( b 1   b 2 + d 1   d 2 )i ( b 0   b 3 + d 0   d 3 )  ],     L 1 3 =2[  ( b 1   b 3 + d 1   d 3 )+i ( b 0   b 2 + d 0   d 2 )  ], L 2 0 =2[ ( b 3   d 1 b 1   d 3 )+i ( b 0   d 2 b 2   d 0 )  ],    L 2 1 =2[  ( b 1   b 2 + d 1   d 2 )+i ( b 0   b 3 + d 0   d 3 )  ], L 2 2 =2[  ( b 0 2 b 2 2 )+( d 1 2 + d 3 2 )  ]1,                L 2 3 =2[ ( b 2   b 3 + d 2   d 3 )i ( b 0   b 1 + d 0   d 1 )  ], L 3 0 =2[ ( b 1   d 2 b 2   d 1 )+i ( b 0   d 3 b 3   d 0 )  ],    L 3 1 =2[  ( b 1   b 3 + d 1   d 3 )i ( b 0   b 2 + d 0   d 2 )  ], L 3 2 =2[ ( b 2   b 3 + d 2   d 3 )+i ( b 0   b 1 + d 0   d 1 )  ],    L 3 3 =2[  ( b 0 2 b 3 2 )+( d 1 2 + d 2 2 )  ]1,                                         (11) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@C57C@

which allow to obtain L if we have the expansion [31]:

S= b 0  I+i d 0  γ 5 + b 1   σ 23 + b 2   σ 31 + b 3  σ 12 +  j=1  3 d j σ oj .     (12) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7C5F@

However, here we have the inverse problem, that is, to obtain b μ  &  d μ , μ=0, , 3 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbqaqaaaaaaaaaWdbiaadkgajuaGpaWaaSbaaSqaaKqzaeWdbiabeY7aTbWcpaqabaqcLbqapeGaaiiOaiaacAcacaGGGcGaamizaKqba+aadaWgaaWcbaqcLbqapeGaeqiVd0gal8aabeaajugab8qacaGGSaGaaiiOaiabeY7aTjabg2da9iaaicdacaGGSaGaaiiOaiabgAci8kaacYcacaGGGcGaaG4maaaa@4DAE@ verifying (11) for a given Lorentz matrix. Our answer is the following:

b 0 = 1 4 ( α+ α +δ+ δ ),   b 1 = 1 4 ( β β+ γ γ ), b 2 = i 4  ( β+ β γ γ ),   b 3 = 1 4 ( α α+δ δ ), d 0 = i 4 ( α α +δ δ ),   d 1 = i 4 ( β +β+ γ +γ ), d 2 = 1 4  ( β β+γ γ ),   d 3 = i 4 ( δ +δα α ),      (13) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@1DEF@

hence the expressions (8) are deduced if we apply (13) into (11). Besides, with (13) the matrix (12) acquires the structure:

S=( A E E A ),          A= 1 2 ( α +δ β γ γ β α+ δ ),          E= 1 2 ( α δ β +γ γ +β δ α ) .      (14) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AB84@

Therefore, for a given Lorentz transformation first we employ (9) to determine α,β,γ,δ, then S is immediate via (14); this approach is an alternative to the process showed in [31] and to the explicit general formula obtained by Macfarlane [30]:

S= 1 4 G  [G I+ i 2   ε μναβ   L μν   L αβ   γ 5 +i Γ( L 2 )i ( 2+tr L ) Γ( L )]      (15) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7960@

G=2 ( 1+tr L )+ 1 2 [   (tr L) 2 tr  L 2  ],       tr L=  μ=0  3 L μ μ ,        tr  L 2 = ν,α=0  3 L ν α L α ν  , Γ( L )= μ,ν=0  3 L μν  σ μν ,      Γ( L 2 )= α,μ,ν=0  3 L μα L α ν  σ μν ,                                                   (16) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@FBE4@

however, the possible physical applications are not evident in it. In our procedure, for example, the relations (9) are of great interest for the physicists working on supersymmetry [29], and the expressions (14) are very useful to study the relativistic motion of a classical point particle [28].

Conclusion

The Dirac equation is relativistic if the corresponding 4-spinor verifies the transformation (7) under Lorentz mappings, with the matrix S satisfying the condition (6). Here we showed a procedure to construct S for a given Lorentz matrix.

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