chapter 4 the dirac field 6.1 properties 6.2 hydrogen atom
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MANDL AND SHAW
CHAPTER 4 THE DIRAC FIELD4.1 The number representation for fermions ✔4.2 The Dirac equation4.3 Second quantization4.4 The fermion propagator4.5 The electromagnetic interaction and gauge invariancePROBLEMS; 4.1 4.2 4.3 4.4 4.5
APPENDIX A THE DIRAC EQUATION A1 A2 A3 A4 A5 A6 A7 A8PROBLEMS; A.1 A.2
MAIANI AND BENHAR
CHAPTER 6 THE DIRAC EQUATION6.1 Properties6.2 Hydrogen atom (skip)6.3 Traces
CHAPTER 7 QUANTIZATION OF THE DIRAC FIELD7.1 Particles and Antiparticles7.2 Second quantization7.3 Canonical quantization7.4 Lorentz group7.5 Microcausality7.6 Spin and statistics
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The Dirac Equation(Read Appendix A and Sec 4.2)
/1/ Recall the Schroedinger equation
The plane wave solutions are
Ψ(x,t) = C e i ( p.x − E t ) (ħ = 1)
H Ψ = E Ψ ⇒ E = p2 /2m
(nonrelativisic)
P = – i ∇, so P Ψ = p Ψ(x,t )
The plane wave is an eigenstate ofmomentum.
Notations, conventions and units❏ Section 1.2: RATIONALIZED gaussian
electromagnetic units❏ Section 2.1: Relativity notations❏ Section 6.1: Natural units
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x μ = ( x 0, x 1, x2 , x3 ) ; x 0 = c t
g μν = g μν = DIAG( 1 , −1, −1, −1 )
xμ = g μν x ν ; ( x0 , xi ) = (x0 , xi)
p.x = g μν p μ x ν = p0 x0 − pi xi
p2 = p.p = E2 − p2 = m2
∂μ = ∂ /∂xμ = ( ∂ /∂x0 , ∂ /∂xi )
ħ = 1 and c = 1
α = e2 /(4πħc) = 1 / 137.037
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● To be consistent with relativity,t and ( x, y, z ) should be treated similarly; because the Lorentz transformations mix t and ( x, y, z).
So let’s try
i ∂Ψ/∂t = ( α.P + β m ) Ψ
such that
( α.P + βm ) 2 = P 2 + m2
The quantities β and ( αx , αy , αz ) will be
matrices.
Now try
Ψ(x,t) ∝ e i ( p.x − E t ) u .
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/2/ The Dirac equationWe need an equation with these properties:
(i) linear in time, (unlike Klein Gordon)
(ii) with plane wave solutions,(iii) with E = .
Ψ(x,t) ∝ e i ( p.x − E t ) = e − i p.x
i ∂Ψ/∂t = H Ψ
Should we try
H Ψ = Ψ ;
i.e., H = ?
But that is a nonlocal operator.
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√ p2+m2
√ p2+m2
√ P2+m2
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sqrt of P2+m2
Four - vector notations (Appendix A)Define ɣ0 = β;also, ( ɣ1 , ɣ2 , ɣ3 ) = ( βαx , βαy , βαz )
UPPER AND LOWER INDICES:{x0 , x1 , x2 , x3 } = { ct, x , y, z} ( c = 1 ){x0 , x1 , x2 , x3 } = { ct, −x , −y, −z}gμν = diag(1,−1,−1,−1)
{ɣ0 , ɣ1 , ɣ2 ,ɣ3 } = β { 1 , αx , αy , αz }{ɣo , ɣ1 , ɣ2 ,ɣ3 } = β { 1 , −αx , −αy , −αz }ɣ . A = ɣμ Aμ = ɣμ A
μ = ɣ0 A0− ɣi Ai
β and ( αx , αy , αz )
Since they don’t commute, they must be matrices.
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/3/ The gamma matricesWhat are the gamma matrices?They are not unique.
The gamma matrices are 4 X 4 matrices, defined by certain anticommutation relations:
Thus, the defining equation is
{ γμ , γν } = 2 gμν (★)
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i ( γ0 ∂0 + γi ∂i ) ψ − m ψ = 0
i γμ ∂μψ − m ψ = 0
i γ • ∂ψ − m ψ = 0
i ∂ ψ − m ψ = 0
That is the Dirac equation.
Various notations may be used
Slash notation
Q = γμ Qμ .
{ αi , αj } = 2 δij
{ γi , γj } = { βαi , βαj } = − 2 δij
{ γi , γ0 } = { βαi , β } = 0
{ γ0 , γ0 } = 2 (γ0)2 = 2
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● The standard representation (“Dirac rep.”) for the gamma matrices is
Exercise. Verify (★).
● The Majorana representation
which is sometimes convenient.
● (Peskin and Schroeder use yet a different representation.)
(We never raise the index on a Pauli matrix!)
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Theorem. If { γμ , γν } = 2 gμν , and U is a unitary matrix (U♰U = 1) ,then { γ′ μ , γ′ ν } = 2 gμν where γ′ μ = U γμ U♰ .
Proof.
Exercise.Find U such that γM
μ = U γμ U♰ .
For most calculations, we don’t need to use any specific representation of the gamma matrices. Instead we can use some identities that are true for all representations.
{ γμ , γν } = 2 gμν (★)
/4/ Examples of gamma matrix identities
∎ Trace ( γμ γν )
Lemma. Trace(BA) = Trace(AB).
Proof. Trace(BA)
= ∑ Brs Asr = ∑ Asr Brs
= Trace(AB).
Even if A and B do not commute,i.e., BA ≠ AB, always Tr(BA) = Tr(AB).
∎ Trace ( γμ γν γρ γσ )
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∎ γμ γρ γμ
∎ Etc.
We’ll use many such identities.See the Appendix, Sections A.2 and A.3.
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/5/ The Dirac spinors
▶ Plane wave solutions of the Dirac equation
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▶ In the standard (Dirac) representation:
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4 Normalization choiceThis can be done in different ways.We’ll follow Mandl and Shaw ;eq. (A.27) ;
Also, define
and
completeness
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Check the normalization:“ENERGY PROJECTION OPERATORS” ;Section A.5 ;
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Homework Problems due Friday, Feb. 17
Problem 24.A. Determine the Dirac spinors v1(p) and v2(p) for
antiparticles.
B. Determine the polarization sum Λ−(p) for antiparticles.
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HW