From Quantum Mechanics to Maxwell’s Equationsdcross/academics/groups/qm_em_talk.pdf · The Maxwell Equations References Fin Fields Quantum Field An object which is an arbitrary

From Quantum Mechanicsto Maxwell’s Equations

The Work of Dr. Jairzinho Ramos

Daniel J. Cross

Department of PhysicsDrexel University

Philadelphia, PA 19104

July 24, 2009

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Daniel J.Cross

Introduction

Groups andRepresenta-tions

The MaxwellEquations

References

Fin

Outline

Fields in Physics

The Poincare group

Group Representations

Representations of the Poincare group

The Maxwell Equations

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Fields

Quantum Field

An object which is an arbitrary linear superposition of basis statesin a Hilbert space which transforms under the Poincare group.There is no constraining equation.

↓ ?

Classical Field

An object that transforms (in a manifestly covariant way) underthe Poincare group (scalar, vector, spinor,. . . ) and is constrainedto be a solution of some differential (field) equation.

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Fields

Quantum Field


↓ ?

Classical Field


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Fields

Quantum Field


↓ ?

Classical Field


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The Poincare Group

The full invariance group of Special Relativity contains

Boosts

Rotations

Parity and Time-reversal

Translations

(Λ, a)v = Λv + a

(Λ, a)v =(

Λ a0 1

)(v1

)=(

Λv + a1

)

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The Poincare Group


Boosts

Rotations


Translations

(Λ, a)v = Λv + a

(Λ, a)v =(

Λ a0 1

)(v1

)=(

Λv + a1

)

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The Poincare Group


Boosts

Rotations


Translations

(Λ, a)v = Λv + a

(Λ, a)v =(

Λ a0 1

)(v1

)=(

Λv + a1

)

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The Poincare Group

(Λ2, a2)(Λ1, a1) =(

Λ2 a2

0 1

)(Λ1 a1

0 1

)

=(

Λ2Λ1 Λ2a1 + a2

0 1

)= (Λ2Λ1,Λ2a1 + a2)

(I, a)(Λ, 0) = (Λ, a) and (Λ, 0)(I, b) = (Λ,Λb)

(I, a)(Λ, 0) = (Λ, a) = (Λ, 0)(I,Λ−1a),

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The Poincare Group

(Λ2, a2)(Λ1, a1) =(

Λ2 a2

0 1

)(Λ1 a1

0 1

)=

(Λ2Λ1 Λ2a1 + a2

0 1

)

= (Λ2Λ1,Λ2a1 + a2)

(I, a)(Λ, 0) = (Λ, a) and (Λ, 0)(I, b) = (Λ,Λb)

(I, a)(Λ, 0) = (Λ, a) = (Λ, 0)(I,Λ−1a),

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The Poincare Group

(Λ2, a2)(Λ1, a1) =(

Λ2 a2

0 1

)(Λ1 a1

0 1

)=

(Λ2Λ1 Λ2a1 + a2

0 1

)= (Λ2Λ1,Λ2a1 + a2)

(I, a)(Λ, 0) = (Λ, a) and (Λ, 0)(I, b) = (Λ,Λb)

(I, a)(Λ, 0) = (Λ, a) = (Λ, 0)(I,Λ−1a),

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The Poincare Group

(Λ2, a2)(Λ1, a1) =(

Λ2 a2

0 1

)(Λ1 a1

0 1

)=

(Λ2Λ1 Λ2a1 + a2

0 1

)= (Λ2Λ1,Λ2a1 + a2)

(I, a)(Λ, 0) = (Λ, a) and (Λ, 0)(I, b) = (Λ,Λb)

(I, a)(Λ, 0) = (Λ, a) = (Λ, 0)(I,Λ−1a),

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The Poincare Group

(Λ2, a2)(Λ1, a1) =(

Λ2 a2

0 1

)(Λ1 a1

0 1

)=

(Λ2Λ1 Λ2a1 + a2

0 1

)= (Λ2Λ1,Λ2a1 + a2)

(I, a)(Λ, 0) = (Λ, a) and (Λ, 0)(I, b) = (Λ,Λb)

(I, a)(Λ, 0) = (Λ, a) = (Λ, 0)(I,Λ−1a),

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Definition

A representation of a group G is a mapping of G into a setof matrices which preserves group multiplication, M(gh) =M(g)M(h).

scalar: φ→ 1φvector: va → Λacv

c

tensor: T ab → ΛacΛbdT

cd

G invariance → states transform under representations of G.

representations↔ physical states

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Definition



c


cd



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Definition



c


cd



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Representations of Quantum States

Unitary - preserve inner products (observables)〈ψ′|ψ′〉 = 〈ψ|U †U |ψ〉 = 〈ψ|ψ〉 → U †U = 1

Irreducible - not made of smaller parts

M =(M1 00 M2

)- leaves no non-trivial subspace invariant.↔ elementary particle.

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Representations of Quantum States

Unitary - preserve inner products (observables)〈ψ′|ψ′〉 = 〈ψ|U †U |ψ〉 = 〈ψ|ψ〉 → U †U = 1Irreducible - not made of smaller parts

M =(M1 00 M2

)- leaves no non-trivial subspace invariant.↔ elementary particle.

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UIR of SO(3) & Angular Momentum

R |ψ〉 =?

RRt = I and R = I +M

M +M t = 0

So

M =

0 θ3 −θ2

−θ3 0 θ1

θ2 −θ1 0

= θ · J

[Ji, Jj ] = −εijkJk

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R |ψ〉 =?


M +M t = 0

So

M =

0 θ3 −θ2

−θ3 0 θ1

θ2 −θ1 0

= θ · J


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R |ψ〉 =?


M +M t = 0

So

M =

0 θ3 −θ2

−θ3 0 θ1

θ2 −θ1 0

= θ · J


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2j + 1-dimensional, j = 0, 1/2, 1, 3/2, . . .

em =∣∣jm

⟩, −j ≤ m ≤ j

Dj(θ) = exp{J(j) · θ

}J(j) are 2j + 1× 2j + 1 matrices (i/2σ for j = 1/2).

Dj∣∣jm

⟩=∣∣∣jm′⟩Dj

mm′ ,∣∣jm

⟩↔ Y j

m ↔ spherical tensor of rank j

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UIR of Abelian Groups

Γ(a) has eigenvector |ψa〉 with eigenvalue λa

Γ(a)Γ(b) |ψa〉 = Γ(b)Γ(a) |ψa〉= λaΓ(b) |ψa〉

Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉

Eigenspace invariant → whole space and

Γ(a) = λaI

Every matrix is diagonal and every subspace invariant, hencethe space is 1-dimensional and Γ(a) = exp(ik(a)).

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Γ(a)Γ(b) |ψa〉 = Γ(b)Γ(a) |ψa〉

= λaΓ(b) |ψa〉Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉


Γ(a) = λaI


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Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉


Γ(a) = λaI


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Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉


Γ(a) = λaI


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Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉


Γ(a) = λaI


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Γ(a) {Γ(b) |ψa〉} = λa {Γ(b) |ψa〉}

Γ(c)Γ(b) |ψa〉 = Γ(c+ b) |ψa〉


Γ(a) = λaI


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Γ(a+ b) = Γ(a)Γ(b)

eik(a+b) = eik(a)eik(b)

= eik(a)+ik(b)

so k(a) = kµaµ = k · a.

1 =(eik·a

)(eik·a

)†= eia·(k−k

∗) → k real

Γ(a) |k〉 = eik·a |k〉

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Γ(a+ b) = Γ(a)Γ(b)eik(a+b) = eik(a)eik(b)

= eik(a)+ik(b)


1 =(eik·a

)(eik·a

)†= eia·(k−k

∗) → k real


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= eik(a)+ik(b)


1 =(eik·a

)(eik·a

)†= eia·(k−k

∗) → k real


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= eik(a)+ik(b)


1 =(eik·a

)(eik·a

)†= eia·(k−k

∗) → k real


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= eik(a)+ik(b)


1 =(eik·a

)(eik·a

)†= eia·(k−k

∗) → k real


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UIR of the Lorentz Group

Lorentz Group defined by

ΛtηΛ = η, η = diag(−1, 1, 1, 1)

Λ = I +M → M tη + ηM = 0(−A Ct

−Bt Dt

)+(−A −BC D

)= 0

So A = 0, Bt = C, and D = −Dt.

M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

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ΛtηΛ = η, η = diag(−1, 1, 1, 1)

Λ = I +M → M tη + ηM = 0(−A Ct

−Bt Dt

)+(−A −BC D

)= 0


M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

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ΛtηΛ = η, η = diag(−1, 1, 1, 1)

Λ = I +M → M tη + ηM = 0(−A Ct

−Bt Dt

)+(−A −BC D

)= 0


M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

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Infinitesimal operation M = θ · J + b ·K[Ji, Jj ] = −εijkJk

[Ki,Kj ] = +εijkJk[Ji,Kj ] = −εijkKk

J(1) =12

(J + iK)

J(2) =12

(J− iK)[J

(1)i , J

(1)j

]= −εijkJ

(1)k[

J(2)i , J

(2)j

]= −εijkJ

(2)k[

J(1)i , J

(2)j

]= 0

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Infinitesimal operation M = θ · J + b ·K[Ji, Jj ] = −εijkJk

[Ki,Kj ] = +εijkJk[Ji,Kj ] = −εijkKk

J(1) =12

(J + iK)

J(2) =12

(J− iK)[J

(1)i , J

(1)j

]= −εijkJ

(1)k[

J(2)i , J

(2)j

]= −εijkJ

(2)k[

J(1)i , J

(2)j

]= 0

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J(1) → Dj and J(2) → Dj′

Djj′ = DjDj′ and∣∣∣jj′µµ′⟩ =

∣∣jµ

⟩ ∣∣∣j′µ′⟩ .Djj′(θ·J+b·K) = exp

{(θ − ib) · J(j)

}exp

{(θ + ib) · J(j′)

}

Since J = J(1) + J(2), a jj′ states is reducible under SO(3).

∣∣JM

⟩=∑m,m′

⟨j j′

mm′

∣∣∣ JM⟩ ∣∣∣jm ⟩ ∣∣∣j′m′⟩ , |j − j′| ≤ J ≤ |j + j′|.

SO(3) irreducible representations Dj0 and D0j .

D12

12 ↔ E&M potential. D10 and D01 ↔ E&M field.

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J(1) → Dj and J(2) → Dj′


∣∣jµ

⟩ ∣∣∣j′µ′⟩ .Djj′(θ·J+b·K) = exp

{(θ − ib) · J(j)

}exp

{(θ + ib) · J(j′)

}Since J = J(1) + J(2), a jj′ states is reducible under SO(3).

∣∣JM

⟩=∑m,m′

⟨j j′

mm′

∣∣∣ JM⟩ ∣∣∣jm ⟩ ∣∣∣j′m′⟩ , |j − j′| ≤ J ≤ |j + j′|.


D12


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J(1) → Dj and J(2) → Dj′


∣∣jµ

⟩ ∣∣∣j′µ′⟩ .Djj′(θ·J+b·K) = exp

{(θ − ib) · J(j)

}exp

{(θ + ib) · J(j′)

}Since J = J(1) + J(2), a jj′ states is reducible under SO(3).

∣∣JM

⟩=∑m,m′

⟨j j′

mm′

∣∣∣ JM⟩ ∣∣∣jm ⟩ ∣∣∣j′m′⟩ , |j − j′| ≤ J ≤ |j + j′|.


D12


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UIR of the Poincare Group

|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩

(I, a) |k, σ〉 = eik·a |k, σ〉 .

(I, a)(Λ, 0) |k, σ〉 = (Λ, 0)(I,Λ−1a) |k, σ〉= eik·Λ

−1a(Λ, 0) |k, σ〉= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ

∣∣Λk, σ′⟩ ,Manifestly covariant are reducible!

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|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩(I, a) |k, σ〉 = eik·a |k, σ〉 .


−1a(Λ, 0) |k, σ〉= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ


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|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩(I, a) |k, σ〉 = eik·a |k, σ〉 .

(I, a)(Λ, 0) |k, σ〉 = (Λ, 0)(I,Λ−1a) |k, σ〉

= eik·Λ−1a(Λ, 0) |k, σ〉

= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ


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|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩(I, a) |k, σ〉 = eik·a |k, σ〉 .


−1a(Λ, 0) |k, σ〉

= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ


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|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩(I, a) |k, σ〉 = eik·a |k, σ〉 .


−1a(Λ, 0) |k, σ〉= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ

∣∣Λk, σ′⟩ ,

Manifestly covariant are reducible!

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|k, σ〉 = |k〉∣∣∣jm j′

m′

⟩(I, a) |k, σ〉 = eik·a |k, σ〉 .


−1a(Λ, 0) |k, σ〉= eiΛk·a(Λ, 0) |k, σ〉 ,

(Λ, 0) |k, σ〉 = Mσ′,σ


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The “Little” Group

Reference 4-momentum ko and boost Λ0(k) : k0 → k.

Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉=

[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉

=[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉=

[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉=

[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉=

[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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Λ |k, σ〉 = ΛΛ0(k) |k0, σ〉=

[Λ0(Λk)Λ−1

0 (Λk)]︸︷︷︸

1

ΛΛ0(k) |k0, σ〉

= Λ0(Λk)[Λ−1

0 (Λk)ΛΛ0(k)]|k0, σ〉 .

H : k0Λ0(k)→ k

Λ→ ΛkΛ−1

0 (Λk)→ k0,

H |k0, σ〉 = Dσ′,σ

∣∣k0, σ′⟩

Λ |k, σ〉 = Dσ′,σ(Λ, k0)Λ0(Λk)∣∣k0, σ

′⟩ ,

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−k · k = m2 ≥ 0

m > 0, k0 = (±1, 0, 0, 0)m = 0, k0 = (±1, 0, 0, 1)

Rotations about z-axis leave k0 invariant.Others → “infinite” spin.SO(2) abelian

Γ(φ) |ξ〉 = eiφξ |ξ〉 ,

ξ = 0, 1/2, 1, . . .Dξ′,ξ = eiφξδξ′ξ

“Helicity index”. States |k, ξ〉.

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−k · k = m2 ≥ 0

m > 0, k0 = (±1, 0, 0, 0)m = 0, k0 = (±1, 0, 0, 1)

Rotations about z-axis leave k0 invariant.Others → “infinite” spin.

SO(2) abelianΓ(φ) |ξ〉 = eiφξ |ξ〉 ,

ξ = 0, 1/2, 1, . . .Dξ′,ξ = eiφξδξ′ξ


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−k · k = m2 ≥ 0

m > 0, k0 = (±1, 0, 0, 0)m = 0, k0 = (±1, 0, 0, 1)

Rotations about z-axis leave k0 invariant.Others → “infinite” spin.SO(2) abelian

Γ(φ) |ξ〉 = eiφξ |ξ〉 ,

ξ = 0, 1/2, 1, . . .Dξ′,ξ = eiφξδξ′ξ


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Λk0 = (I +M)k0 = k0 →Mk0 = 0

M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

±1001

=

b3

−θ2 ± b1θ1 ± b2±b3

= 0

M =

0 ±θ2 ∓θ1 0±θ2 0 θ3 −θ2

∓θ1 −θ3 0 θ1

0 θ2 −θ1 0

↔ 0 θ3 −θ2

−θ3 0 θ1

0 0 0

Group ISO(2) of rotations and translations in R2.

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Λk0 = (I +M)k0 = k0 →Mk0 = 0

M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

±1001

=

b3

−θ2 ± b1θ1 ± b2±b3

= 0

M =

0 ±θ2 ∓θ1 0±θ2 0 θ3 −θ2

∓θ1 −θ3 0 θ1

0 θ2 −θ1 0

↔ 0 θ3 −θ2

−θ3 0 θ1

0 0 0


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Λk0 = (I +M)k0 = k0 →Mk0 = 0

M =

0 b1 b2 b3b1 0 θ3 −θ2

b2 −θ3 0 θ1

b3 θ2 −θ1 0

±1001

=

b3

−θ2 ± b1θ1 ± b2±b3

= 0

M =

0 ±θ2 ∓θ1 0±θ2 0 θ3 −θ2

∓θ1 −θ3 0 θ1

0 θ2 −θ1 0

↔ 0 θ3 −θ2

−θ3 0 θ1

0 0 0


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Helicity and Spin States

Dj0 representation of H:

exp [(θ − ib)J] = exp [(θ3J3 + (θ1 − iθ2)J1 + (θ2 + iθ1)J2]

= exp [(θ3J3 + (θ1 − iθ2)(J1 + iJ2)]= exp [(θ3J3 + (θ1 − iθ2)(J+)]

=

eijθ3 ∗ · · · ∗

0 ei(j−1)θ3 · · · ∗...

.... . .

...0 0 · · · e−ijθ3

Dj0 [(θ − ib)J]

∣∣∣j0j0⟩ = eijθ3∣∣∣j0j0⟩

Helicity state |ξ〉 for j = ξ > 0.

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exp [(θ − ib)J] = exp [(θ3J3 + (θ1 − iθ2)J1 + (θ2 + iθ1)J2]= exp [(θ3J3 + (θ1 − iθ2)(J1 + iJ2)]

= exp [(θ3J3 + (θ1 − iθ2)(J+)]

=

eijθ3 ∗ · · · ∗

0 ei(j−1)θ3 · · · ∗...

.... . .

...0 0 · · · e−ijθ3

Dj0 [(θ − ib)J]

∣∣∣j0j0⟩ = eijθ3∣∣∣j0j0⟩


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Daniel J.Cross

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exp [(θ − ib)J] = exp [(θ3J3 + (θ1 − iθ2)J1 + (θ2 + iθ1)J2]= exp [(θ3J3 + (θ1 − iθ2)(J1 + iJ2)]= exp [(θ3J3 + (θ1 − iθ2)(J+)]

=

eijθ3 ∗ · · · ∗

0 ei(j−1)θ3 · · · ∗...

.... . .

...0 0 · · · e−ijθ3

Dj0 [(θ − ib)J]

∣∣∣j0j0⟩ = eijθ3∣∣∣j0j0⟩


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Daniel J.Cross

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=

eijθ3 ∗ · · · ∗

0 ei(j−1)θ3 · · · ∗...

.... . .

...0 0 · · · e−ijθ3

Dj0 [(θ − ib)J]∣∣∣j0j0⟩ = eijθ3

∣∣∣j0j0⟩Helicity state |ξ〉 for j = ξ > 0.

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=

eijθ3 ∗ · · · ∗

0 ei(j−1)θ3 · · · ∗...

.... . .

...0 0 · · · e−ijθ3

Dj0 [(θ − ib)J]

∣∣∣j0j0⟩ = eijθ3∣∣∣j0j0⟩


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Daniel J.Cross

Introduction



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Fin


D0j′ representation of H:

exp [(θ + ib)J] = exp [(θ3J3 + (θ1 + iθ2)(J−)]

=

eij′θ3 0 · · · 0∗ ei(j

′−1)θ3 · · · 0...

.... . .

...

∗ ∗ · · · e−ij′θ3

D0j′ [(θ + ib)J]

∣∣∣0 j′

0−j′⟩

= e−ijθ3∣∣∣0 j′

0−j′⟩

Helicity state |ξ〉 for −j′ = ξ < 0.

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Physical States

For physical state |ψ〉,⟨k, jm

00

∣∣ψ⟩ = 0, and⟨k, 0

0j′

−m′

∣∣∣ψ⟩ = 0,

when m 6= j and m′ 6= j′.

{J · k− jI

} ⟨k, jm

00

∣∣ψ⟩ = 0,

k · k = 0→ ||k||2 = k21 + k2

2 + k23 = k2

t so

{J · k− jIkt}⟨k, jm

00

∣∣ψ⟩ = 0F ↓

−i{J · ∇ − j ∂

∂tI

}ψjm(x) = 0

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Daniel J.Cross

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Physical States


00

∣∣ψ⟩ = 0, and⟨k, 0

0j′

−m′

∣∣∣ψ⟩ = 0,

when m 6= j and m′ 6= j′.{J · k− jI

} ⟨k, jm

00

∣∣ψ⟩ = 0,

k · k = 0→ ||k||2 = k21 + k2

2 + k23 = k2

t so


00

∣∣ψ⟩ = 0F ↓

−i{J · ∇ − j ∂

∂tI

}ψjm(x) = 0

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Physical States


00

∣∣ψ⟩ = 0, and⟨k, 0

0j′

−m′

∣∣∣ψ⟩ = 0,


} ⟨k, jm

00

∣∣ψ⟩ = 0,

k · k = 0→ ||k||2 = k21 + k2

2 + k23 = k2

t so


00

∣∣ψ⟩ = 0

F ↓

−i{J · ∇ − j ∂

∂tI

}ψjm(x) = 0

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Daniel J.Cross

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Physical States


00

∣∣ψ⟩ = 0, and⟨k, 0

0j′

−m′

∣∣∣ψ⟩ = 0,


} ⟨k, jm

00

∣∣ψ⟩ = 0,

k · k = 0→ ||k||2 = k21 + k2

2 + k23 = k2

t so


00

∣∣ψ⟩ = 0F ↓

−i{J · ∇ − j ∂

∂tI

}ψjm(x) = 0

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Daniel J.Cross

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Physical States

E&M spin j = 1 and

J1 =i√2

0 1 01 0 10 1 0

, J2 =1√2

0 1 0−1 0 10 −1 0

,

J3 = i

1 0 00 0 00 0 −1

.

∂3 − ∂t ∂−1 0∂+1 −∂t ∂−1

0 ∂+1 −∂3 − ∂t

ψ1

ψ0

ψ−1

= 0

∂±1 = ∓(∂x ± i∂y)/√

2.

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Daniel J.Cross

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Physical States

Cartesian basis:

Jx =

0 0 00 0 10 −1 0

, Jy =

0 0 −10 0 01 0 0

,

Jz =

0 1 0−1 0 00 0 0

−i∂t ∂z −∂y−∂z −i∂t ∂x∂y −∂x −i∂t

ψxψyψz

= 0,

ψ0 = ψz and ψ±1 = ∓(ψx ± iψy)/√

2.

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Faraday & Ampere-Maxwell Equations

∇×ψ = −i ∂∂tψ, ψ = (ψx, ψy, ψz)

ψ = B + iE→

∇× (E− iB) = −i ∂∂t

(E− iB)

∇×E = −∂B∂t

∇×B = +∂E∂t,

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Fin


∇×ψ = −i ∂∂tψ, ψ = (ψx, ψy, ψz)

ψ = B + iE→

∇× (E− iB) = −i ∂∂t

(E− iB)

∇×E = −∂B∂t

∇×B = +∂E∂t,

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Daniel J.Cross

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Fin


∇×ψ = −i ∂∂tψ, ψ = (ψx, ψy, ψz)

ψ = B + iE→

∇× (E− iB) = −i ∂∂t

(E− iB)

∇×E = −∂B∂t

∇×B = +∂E∂t,

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Daniel J.Cross

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References

Fin

Gauß Equations

For k0 state, only ψ1 = (−ψx + iψy)/√

2 6= 0and k · ψ1 = 0 or k · ψ1 = 0.

F {k · ψ1 = 0} → ∇ · ψ1 = ∇ · (B + iE) = 0.

∇ ·E = 0∇ ·B = 0

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Daniel J.Cross

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Fin

Gauß Equations

For k0 state, only ψ1 = (−ψx + iψy)/√

2 6= 0and k · ψ1 = 0 or k · ψ1 = 0.

F {k · ψ1 = 0} → ∇ · ψ1 = ∇ · (B + iE) = 0.

∇ ·E = 0∇ ·B = 0

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Daniel J.Cross

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Fin

References

Jairzinho Ramos and Robert Gilmore.Derivation of the source-free maxwell and gravitationalradiation equations by group theoretical methods.Int. J. Mod. Phys. D, 15(4):505–519, 2006,arXiv:gr-qc/0604023.

Robert Gilmore.Lie Groups, Physics, and Geometry.Cambridge, Cambridge, UK, 2008.

Steven Weinberg.Photons and gravitons in perturbation theory: Derivationof maxwell’s and einstein’s equations.Physical Review, 138(4B):988–1002, 1965.

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From Quantum Mechanics to Maxwell’s Equationsdcross/academics/groups/qm_em_talk.pdf · The Maxwell Equations References Fin Fields Quantum Field An object which is an arbitrary