The Lorentz and Poincaré Groups in Relativistic Field Theory

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Lorentz Group

Poincar´e Group

The Lorentz and Poincar´eGroups in Relativistic Field Theory

Term Project

NicolásFernándezGonzález

University of California – Santa Cruz

June 2015

1 / 14 Using our group theory language the Lorentz group is just the indeﬁnite orthogonal group O(1, 3)

Lorentz Group Lorentz Group Generators of the Lorentz Group Our ﬁrst encounter with the Lorentz group is in special relativity it Boost and Rotations composed of the transformations that preserve the line element in Lie Algebra 2 µ ν µ 0µ µ ν of the Minkowski space (s = ηµν x x ). In particular, for x → x = Λν x , Lorentz Group we get: Poincar´e Λµη Λν = η Group ρ µν σ ρσ Or in matrix notation Λ satisfy the following relation:

Λ|ηΛ = η =⇒ O|ηO = η

Where η is the matrix:

η = diag(1, −1, −1, −1)

2 / 14 Lorentz Group

Λ|ηΛ = η =⇒ O|ηO = η

Where η is the matrix:

η = diag(1, −1, −1, −1)

Using our group theory language the Lorentz group is just the indeﬁnite orthogonal group O(1, 3)

2 / 14 There are two discrete transformations P and T

↑ ↑ ↓ ↓ L = L+ ∪ L− ∪ L+ ∪ L− ↑ ↑ ↑ ↑ = L+ ∪ P L+ ∪ T L+ ∪ PT L+

↑ We get the decomposition of L into cosets of the restricted group L+.

4 Islands, 2 Boats

Lorentz Group Lorentz Group The Lorentz group consists of four separated components: Generators of the Lorentz Group ↑ 0 Boost and L : det Λ = 1 and Λ ≥ 1. Rotations + 0 Lie Algebra of the ↑ 0 Lorentz L : det Λ = −1 and Λ ≥ 1. Group − 0 Poincar´e Group ↓ 0 L+: det Λ = 1 and Λ0 ≤ −1.

↓ 0 L−: det Λ = −1 and Λ0 ≤ −1.

The subgroup of the Lorentz group that exclude spatial reﬂections and time reversal is called the proper orthochronous Lorentz group SO+(1, 3).

3 / 14 4 Islands, 2 Boats

↓ 0 L−: det Λ = −1 and Λ0 ≤ −1.

The subgroup of the Lorentz group that exclude spatial reﬂections and time reversal is called the proper orthochronous Lorentz group SO+(1, 3). There are two discrete transformations P and T

↑ ↑ ↓ ↓ L = L+ ∪ L− ∪ L+ ∪ L− ↑ ↑ ↑ ↑ = L+ ∪ P L+ ∪ T L+ ∪ PT L+

↑ We get the decomposition of L into cosets of the restricted group L+. 3 / 14 Peter Woit has a nice blog entry about this: ”...physicists dont appreciate the mathematical concept of real structure or complexiﬁcation.” and ”...pseudo-Majorana spinors a nebulous concept best kept undisturbed.” Why not both? UCSC does not discriminate against metrics.

Link: http://www.math.columbia.edu/~woit/wordpress/

The West Coast or East Coast?

4 / 14 Why not both? UCSC does not discriminate against metrics.

Link: http://www.math.columbia.edu/~woit/wordpress/

The West Coast or East Coast?

Lorentz Group Lorentz Usually we have two possible choices for the Minkowski metric: Group Generators of the Lorentz Group West Coast metric (1, −1, −1, −1). Feynman’s favorite Boost and Rotations Lie Algebra East Coast metric (−1, 1, 1, 1). The choice of Weinberg and of the Lorentz Schwinger (also Ross and Joey?) Group Poincar´e Group Peter Woit has a nice blog entry about this: ”...physicists dont appreciate the mathematical concept of real structure or complexiﬁcation.” and ”...pseudo-Majorana spinors a nebulous concept best kept undisturbed.”

4 / 14 Link: http://www.math.columbia.edu/~woit/wordpress/

The West Coast or East Coast?

4 / 14 The West Coast or East Coast?

Link: http://www.math.columbia.edu/ woit/wordpress/ 4 / 14 ~ We already know these six parameters: 3 rotations about orthogonal directions in space and 3 boosts along these same directions.

Generators of the Lorentz group

Lorentz Group Lorentz Group µ Generators of An infinitesimally Lorentz transformation Λν should be of form, the Lorentz Group Boost and µ µ µ Rotations Λν = δν + ων Lie Algebra of the Lorentz µ Group Where with ων is a matrix of infinitesimal coefficients. Plugging in our 2 Poincaré definition of Lorentz group and keeping O(ω ), we have: Group µ ν ηρσ = Λρ ηµν Λσ µ µ ν ν ηρσ = δρ + ωρ ηµν (δσ + ωσ)

ηρσ = ηρσ + ωρσ + ωσρ

0 = ωρσ + ωσρ

We conclude that ωρσ = −ωσρ Thus the generators ω are 4 × 4 antisymmetric matrices characterized by six parameters so that there are six generators.

5 / 14 Generators of the Lorentz group

ηρσ = ηρσ + ωρσ + ωσρ

0 = ωρσ + ωσρ

We conclude that ωρσ = −ωσρ Thus the generators ω are 4 × 4 antisymmetric matrices characterized by six parameters so that there are six generators. We already know these six parameters: 3 rotations about orthogonal directions in space and 3 boosts along these same directions.

5 / 14 Inﬁnitesimal rotation for x,y and z:

0  0  0   0   0 1  0 −1  J1 = i   J2 = i   J3 = i    0 −1  0   1 0  1 0 −1 0 0

Inﬁnitesimal boost:

0 −1   0 −1   0 −1 1 0   0   0  K1 = i   K2 = i   K3 = i    0  −1 0   0  0 0 −1 0

These matrices are the generators of the Lorentz Group (4-vector basis)

Boost and Rotations

6 / 14 Inﬁnitesimal boost:

0 −1   0 −1   0 −1 1 0   0   0  K1 = i   K2 = i   K3 = i    0  −1 0   0  0 0 −1 0

These matrices are the generators of the Lorentz Group (4-vector basis)

Boost and Rotations

Lorentz Group Lorentz Group Generators of the Lorentz The rotations can be parametrized by a 3-component vector θi with Group Boost and |θi| ≤ π, and the boosts by a three component vector βi (rapidity) Rotations Lie Algebra with |β| < ∞. Taking a infinitesimal transformation we have that: of the Lorentz Group Infinitesimal rotation for x,y and z: Poincaré Group 0  0  0   0   0 1  0 −1  J1 = i   J2 = i   J3 = i    0 −1  0   1 0  1 0 −1 0 0

6 / 14 Boost and Rotations

Inﬁnitesimal boost:

0 −1   0 −1   0 −1 1 0   0   0  K1 = i   K2 = i   K3 = i    0  −1 0   0  0 0 −1 0

These matrices are the generators of the Lorentz Group (4-vector basis)

6 / 14 We can be even more eﬃcient writing as a rank-2 tensor representation

[Vµν ,Vρσ] = i (Vµσηνρ + Vνρηµσ − Vµρηνσ − Vνσηµρ) .

Where Vµν is   0 K1 K2 K3 −K1 0 J3 −J2 Vµν =   −K2 −J3 0 J1  −K3 J2 −J1 0 So now we can deﬁne the Lorentz group as the set of transformations generated by these generators.

Lie Algebra of the Lorentz Group

Lorentz Group Lorentz Group Having the generation of any representation we can know the Lie Generators of the Lorentz Group algebra of the Lorentz Group. Boost and Rotations Lie Algebra [J ,J ] = i J of the i j ijk k Lorentz Group [Ji,Kj] = iijkKk Poincar´e Group [Ki,Kj] = −iijkJk

7 / 14 Lie Algebra of the Lorentz Group

We can be even more eﬃcient writing as a rank-2 tensor representation

[Vµν ,Vρσ] = i (Vµσηνρ + Vνρηµσ − Vµρηνσ − Vνσηµρ) .

7 / 14 Which satisfy the commutation relations,

+ + + Ji ,Jj = iijkJk − − − Ji ,Jj = iijkJk + − Ji ,Jj = 0 Black magic! We now see that the algebra of these generators decouple into two SU(2) algebras.

Generators of the Lorentz Group

Lorentz Group Lorentz Group Generators of the Lorentz Group We can deﬁne a ”suspicious” combinations of these two sets of Boost and Rotations Lie Algebra generators as of the Lorentz Group + 1 − 1 Poincar´e Ji = (Ji + iKi) Ji = (Ji − iKi) Group 2 2

8 / 14 Generators of the Lorentz Group

+ + + Ji ,Jj = iijkJk − − − Ji ,Jj = iijkJk + − Ji ,Jj = 0 Black magic! We now see that the algebra of these generators decouple into two SU(2) algebras.

8 / 14 1 1 ( 2 , 0) ⊕ (0, 2 ) is a Dirac spinor

(1, 0) ⊕ (0, 1) is the electromagnetic ﬁeld tensor Fµν

We can label the representations of so(3, 1) by the weights (quantum numbers) of su(2) ⊗ su(2)

Representations of the Lorentz group

Lorentz Group Lorentz Group Generators of Representations of the Lie algebra of the Lorentz group the Lorentz + − Group j j Reps Dim Field Boost and Rotations 0 0 (0, 0) 1 Scalar or Singlet Lie Algebra of the 1 1 Lorentz 2 0 ( 2 , 0) 2 Left handed Weyl spinor Group 1 1 0 2 (0, 2 ) 2 Right handed Weyl spinor Poincaré 1 1 1 1 Group 2 2 ( 2 , 2 ) 4 Vector 1 0 (1, 0) 3 Self-dual 2-form field 0 1 (0, 1) 3 Anti-self-dual 2-form field 1 1 (1, 1) 9 Traceless symmetric tensor field.

9 / 14 Representations of the Lorentz group

1 1 ( 2 , 0) ⊕ (0, 2 ) is a Dirac spinor

(1, 0) ⊕ (0, 1) is the electromagnetic ﬁeld tensor Fµν

We can label the representations of so(3, 1) by the weights (quantum numbers) of su(2) ⊗ su(2) 9 / 14 It all comes from the factor of i associated with boosts. Or in other words, The precise relationship between the two groups are that the complex linear combinations (complexiﬁcation) of the generators of the Lorentz algebra to SU(2) × SU(2).

We have a problem

10 / 14 We have a problem

Lorentz Group Lorentz Group Generators of the Lorentz Group Boost and Rotations Lie Algebra of the Lorentz Group Lorentz group are not unitary and that is a deal braking for Physics Poincar´e Group because we really really like unitary for Quantum Field Theory. Where we were wrong? It all comes from the factor of i associated with boosts. Or in other words, The precise relationship between the two groups are that the complex linear combinations (complexiﬁcation) of the generators of the Lorentz algebra to SU(2) × SU(2).

10 / 14 Poincar´eGroup

Lorentz Group

Poincar´e Group Casimir Operators The Poincar´egroup, as the Lorentz group, preserve the line element in Minkowski space but now with the translations, : µ 0µ µ ν µ x → x = Λν x + a

(Λ1, a1) · (Λ2, a2) = (Λ2Λ1, Λ2a1 + a2) The Poincar´egroup has 10 parameters: 6 from a Lorentz and 4 from translations

 a0  a1   η(a, Λ) =  Λ a2    a3 0 0 0 0 1

11 / 14 Poincar´eGenerators

Lorentz Group

Poincar´e Group Casimir Operators

The Poincar´egroup is represented by the algebra

[Vµν ,Vρσ] = i (Vµσηνρ + Vνρηµσ − Vµρηνσ − Vνσηµρ)

[Pµ,Pν ] = 0 µ νσ ν µσ [Vµν ,Pσ] = i (P η − P η )

12 / 14 Representations of the Poincar´egroup

Lorentz Group

Poincaré Group Casimir Operators Wigner, on a paper 1939, gave us a method to classify all reducible representations of the Poincarégroup by considering the little group of a quantum eigenstate with definite momentum P 2. In the literature is called: Classification of representations by orbits.

P 0 P 2 P µ Little Group m m2 (m, 0, 0, 0) SO(3) −m m2 (−m, 0, 0, 0) SO(3) |P | 0 (|P |, 0, 0, |P |) E(2) −|P | 0 (−|P |, 0, 0, |P |) E(2) 0 −m2 (0, 0, 0, m) SO(2, 1) 0 0 (0, 0, 0, 0) SO(3, 1)

13 / 14 Casimir Operators

Lorentz Group

Poincaré Group Casimir Operators Finally, there are only two invariants in the Poincarégroup that commute with all generators. The first Casimir invariant C1 is associated with the mass invariance, and the second Casimir invariant C2 refers to spin invariance.

µ C1 = P Pµ µ C2 = WµW

Where Wµ is the Pauli-Lubanski (pseudo) vector deﬁned by 1 W = P ν V ρσ µ 2 µνρσ A particle with ﬁx energy | m, s; P µ, σi

14 / 14