7. Relativity Wave Equations and Field Theories 7.1. The Klein-Gordon Equation 7.2. Scalar Field Theory for Free Particles 7.3. The Dirac Equation and spin-1/2 Particles 7.4. Spinor Field Theory 7.5. Weyl and Marjorana Spinors 7.6. Particles of Spin 1 and Wave Equations in Curved Spacetime

Natural Units c = 1→[ L ] = [ T ]  = 1→[ E ] = [ T ]  1 Choosing [ E ] = MeV, we have [ M ] = [ P ] = [ E ] = MeV [ L ] = [ T ] = [ E ]  1 = (MeV)  1 c =  10 8 m/s h =  10  22 MeV  s 1 MeV =  10  13 J → T( s ) =  10  22 T( MeV  1 ) L( m ) =  10  13 L( MeV  1 ) M( kg ) =  10  30 M( MeV )

7.1. The Klein-Gordon Equation → Klein-Gordon equation D’Alembertian  μ  μ is a Lorentz scalar →  must be a Lorentz tensor Only case  = scalar considered.

Conserved Current Density Statistical interpretation of quantum mechanics requires existence of a conserved probability density . 4-current density: Equation of continuity: →  is conserved. However, j μ cannot be the 4-current probability density because is not positive definite.

Plane Wave Solutions Plane wave with 4-wavevector k = (k 0, k ): →  ϕ is a solution to the K-G eq only if → energy of the particle is E < 0 solutions are unphysical since the vaccuum is at E = 0. If E < 0 solutions are allowed, ground state is E → . However, E < 0 solutions are needed to form a complete set of basis functions. These problems are resolved through the concept of anti-particles.

Precursor to a Quantum Field Problems of the K-G eq can be fully resolved only by switching to QFT. Consistent 1-particle relativistic quantum theory does not exist ( particle- antiparticle pairs always emerge spontaneously at sufficiently high E ) Action for the K-G eq: Conjugate momenta : General solution:

where

The following aims to find a and c : (a) (b) Add: Subtract:

7.2. Scalar Field Theory for Free Particles Scalar field → particles are bosons.  Quantization is expressed by equal-time commutators : c kills anti-particle of E > 0.

Covariant Normalization changes the normalization of the 1-P eigenstates. Factor (2π) 3 2ω k in where For a system with exactly one “particle”, the states { | k  } are complete : | k  is a free particle state → 

→→ transforms like k 0  is a Lorentz covariant normalization.

Hamiltonian Ignoring the possible ambiguity that may arise from we write

→ Vacuum energy:

Normal Ordering Determination of structure of spacetime by matter distribution requires E 0 = 0. This is accomplished by writing H in normal ordering : … :, which means all creation operators are to the left of all annihilation operators. The technique should be applied to all “total” operators. Total number operator: j 0 is the net probability density so that net number of “particles” is conserved Some neutral particles are identical to their anti-particles, e.g., γ, π 0 … → Number of γ is not conserved ( necessary if they’re the quanta of EM fields )

Summary

7.3. The Dirac Equation and spin-1/2 Particles K-G eq. is 2 nd order in t partials → E < 0 solutions. Remedy: find eq. that’s 1 st order in t partials. Dirac’s choice: Correct 4-momentum →  must satisfy the K-G eq. → → Set of all linear combinations of   is called a Dirac algebra. It is a special case C(V 4 (1) ) of the Clifford algebra C(V n (s) ) [see Choquet]. E.g., C(V 1 (0) ) = C, C(V 2 (0) ) = quarternions, C(V 3 (3) ) = Pauli algebra

Standard representation: Pauli matrices: For j = ½, Angular momenta commutator Pauli algebra Spin ½ Dirac eq.:

Lorentz Covariance and Spin Transformation of the  Matrices : Dirac eq. is Lorentz covariant if  is a scalar.   are 4  4 matrices →  must be a 4  1 matrix → covariance not automatic. Components of γ matrices: ( γ μ ) αβ with μ = 0,1,2,3 and α, β = 1,2,3,4. Lorentz transformation:  Covariance →  →  →→ spinor

The Matrix S Infinitesimal transformation: ( No need to distinguish primed and unprimed indices in  since such information does not appear explicitly in S. ) → ( μ → λ )

→ → → Simplest ansatz: →

Generators Scalar  : Infinitesimal (inverse) Lorentz transformation:

Spinor  : → ½ (4 2  4) = 6 independent components Divided into 2 groups :

Spin ω 0 j ~ boost → K j = generator of boost along j-axis. ω i j ~ rotation→ J i = rotation generator in the j-k plane ( i j k cyclic ) →

→ J = total angular momentum, L = orbital angular momentum →  = spin ( ½)

Pauli-Lubanski 4-Vector A simple 4-vector description of the spin is the Pauli-Lubanski 4-vector since W μ is a pseudovector since ε μνλσ is a tensor density of weight +1.

Caution: T iμ a μ is the ith component of a 4-vector, but T i j a j is not. → has no tensorial meaning in Minkowski space. → Let a μ be a 4-vector with spatial part Then W 2 = W μ W μ is invariant → it can be evaluated in any convenient coordinate system. Consider W 2 acting on the rest frame of a plane wave with k μ = ( m, 0 ). → →  is the spin operator in the rest frame of the particle with eigenvalue s(s + 1).

Some Properties of the γ Matrices → → → → →

for Define( not a Lorentz invariant since γ is not a 4-vector ) Dirac equation: Pauli-Lubanski 4-vector: where Ex 7.6

Conjugate Wavefunction and the Dirac Action → → Conjugate wavefunction Conjugate eq. Dirac action

Probability Current →→ γ μ is not a 4-vector → need to show j μ is.

→ 4-vector. ~ probability density

Bilinear Covariants ( Tensors ) Tensor TypeBilinearsTransformation Scalar Pseudoscalar Vector Pseudovector Tensor Dirac algebra is spanned by 16 basis “numbers”, e.g., I, γ 5, γ μ, γ μ γ 5, σ μν.

Covariant Spin Polarization Spin 1/2 particle in its rest frame: n = unit vector along axis of spin polarization. → In a frame in which the particle is moving with momentum p = k, Since we have →

By symmetry, n can involve only n and k → Square: →→ b = +1 since n = n when k = 0.  → Rest frame: →

Plane Wave Solutions Free particle (plane wave) solution:spinor → →  → → →  → →

Dirac equation :

Interpretation Rest frame: Choose u ks and v ks to be the eigenstates of the spin Σ  n : → where Frame p μ = k μ : →

Charge Conjugation Charge conjugation : particle ↔ antiparticle Principle of minimal coupling : Charge conjugation:  →  C Conjugate equation → Set→  Standard representations → Exercise:

Massless Spin 1/2 Particles Dirac equation for massless particle: No rest frame → spin polarization specified by helicity Massless particle: → Plane wave  is eigenfunction of W  if it is an eigenfunction of γ 5. →

If  is a plane wave with wavevector k μ, then  R and  L are eigenfunctions of W  with eigenvalue ½ k μ and  ½ k μ, resp. m = 0 → → chiral projections γ 5 = chirality operator Only for massless particles do the chiral projections have definite helicities.

7.4. Spinor Field Theory Dirac eq: ρ  0 but k 0 =  ω k → 2 nd quantization needed for proper interpretation. Dirac action: Momentum conjugate to  i : Hamiltonian density : Hamiltonian :

2nd Quantization Normalization: → → → → Ex 7.4

Non-negativeness of H requires anticommutation relations: Using commutation relations leads to causality violation (operators with space-like separation would not commute)  removed by normal ordering.with

Field Operator Version Spatial parts of the Dirac wavefunctions should be anticommutating. ( Not positive-definite )

7.5. Weyl and Marjorana Spinors Weyl (Chiral) Representation ( for Massless Particles ) : →

Chiral Solutions Weyl representation: m = 0 → ( u and v are linearly dependent ) → E.g., k = ( 0,0,k )

General k → Writing we have Simplest solution: → → c = normalization constant.

Normalization For c real,

→

→ 

Setting →  orthonormality

Charge Conjugation The chiral solution are also related by charge conjugation:

2 nd Quantization Matrix form:Weyl representation: Dirac equation:

→ 

Weyl Spinors From Ex.7.9: m = 0 → Action: (  R and  L decoupled ) → Either  R or  L alone describes a self-consistent theory. Spinors in this reduced theory are called Weyl spinors. E.g., in a theory involving only  L, the operators are → there are only left-handed particles and right-handed antiparticles. For a theory involving only  R, the opposite is true. Both theories are entirely equivalent, physically as well as mathematically.

Majorana Spinors A spin ½ particle which is its own anti-particle can access only 2 of the 4 spinor states allowed by the Dirac equation. The field operator  M for such a particle is called a Majorana spinor. By definition If the particle is also massless, we have →→ → A, B  R, L Proof ?

For non-interacting massless particles, Weyl & Majorana spinors are equivalent. E.g. Corresponding action: This equivalence is broken in the presence of interactions.

7.6. Particles of Spin 1 and 2 Maxwell equations in a source-free region: → Proca equation: → → ( Klein-Gordon eq for A μ ) Consider the gauge transformation: Settinggives In this context, is called the Lorentz condition. A μ is said to be in the Lorentz gauge.

Spin Lorentz transformation: or Comparing withgives Hence, the “spin” part of the infinitesimal generators M  are simply the generators of the Lorentz transformation on x . The spin part of K i (  i ) denotes a boost (rotation):

Plane wave solution: polarization vector ε μ is a column matrix Lorentz condition → → For a massive particle, k μ = ( m, 0 ) in the rest frame. The Pauli- Lubanski vector becomes → →→

Secular equations for all three matrices  i are the same, i.e., so that The corresponding (orthonormalized) eigenvectors for  3 are ε 0 is not an acceptable solution since ( Lorentz condition violated )  form a basis for the polarization of the spin 1 particle. Wrt this basis, Σ  Σ is effectively a unit matrix.

For massless particles → → The last condition is satisfied by → polarization ε L is longitudinal → corresponding plane wave is pure gauge ( ~ A μ = 0 ) → photon has only 2 polarizations h =  1 → complications in quantization scheme → path integral

Gravitons Source free regions : Field eq. → To 1 st order in h  (x) : Field eq. →

Gauge Invariance g  symmetric → at most 10 independent components in h . 2 different h  are equivalent if they lead to the same g  in different coordinates. i.e.,  (gauge) transformations on g  that leave all physical properties unchanged. Gauge invariance further reduces number of independent components of h . Infinitesimal coordinate transformation:

Set To 1 st order terms in both h  and  , gauge transformation

Harmonic Gauge Condition Field eq. where → Gauge transformation:

 →harmonic gauge condition → to 1 st order in h  in harmonic gauge Field eq. →( massless K-G eq ) → graviton is massless ~ 0

Spin Field eq. Plane wave solution:   is a symmetric polarization tensor (at most 10 independent components) In the harmonic gauge, → Gauge condition → 4 constraints on   for transformation between harmonic gauges → E.g. with

→ → Writing   further reduce the number of independent components of   to 2. ~ helicity states of h =  2. → Gravition is massless, spin 2. Helicity states of h = 0,  1 correspond to purely gauge degrees of freedom and have no physical significance. Experimental proof of gravitational waves : binary pulsar No graviton has yet been detected.

7.7. Wave Equations in Curved Spacetime Scalar Field  = dimensionless coupling constant, R(x) = Ricci curvature scalar, Λ = 0. L field is the only choice that allows a dimensionless coupling constant. Note: but

Euler-Lagrange equations For the  degrees of freedom: Covariant Gauss’ theorem [see eq(A.23), appendix A.4] → Hence, the covariant Euler-Lagrange equations are → δ  = 0 on S

→ → There is no known physical principle that can be used to determine . Effects of spacetime curvature are too small for measurement of . The arbitrary case ξ = 0 is called minimal coupling. Conformal transformation :  (x) = arbitrary real function Conformal coupling: For ξ= 1/6 and m = 0, is invariant under a conformal transformation if

Vierbeins Let y a be local coordinates at point X with large-scale coordinates x μ = X μ. Spacetime is locally flat → g(y) = η. Transformation matrices : Orthogonality: Local inertial frames should vary smoothly from point to point. → For a fixed, e μ a (x) is a vector field specifying the y a axis at every point x. is called a vierbein, a tetrad, or a frame field. Likewise

Verbein e is a 2 nd rank tensor whose  and a indices are associated with g  (x) &  ab. The 16 components of e carry 2 kinds of information. 10 components specifying g . 6 components specifying boost & rotation relating each local frame to a fixed reference Minkowkian frame. coordinate vector Lorentz vector

Spin Connection Parallel transport of a Lorentz vector such as = spin connection →

→ → → → (a) →

Vierbein, like g, is invariant under parallel transport Reminder: e serves the role of g that converts between a and  types of indices. → is the compatibility condition between ω & η. → magnitudes and angles be invariant under parallel transport: L.H.S. =→ To 1 st order in dx: →→ This can be shown to implyi.e., Γ = metric connection.

Spinors Parallel transport of a spinor : → Ω = 4  4 spin matrix Scalar field: To 1 st order in dx : S is invariant →

Lorentz vector : → →

Consider ansatzfor → → → →

→is automatically satisfied since ω is real →

Dirac Equation  μ are defined only for inertial frames →   must be mediated by e.  Covariant Dirac equation is Set→ ( Covariant Clifford / Dirac algebra ) Covariant action : There’s no field term because the coupling constant cannot be dimensionless. E.g., a term like requires  to have the dimension of length.

Vacuum State Problem Concept of the vacuum is problematic even in Minkowkian spacetime: The vacuum state, which by definition contains no particle to an inertial observer, will appear to an accelerating observer as a thermal bath of particles with temperature proportional to the acceleration. We’ll demonstrate it for the special case of a massless 2-D Hermitian scalar field. A complete proof of this statement is rather involved.

Rindler Coordinates Massless spin 0 particles in a 2-D Minkowskian spacetime. Inertial frame: (t, x)Rindler coordinates: (η,ξ) α > 0 →→for x > 0Rindler wedge Inverse: →

→ → For an observer at fixed . The eq of his worldline in an inertial frame is →  His velocity in the inertial frame is His acceleration: →→

Let inertial frame S be moving with velocity v with respect to S. Then If S coincides instantaneously with the rest frame of the particle, then v = u, ( see Ex.2.2 ) → a p is the proper acceleration of the observer. Proper time of the observer is obtained by setting dξ= 0. →→ With a proper choice of coordinate origins, we have

2  D Massless Hermitian Scalar Field  (x, t) Klein-Gordon equation inside the Rindler wedge : Plane wave solution : →

Expanding in terms of  k (ξ,η), → Caution: b k and b k + are confined in the Rindler wedge but not a q and a q +. In general, transformation that relates different sets of solutions to a wave equation is called a Bogoliubov transformation.

Average Particle Numbers | 0  = vacumm state in Minkowskian spacetime → = number of particles with momentum between k and k+dk as seen by a Rindler observer. Problem: If volume → , then N(k,k) →  even if the density of particles is finite. Remedy: Use N(k,k) then take limit k → k. Involved manipulation →

= Bose-Einstein occupation number with Consider the observer at fixed  = ξ 0. His proper time is A positive energy plane wave is which represents a particle with energy → → QED