A Semiclassical Kinetic Theory of DiracParticles and Thomas Precession

O. F. DAYI

Istanbul Technical University, Physics Department, Turkey

Selected Topics in Theoretical High Energy Physics,September 21-27, 2015, Tbilisi

with Eda Kilincarslan,

PLB 749, 2015

I We deal with the kinetic theory of Dirac fermions in theexternal electromagnetic fields within the matrix valueddifferential forms method. Based on the symplectic formderived by employing the semiclassical wave packet build ofthe positive energy solutions of the Dirac equation.

I There is a discrepancy between this non-relativisticformulation and relativistic formulations of the kinetic theoryof Dirac particles. In the former case there exists ananomalous velocity term which does not appear in the latter.We show that Thomas precession is the source of it.

I Thomas precession correction can be studied straightforwardlywithin our approach. It contributes on an equal footing withthe Berry gauge fields. In fact in equations of motion iteliminates the terms arising from the Berry gauge fieldsyielding the anomalous velocity term.

Dirac particle interacting with the external electromagnetic fieldsE, B, whose vector and scalar potentials are a(x) and a0(x), isdescribed by

H(p − ea(x)) = βm + α · (p − ea(x)) + ea0(x).

Semiclassical wave packet composed of the positive energysolutions of the Dirac equation.

ψx(pc , xc) =∑α

ξαuα(pc , xc)e−ipc ·x/~.

Center of the wave packet (xc ,pc) coincides with the center ofmass. For simplicity we deal with constant ξα coefficients.The one-form η is defined as∫

[dx ]δ(xc − x)Ψ†x

(−i~d − H0Ddt) Ψx =∑αβ

ξ∗αηαβξβ.

H0D is the block diagonal Hamiltonian which should be derived.

Simple calculations yield

ηαβ = −δαβxc · dpc − aαβ · dxc − Aαβ · dpc − Hαβ0Ddt.

We introduced the matrix valued Berry gauge fields

aαβ = i~u†(α)(pc , xc)∂

∂xcu(β)(pc , xc),

Aαβ = i~u†(α)(pc , xc)∂

∂pcu(β)(pc , xc).

Symplectic two-form is defined as

ω = dη − i

~η ∧ η.

Recall that it is a matrix in spin indices.Derivations do not depend on dimension.

Substitute p → p + ea(x), in the Hamiltonian and consider thefree particle solutions with E =

√p2 + m2.

Now, the positive energy solutions will not possess x dependence.Rename (xc ,pc)→ (x ,p) and set aαβ = 0,

η = pidx i + eaidx i − Aidpi − Hdt.

The repeated indices are summed over. We suppress the matrixindices α, β, and do not write explicitly the unit matrix.The Berry gauge fields calculated

A = −~ σ × p

2E (E + m).

Define G in terms of the Berry curvature:

Gij =

(∂Aj

∂pi− ∂Ai

∂pj− i [Ai ,Aj ]

)= εijkGk .

To obtain the kinetic theory we need velocities of the phase spacevariables. From the equations of motion, applying the methoddevoleped in (Dayi-Elbistan, arXiv:1402.4727) we find

˙x ω1/2 = −f − ~em2E 3

(E × σ +

(E × p)(σ · p)

m(m + E )

)− eB(f · G ),

ω1/2˙p = eE − ef × B − ~e2m

2E 3

(σ +

p(σ · p)

m(m + E )

)(E · B).

Here fi = −∂HD(p)/∂pi − i/~[Ai ,HD(p)].ω1/2 is the Pfaffian of the two-form in phase space indices,obtained as:

ω1/2 = 1− e~m2E 3

(σ · B +

(p · B)(σ · p)

m(m + E )

),

G dependent term in ˙x ω1/2 is known as anomalous velocity.

Thomas Precession

The Lorentz boost λboost(v + dv) can be separated into twosuccessive Lorentz boosts accompanied by the rotation R(dθ)

λboost(v + dv) = R(dθ)λboost(dv)λboost(v). (1)

The infinitesimal angle of rotation is

dθ =γ2

γ + 1v × dv . (2)

γ is the Lorentz factor.In nonrelativistic equations of motion, this rotation yields anangular velocity known as Thomas precession.Let us recall it briefly considering a massive particle at rest att = 0, moving with v at t and moving with v + dv at t + dt :

Relativisticx =Lab frame coordinates at t = 0.x ′ = Co-moving frame coordinates with v at t.x ′′ = Co-moving frame coordinates with v + dv at t + dt.

x ′ = λboost(v)x , x ′′ = λboost(v + dv)x ,

x ′′ = R(dθ)λboost(dv)x ′.

𝜆𝑏𝑜𝑜𝑠𝑡(𝒗)

𝑥

𝒗+ 𝑑𝒗

𝒗

𝑑𝒗 𝑥′

Non-relativisticx =Lab frame coordinates at t = 0.x ′ = Co-moving frame coordinates with v at t.x ′′′ = Co-moving frame coordinates with v + dv at t + dt.

x ′ = λboost(v)x ,

x ′′′ = λboost(dv)x ′,

x ′′′ = R(−dθ)λboost(v + dv)x .

𝜆𝑏𝑜𝑜𝑠𝑡(𝒗)

𝑥

𝒗+ 𝑑𝒗

𝒗

𝑑𝒗 𝑥′

How this phenomena manifests itself in the semiclassical kinetictheory of Dirac particles?In the wave packet formalism one deals with the group velocity

v ≡ ∂E

∂p=

p

E. Lorentz factor : γ = E/m,

dθ =p × dp

m(E + m).

Lab and co-moving reference frames coincide at t = 0 when theparticle is at rest. Hence the solution of the Dirac equation inlaboratory frame at t = 0, is u(0). Our kinetic theory formulationis nonrelativistic, thus

[du(p)]NR = u′′′(p + dp)− u′(p)

= R(−dθ) λboost(v + dv)u(0)− λboost(v)u(0)

= R(−dθ)u(p + dp)− u(p).

Terms at order of p2, are ignored, thus write dv = dp/m.

Thus, taking into account Thomas precession, the Berry gaugefield related term in the η one-form will be

i~u†(p)[du(p)]NR = i~u†(p) [R(−dθ)u(p + dp)− u(p)]

The infinitesimal rotation of spinors is given by

R(dθ) = 1 +D(dθ)

where

D(dθ) =

(σ · dθ 0

0 σ · dθ

).

Then, keeping only the first order terms in dp,

i~u†(p)[du(p)]NR = i~u†(p)∂u(p)

∂p· dp + i~u†(p)D(−dθ)u(p).

The first term is the Berry gauge field calculated before. TheThomas correction term can be shown to be

i~u†(p)D(−dθ)u(p) =(~σ × p)

4m2· dp.

Ignoring the terms at the order of p2 the Berry gauge fields yieldthe same contribution, up to a minus sign.

Conclusion: When the Thomas correction is considered in thevelocities of phase space variables we set the Berry curvatures tozero: The anomalous velocity term disappears.This is in accord with the results obtained in relativisticformulations of Dirac particles.

Thank You!