2 ω. 1 α 1. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
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1 α 2 ω 2 ω 1 α 1 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
2 Berry Curvature, Spin, and Anomalous Velocity Michael Stone Institute for Condensed Matter Theory University of Illinois Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
3 Talk based on: Motivation M.A.Stephanov, Y.Yin, Chiral Kinetic Theory, Phys. Rev. Lett (2012). Our Work MS, V.Dwivedi, A Classical Version of the Non-Abelian Gauge Anomaly Phys. Rev. D (2013). V.Dwivedi, MS, Classical chiral kinetic theory and anomalies in even space-time dimensions, J. Phys. A (2014). MS, V.Dwivedi, T.Zhou, Berry Phase, Lorentz Covariance, and Anomalous Velocity for Dirac and Weyl Particles, arxiv: Also important J.Y.Chen, D.T.Son, M.A.Stephanov, H.U.Yee, Y.Yin, Lorentz Invariance in Chiral Kinetic Theory, arxiv: Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
4 Bruno Zumino Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
5 Outline 1 Covariant Berry Connection Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski 2 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor 3 Massless Case A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations 4 Conclusions Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
6 Covariant Berry Connection Outline 1 Covariant Berry Connection Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski 2 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor 3 Massless Case A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations 4 Conclusions Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
7 Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle anomalous velocity: k = ε(k,x) + e(ẋ B), x ε(k, x) ẋ = ( k Ω). k Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
8 Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle anomalous velocity: k = ε(k,x) + e(ẋ B), x ε(k, x) ẋ = ( k Ω). k Many applications! Want to use for Dirac and Weyl particles Can we make these equations covariant? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
9 Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle anomalous velocity: k = ε(k,x) + e(ẋ B), x ε(k, x) ẋ = ( k Ω). k Many applications! Want to use for Dirac and Weyl particles Can we make these equations covariant? k = e(e + ẋ B) k µ = ef µν ẋ ν, µ = 0,1,2,3. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
10 Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle anomalous velocity: k = ε(k,x) + e(ẋ B), x ε(k, x) ẋ = ( k Ω). k Many applications! Want to use for Dirac and Weyl particles Can we make these equations covariant? k = e(e + ẋ B) k µ = ef µν ẋ ν, µ = 0,1,2,3. ẋ = v ε ( k Ω) ẋ i = v i,ε + Ω ij kj, i = 1,2,3.? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
11 Covariant Berry Connection WKB and Berry Covariant WKB for Dirac Look for WKB solution of Dirac equation as where (i γ µ ( µ + iea µ / ) m)ψ = 0. ψ(x) = a(x)e iϕ(x)/, a = a 0 + a a , a 0 (x) = u α (k(x))c α (x) and u α (k) (and later v α (k) ) are solutions to covariantly normalized so that (γ µ k µ m)u α (k) = 0 (γ µ k µ + m)v α (k) = 0 ū α u β = δ αβ = v α v β Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
12 Covariant Berry Connection WKB and Berry Spin Transport Equation Plug WKB solution into Dirac. Find that ( [δ αβ V µ x µ + 1 V µ 2 x µ ) + ie 2m Sµν αβ F µν ia αβ,ν kν ] C β (x) = 0. where gives Larmor precession, and ie 2m Sµν αβ F µν a αβ,ν =iū α u β k ν, ν = 0,1,2,3 is an unconventional, but covariant Berry connection. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
13 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Covariant Berry Curvature Matrix-valued connection form Curvature form Use Dirac equation to find a αβ,ν dk ν =iū α u β k ν dkν. F = da ia 2. where F αβ = 1 2m 2(S µν) αβ dk µ dk ν, ( ) i (S µν ) αβ = ū α 4 [γ µ,γ ν ] u β = iū α σ µν u β. Note that Dirac k µ S µν = 0. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
14 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Pauli-Lubanski Tensor Use mass-shell condition E 2 k 2 0 = k2 + m 2 to eliminate k 0 and find that F αβ = 1 ( 2m 2 S ij k ) i E S k j 0j S i0 E αβ dk i dk j, Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
15 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Pauli-Lubanski Tensor Use mass-shell condition E 2 k 2 0 = k2 + m 2 to eliminate k 0 and find that F αβ = 1 ( 2m 2 S ij k ) i E S k j 0j S i0 E αβ dk i dk j, Expression in parentheses is a skew-symmetric tensor generalization of the Pauli-Lubanski vector Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
16 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have F = 1 { ( 1 2m 2 σ + γ 2 (k σ)k m 2 (1 + γ) )} (dk dk). Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
17 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have F = 1 { ( 1 2m 2 σ + γ 2 What does this mean this physically? (k σ)k m 2 (1 + γ) )} (dk dk). Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
18 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have F = 1 { ( 1 2m 2 σ + γ 2 (k σ)k m 2 (1 + γ) )} (dk dk). What does this mean this physically? Look at connection a αβ,i ki 1 = m 2 (1 + γ) (k k) ( σ ) 2 γ 2 = 1 + γ (β β) ( σ ) 2, αβ ( σ ) = ω Thomas. 2 αβ αβ β = k/e = k/mγ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
19 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have F = 1 { ( 1 2m 2 σ + γ 2 (k σ)k m 2 (1 + γ) )} (dk dk). What does this mean this physically? Look at connection a αβ,i ki 1 = m 2 (1 + γ) (k k) ( σ ) 2 γ 2 = 1 + γ (β β) ( σ ) 2, αβ ( σ ) = ω Thomas. 2 αβ αβ β = k/e = k/mγ Covariant Berry-transport is Thomas precession Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
20 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Nishina, Thomas, Hund Yoshio Nishina, Llewellyn Thomas, Friedrich Hund Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
21 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Thomas versus Lobachevsky Thomas precession is parallel transport on the positive-energy mass-shell: Z P R Q R X Embedding of three-dimensional Lobachevsky space into four-dimensional Minkowski space. The arrow shows the sterographic parametrization of the embedded space by the Poincaré ball x x2 2 + x2 3 < R2. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
22 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Non-covariant WKB With u αu β = δ αβ = v αv β, have {δ αβ ( t + v + 12 divv ) + N αβ } C β (x,t) = 0, with ( ) { ( e 1 N αβ = i mγ 2 B σ + 1 ) } (k σ)k 2 m 2 ia γ + 1 ki αβ,i, αβ and A αβ,i =iu u β α k i, i = 1,2,3 { ( F αβ = 1 2m 2 γ 3 σ + 1 ) } (k σ)k m 2 (dk dk). γ + 1 αβ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
23 Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Non-covariant WKB With u αu β = δ αβ = v αv β, have {δ αβ ( t + v + 12 divv ) + N αβ } C β (x,t) = 0, with ( ) { ( e 1 N αβ = i mγ 2 B σ + 1 ) } (k σ)k 2 m 2 ia γ + 1 ki αβ,i, αβ and A αβ,i =iu u β α k i, i = 1,2,3 { ( F αβ = 1 2m 2 γ 3 σ + 1 ) } (k σ)k m 2 (dk dk). γ + 1 αβ Berry curvature has opposite sign! Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
24 Relativistic Mechanics of Spinning Particles Outline 1 Covariant Berry Connection Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski 2 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor 3 Massless Case A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations 4 Conclusions Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
25 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Classical action for spinning particle in GR Let λ be a Lorentz transformation in Dirac representation, e µ a and e a µ a frame and co-frame, and define k a = tr {κλ 1 γ a λ}, κ = κ a γ a S ab = tr{σλ 1 σ ab λ}, Σ = 1 2 Σab σ ab where [κ,σ] = 0, so that k a S ab = 0 (Tulczyjew-Dixon condition) Action {ka S[x,λ] = e a µ dx µ tr{σ λ 1 (d + ω)λ} }. where is spin connection one-form. ω = 1 2 σ ab ω ab µdx µ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
26 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Mathisson-Papapetrou-Dixon equations Varying x µ gives us Dk c Dτ S abr ab cdẋ d = 0 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
27 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Mathisson-Papapetrou-Dixon equations Varying x µ gives us Dk c Dτ S abr ab cdẋ d = 0 Varying λ gives DS ab Dτ + ẋ ak b k a ẋ b = 0 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
28 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Mathisson-Papapetrou-Dixon equations Varying x µ gives us Dk c Dτ S abr ab cdẋ d = 0 Varying λ gives DS ab Dτ + ẋ ak b k a ẋ b = 0 Need additional condition such as k a S ab = or n a S ab = 0 for closed system. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
29 Relativistic Mechanics of Spinning Particles Anomalous velocity Anomalous velocity Use k a S ab = 0 to get Dka Dτ S ab = k 2 ẋ b k b (ẋ k). or ẋ a = 1 ( Dk c ) m 2 k a (ẋ k) + S ac. Dτ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
30 Relativistic Mechanics of Spinning Particles Anomalous velocity Anomalous velocity Use k a S ab = 0 to get or Dka Dτ S ab = k 2 ẋ b k b (ẋ k). ẋ a = 1 ( Dk c ) m 2 k a (ẋ k) + S ac. Dτ Chose time so that ẋ 0 = 1, then 1 = 1 { Dk c } m 2 (ẋ k)e + S 0c, Dt Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
31 Relativistic Mechanics of Spinning Particles Anomalous velocity Anomalous velocity Use k a S ab = 0 to get or Dka Dτ S ab = k 2 ẋ b k b (ẋ k). ẋ a = 1 ( Dk c ) m 2 k a (ẋ k) + S ac. Dτ Chose time so that ẋ 0 = 1, then 1 = 1 { Dk c } m 2 (ẋ k)e + S 0c, Dt Eliminate k 0, then ẋ i = k i E + 1 ( k j m 2 S ij S i0 E k ) i Dk j E S 0j Dt Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
32 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor Lab frame energy centroid XL i : { } T 00 d 3 x XL i = x 0 =t x 0 =t x i T 00 d 3 x. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
33 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor Lab frame energy centroid XL i : { } T 00 d 3 x XL i = x 0 =t x 0 =t x i T 00 d 3 x. Angular momentum about x µ A : M µν A = { (x µ x µ A )T ν0 (x ν x ν A)T µ0} d 3 x x 0 =t Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
34 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor Lab frame energy centroid XL i : { } T 00 d 3 x XL i = x 0 =t x 0 =t x i T 00 d 3 x. Angular momentum about x µ A : M µν A = { (x µ x µ A )T ν0 (x ν x ν A)T µ0} d 3 x Therefore M i0 A = x 0 =t x 0 =t = (X i L x i A)E. { (x i x i A)T 00 (x 0 x 0 A)T i0} d 3 x Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
35 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor M i0 = 0 when x i A = Xi L, meaning that angular momentum is about lab-frame energy centroid. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
36 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor M i0 = 0 when x i A = Xi L, meaning that angular momentum is about lab-frame energy centroid. n a M ab = 0 for angular momentum about centroid in frame where n a = (1,0,0,0). Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
37 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor M i0 = 0 when x i A = Xi L, meaning that angular momentum is about lab-frame energy centroid. n a M ab = 0 for angular momentum about centroid in frame where n a = (1,0,0,0). Thus k a S ab = 0 implies that S ab is the intrinsic angular momentum, meaning angular momentum about energy centroid in rest frame where k a = (m,0,0,0). Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
38 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor M i0 = 0 when x i A = Xi L, meaning that angular momentum is about lab-frame energy centroid. n a M ab = 0 for angular momentum about centroid in frame where n a = (1,0,0,0). Thus k a S ab = 0 implies that S ab is the intrinsic angular momentum, meaning angular momentum about energy centroid in rest frame where k a = (m,0,0,0). k a S ab = 0 implies that x µ (τ) in the M-P-D equation is trajectory of centre of mass i.e. energy centroid in rest frame. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
39 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Meaning of conditions on spin tensor M i0 = 0 when x i A = Xi L, meaning that angular momentum is about lab-frame energy centroid. n a M ab = 0 for angular momentum about centroid in frame where n a = (1,0,0,0). Thus k a S ab = 0 implies that S ab is the intrinsic angular momentum, meaning angular momentum about energy centroid in rest frame where k a = (m,0,0,0). k a S ab = 0 implies that x µ (τ) in the M-P-D equation is trajectory of centre of mass i.e. energy centroid in rest frame. Also see that Pauli-Lubansky Berry curvature S µν S µ0 k ν E k µ E S 0ν is angular momentum about lab-frame centroid. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
40 Relativistic Mechanics of Spinning Particles Meaning of Conditions on Spin Tensor Myron Mathisson explaining Spin Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
41 Massless Case Outline 1 Covariant Berry Connection Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski 2 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor 3 Massless Case A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations 4 Conclusions Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
42 Massless Case A Gauge Invariance? Massless case When m 2 = 0 bad things happen! Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
43 Massless Case A Gauge Invariance? Massless case When m 2 = 0 bad things happen! Suppose that k 2 = 0, and S ab satisfies S ab k b = 0, then S ab = S ab + (k a S pb k b S pa )Θ p still satisfies S ab k b = 0. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
44 Massless Case A Gauge Invariance? Massless case When m 2 = 0 bad things happen! Suppose that k 2 = 0, and S ab satisfies S ab k b = 0, then still satisfies S ab k b = 0. S ab = S ab + (k a S pb k b S pa )Θ p If S ab and x a satisfy M-P-D equation for k a = 0, and x a = x a + S pa Θ p, then S ab, x a are also a solution of M-P-D for any time-dependent Θ p (τ). Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
45 Massless Case A Gauge Invariance? Massless case When m 2 = 0 bad things happen! Suppose that k 2 = 0, and S ab satisfies S ab k b = 0, then still satisfies S ab k b = 0. S ab = S ab + (k a S pb k b S pa )Θ p If S ab and x a satisfy M-P-D equation for k a = 0, and x a = x a + S pa Θ p, then S ab, x a are also a solution of M-P-D for any time-dependent Θ p (τ). A gauge invariance? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
46 Massless Case Wigner Translations Wigner Translations Massless reference momentum κ a = (1,0,0,...,0,1). Little group: σ ab with 0 < a,b,< d 1. Generate SO(d 2), together with translations π a = κ b σ ba σ 0a + σ (d 1)a, 0 < a < d 1. [π a,π b ] = 0, [σ ab,π c ] = η bc π a η ac π b. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
47 Massless Case Wigner Translations Wigner Translations Massless reference momentum κ a = (1,0,0,...,0,1). Little group: σ ab with 0 < a,b,< d 1. Generate SO(d 2), together with translations π a = κ b σ ba σ 0a + σ (d 1)a, 0 < a < d 1. [π a,π b ] = 0, [σ ab,π c ] = η bc π a η ac π b. Wigner says that the π a must have no physical effect... Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
48 Massless Case Wigner Translations Wigner Translations Massless reference momentum κ a = (1,0,0,...,0,1). Little group: σ ab with 0 < a,b,< d 1. Generate SO(d 2), together with translations π a = κ b σ ba σ 0a + σ (d 1)a, 0 < a < d 1. [π a,π b ] = 0, [σ ab,π c ] = η bc π a η ac π b. Wigner says that the π a must have no physical effect......but ( d 2 ) λ λexp θ i π i, in S ab = tr{σλ 1 σ ab λ}, i=1 takes S ab S ab + (k a S pb k b S pa )Θ p, Θ p = Λ p iθ i. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
49 Massless Case Wigner Translations Heisenberg, Wigner Heisenberg and Eugene Wigner Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
50 Massless Case Physical Meaning of Wigner Translations Physical Meaning of Wigner Translations S S Head-on collision of massless spinning particles. L = 0, S = 0 J = 0. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
51 Massless Case Physical Meaning of Wigner Translations Physical Meaning of Wigner Translations S S Run towards collision, top view. J = 0, S 0 L 0. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
52 Massless Case Physical Meaning of Wigner Translations Physical Meaning of Wigner Translations S Miss! S Boost towards collision, front view. Miss by δx = L/k Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
53 Massless Case Physical Meaning of Wigner Translations Huh! It s not that weird: Any interaction that occurs in one frame still occurs when viewed from another frame. Cross-sections depend on J = L + S. For massless particles, cannot separate L from S. Means that particle position is frame dependent. A serious problem for any covariant mechanics! Show some Mathematica TM plots to prove that frame dependence is a real phenomenon Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
54 Conclusions Outline 1 Covariant Berry Connection Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski 2 Relativistic Mechanics of Spinning Particles Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor 3 Massless Case A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations 4 Conclusions Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
55 Conclusions Conclusions For massive particles, the Berry-phase equations of motion for relativistic spinning particles are the 3-dimensional reduction of 3+1 Lorentz covariant equations The Berry phase equations of motion for massless particles are not the m 0 limit of the massive-particle equations The Berry phase equations of motion for massless particles are not the 3-dimensional reduction of covariant equations The lack of covariance arises because the position ascribed to a massless particle is the lab-frame centroid, and is frame-dependent Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th
arxiv: v2 [hep-th] 16 Jun 2014
Berry Phase, Lorentz Covariance, and Anomalous Velocity for Dirac and Weyl Particles MICHAEL STONE University of Illinois, Department of Physics arxiv:1406.0354v2 [hep-th] 16 Jun 2014 1110 W. Green St.
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