Download Dirac fermions in arbitrary external classical fields: quantum spin

Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac fermions in arbitrary external classical
fields: quantum spin dynamics
Yuri N. Obukhov
Theoretical Physics Laboratory, IBRAE, Russian Academy of Sciences
Ginzburg Centennial Conference on Physics,
Lebedev Institute, Moscow, 02 June 2017
Based on joint results with O. Teryaev and A. Silenko (JINR)
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Outline
1
Relativistic particles in curved spacetimes
Dynamics of spin and equivalence principle
Geometry of spacetime
2
Spin 12 particle in curved spacetime
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
3
Classical and quantum spin: comparison
Classical spin in external fields
Spin in the gravitational field
4
Conclusions and Outlook
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dynamics of spin and equivalence principle
Geometry of spacetime
Dynamics of spin and equivalence principle
High-energy experiments take place in curved space or in
noninertial frame (for example, on Earth)
Equivalence principle (EP) - a cornerstone of gravity
Newton’s theory ⇒ Einstein’s “falling elevator”
Colella-Overhauser-Werner (1975) and Bonse-Wroblewski
experiments - EP for quantum-mechanical systems:
Measured phase shift due to inertial and gravitational force
Gravity on spin: EP for relativistic particles?
Classical theory of spin: Frenkel (1928), Mathisson (1937),
Papapetrou (1951), Weyssenhoff-Raabe (1947)
Compare classical rotator and quantum spin
Relativistic spin effects not measured yet!
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dynamics of spin and equivalence principle
Geometry of spacetime
Arbitrary Riemannian geometry in 4 dimensions
Let t be time, xa (a = 1, 2, 3) be spatial coordinates:
ds2 = V 2 c2 dt2 − δbabb W ba c W b d (dxc − K c cdt) (dxd − K d cdt)
b
V and K a , and 3 × 3 matrix W ba b depend arbitrarily on t, xa .
Their number 1 + 3 + 9 = 13 but rotation W ba b −→ Lba bc W bc b
is allowed with arbitrary Lba bc (t, x) ∈ SO(3): =⇒ 13 − 3 = 10
Coframe eαi with gαβ eαi eβj = gij , gαβ = diag(c2 , −1, −1, −1):
b
ei0 = V δi0 ,
ebai = W ba b δib − cK b δi0 ,
a = 1, 2, 3
Exact metric of flat spacetime in noninertial frame
V =1+
a·r
,
c2
W ba b = δba ,
Yuri N. Obukhov
1
K a = − (ω × r)a
c
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
Dirac particle in gravitational & electromagnetic field
be~
Fermion with moments (AMM µ0 = (g−2)e~
& EDM δ 0 = 2mc
)
4m
0
0
δ
µ
i~γ α Dα − mc + σ αβ Fαβ + σ αβ Gαβ ψ = 0
2c
2
Spinor covariant derivative (with σαβ = iγ[α γβ] )
ie
i
Dα = eiα Di ,
Di = ∂i − Ai + σ αβ Γi αβ
~
4
Connection for general spacetime geometry
c2 b
c
b
b
Γi bab0 =
W ba ∂b V ei 0 − Q(babb) ei b ,
V
V
c
b
Γi babb =
Q[babb] ei 0 + Cbabbbc + Cbabcbb + Cbcbbba ei bc
V
Here anholonomity Cbabb bc = W d ba W ebb ∂[d W bc e] and
1
Qbabb = gbabc W dbb Ẇ bc d + K e ∂e W bc d + W bc e ∂d K e
c
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
Dirac Hamiltonian
Naive Hamiltonian
is not Hermitian. Rescale wave function
√
1
2
ψ −→
ψ and recast Dirac wave equation into
−geb00
Schrodinger form i~ ∂ψ
∂t = Hψ
Dirac Hamiltonian (with F b a = V W b ba and π = −i~∇ − eA)
H = βmc2 V + eΦ + 2c πb F b a αa + αa F b a πb
+ 2c (K · π + π · K) +
~c
4
(Ξ · Σ − Υγ5 )
−βV (Σ · M + iα · P)
Here β =
b
γ0,
αa
=
b
γ 0 γ ba ,
γ5 =
Υ = V babbc Γbabbbc = −V babbc Cbabbbc ,
b
b
Yuri N. Obukhov
0
−1
−1
0
Ξba =
,Σ=
σ
0
0
σ
,
V
b
b
b Γbbbc = babbbc Qbbc
c babbc 0
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
Electrodynamics in curved spacetime
Gravity is universal: affects also electromagnetism. How?
Basic objects: field strength F , excitation H and current J
Maxwell’s theory – without coordinates and frames
p
dF = 0,
dH = J,
H = λ0 ?F,
λ0 = ε0 /µ0
Coordinates xi : F = 12 Fij dxi ∧ dxj , H = 21 Hij dxi ∧ dxj ,
and J = 61 Jijk dxi ∧ dxj ∧ dxk are (1 + 3) decomposed:
Ea = {F10 , F20 , F30 },
B a = {F23 , F31 , F12 }
Ha = {H01 , H02 , H03 },
D a = {H23 , H31 , H12 }
J a = {− J023 , − J031 , − J012 },
ρ = J123
Maxwell equations are recast into standard form
∇ × E + Ḃ = 0,
∇ · B = 0,
∇ × H − Ḋ = J ,
Yuri N. Obukhov
∇·D =ρ
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
Gravity/inertia encoded in constitutive relation H = H(F )
w
ε0 w ab
g Eb − λ0 g ad bcd K c B b ,
Da =
V
V
1 2
w
Ha =
(V − K 2 )g ab + Ka Kb B b − λ0 adc K c g db Eb
µ0 wV
V
Here Ka = g ab K b , K 2 = g ab K a K b and w = det W bc d .
Frame eαi needed for fermions =⇒ Fαβ = eiα ejβ Fij
Components: Ea = {Fb1b0 , Fb2b0 , Fb3b0 } & Ba = {Fb2b3 , Fb3b1 , Fb1b2 }
Relation between holonomic and anholonomic fields
1
1
Ea = W b ba (E + cK × B)b , Ba = W ba b B b
V
w
Nonminimal coupling −βV (Σ · M + iα · P) governed by
Ma = µ0 Ba + δ 0 Ea ,
Yuri N. Obukhov
P a = cδ 0 Ba − µ0 Ea /c.
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Dirac Hamiltonian for arbitrary metric
Electrodynamics in curved spacetime
Foldy-Wouthuysen Hamiltonian and equations of motion
Foldy-Wouthuysen transform needed to reveal physics
(1)
(2)
FW Hamiltonian
HF W = HF W + HF W . E.g. for Earth:
√
with = m2 c4 + π 2 c2 and J = ∇ × M + ∂P
c∂t we find
(1)
HF W
(2)
HF W
~
q~c2 1
= β + qΦ − ω · (r × π) − ω · Σ −
,Π · B
2
4
h
i
1
q~c2
,
Σ
·
(π
×
E
−
E
×
π)
−
~∇
·
E
,
+
8
( + mc2 )
h
i
c 1
= −
, Σ · (π × P − P × π) − ~∇ · P
−Π·M
4 h
c2
1
+
,
(Π · π)(π · M) + (M · π)(Π · π)
4 ( + mc2 )
oi
~
~n
+ β (π·J + J ·π) − β
[ω × r] · ∇ , (π · P)
2
2c
Here { , } anticommutators, T = 22 + {, mc2 V }, Π = βΣ,
This result is exact – no (weak field etc) approximations for
V, W ba b , K a . Planck ~ is the only small parameter
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Classical spin in external fields
Spin in the gravitational field
Classical spin in external fields
Dynamics of spinning particle in external classical fields
dU α
dS α
= F α,
= Φα β S β
dτ
dτ
Physical spin is defined in rest frame of particle uα = δ0α
Local Lorentz transformation U α = Λα β uβ
γvb /c2
γ
α
Λ β=
γv a δba + (γ − 1)v a vb /v 2
Dynamics of physical spin sα = (Λ−1 )α β S β
dsα
dτ
= Ω α β sβ ,
Ωα β = (Λ−1 )α γ Φγ δ Λδ β − (Λ−1 )α γ
Yuri N. Obukhov
Quantum spin dynamics
d γ
Λ β
dτ
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Classical spin in external fields
Spin in the gravitational field
Mathisson-Papapetrou theory
In curved spacetime and electromagnetic field
e
F α = − Γγβ α uγ uβ − g αβ Fβγ uγ ,
m
e αγ
α
α γ
Φ β = − Γγβ u − g Fγβ
m
2
1
α
α γ
αγ
−
M β + 2 (Mβγ u u − M uβ uγ )
~
c
Spin is affected by force due to “polarization” tensor
∗
Mαβ = µ0 Fαβ + cδ 0 Fαβ
.
Dimensionless parameters a = g−2
2 and b characterize
e~
e~
0
magnitude of AMM and EDM: µ = a 2m
and δ 0 = b 2mc
In (1 + 3)-decomposed form, we recover
Mb0ba = cPa ,
Mbabb = abc Mc
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Classical spin in external fields
Spin in the gravitational field
ds
dt
=Ω×s
p
r, γ = 1/ 1 − v 2 /c2 )
Physical spin s precesses wrt rest frame:
Spin dynamics on Earth (with g = − GM
r3
1
1 v×E
e
2γ + 1 v × g
− B+
Ω =
−ω+
2
m
γ
γ+1 c
γ + 1 c2
2µ0
v×E
B·v
γ
−
B−
v 2
−
2
~
c
γ+1
c
0
E·v
2δ
γ
v 2
−
E+v×B−
~
γ+1
c
Analysis of manifestations of terrestrial rotation and gravity
in precision high-energy physics: influence not negligible
E.g.: Earth’s gravity produces same effect as deuteron’s
EDM of δ 0 = 1.5 × 10−29 e·cm in planned dEDM experiment
with magnetic focusing (AGS proposal EDM Collaboration)
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Classical spin in external fields
Spin in the gravitational field
Comparison: quantum vs. classical dynamics
Quantum (semiclassical) precession velocity
1 ac
c2 d
c
a
akl
abc
k
Ω(1) = F c pd
Υδ − V Ckl +
W bb ∂d V ,
2
+ mc2 V
c3
c
abc Q(bd) δ dn F k n pk F l c pl
Ωa(2) = Ξa −
2
( + mc2 V )
Classical precession velocity
γ abc d
γ 1
b
a
Υ vba − abc V Cbbbc d vdb +
W bb ∂d V vbc
Ω =
V 2
γ+1
!
b
v d vbc
c
γ abc
+ Ξba −
Q(bbd)
b
2
γ+1
c
1st comes from Dirac fermion theory; 2nd from Hamilton
particle mechanics of Mathisson and Papapetrou
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Classical spin in external fields
Spin in the gravitational field
Quantum (semiclassical) FW Hamiltonian
p
~
~
HF W =β m2 c4 V 2 +c2 δ cd F a c F b d pa pb +cK·p+ Π·Ω(1) + Σ·Ω(2)
2
2
Classical particle with spin
p
Hclass = m2 c4 V 2 + c2 δ cd F a c F b d pa pb + cK · p + sa Ωa
Semiclassical velocity operator from ~i [HF W , x]:
c2 b
F a p b = va
=⇒
δ cd F a c F b d pa pb = 2 v 2 /c2
Hence 2 = m2 c4 V 2 + 2 v 2 /c2 , or = γ mc2 V .
This yields a direct correspondence
β
γ
=
,
2
+ mc V
1+γ
c3
γ vn vc
F k n pk F l c pl =
2
( + mc V )
1+γ c
Perfect agreement of quantum and classical dynamics!
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Conclusions and Outlook
Relativistic Dirac theory governs quantum dynamics of
fermions (also with dipole moments) in curved spacetime
Earlier results for weak field and stationary configurations
[Obukhov, Silenko, Teryaev, Phys. Rev. D80 (2009)
064044; Phys. Rev. D84 (2011) 024025; Phys. Rev. D88
(2013) 084014; Phys. Rev. D90 (2014) 124068; Phys. Rev.
D94 (2016) 044019] extended to arbitrary strong and timedependent gravitational, inertial and electromagnetic fields
Exact Foldy-Wouthuysen transformation constructed
Quantum and semiclassical equations of motion agree
with classical Mathisson-Papapetrou theory of spin
Possible applications include analysis of spin dynamics
in (weak and strong) gravitational wave – new detectors?
Yuri N. Obukhov
Quantum spin dynamics
Relativistic particles in curved spacetimes
particle in curved spacetime
Spin 1
2
Classical and quantum spin: comparison
Conclusions and Outlook
Thanks !
Yuri N. Obukhov
Quantum spin dynamics