Grassmann dynamics of classical spin in nonabelian gauge fields
Using Grassmann variant of classical mechanics, we construct Lagrangian dynamics of classical spinning particle in (possibly non-abelian) gauge fields. Quantization of this model is briefly discussed. На основе грассманова варианта классической механики построена лагранжева динамика классической час...
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| Veröffentlicht in: | Вопросы атомной науки и техники |
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| Datum: | 2012 |
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| Sprache: | Englisch |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2012
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| Zitieren: | Grassmann dynamics of classical spin in nonabelian gauge fields / S.A. Pol'shin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 37-38. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859676262361989120 |
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| author | Pol’shin, S.A. |
| author_facet | Pol’shin, S.A. |
| citation_txt | Grassmann dynamics of classical spin in nonabelian gauge fields / S.A. Pol'shin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 37-38. — Бібліогр.: 8 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Using Grassmann variant of classical mechanics, we construct Lagrangian dynamics of classical spinning particle in (possibly non-abelian) gauge fields. Quantization of this model is briefly discussed.
На основе грассманова варианта классической механики построена лагранжева динамика классической частицы со спином в калибровочных полях (в т.ч. неабелевых). Кратко рассмотрено квантование предложенной модели.
На основi грасманова варiанта класичної механiки побудовано лагранжеву динамiку класичної частинки зi спiном у калiбрувальних полях (в т.ч. неабелевих). Коротко розглянуто квантування запропонованої моделi.
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| first_indexed | 2025-11-30T16:00:30Z |
| format | Article |
| fulltext |
GRASSMANN DYNAMICS OF CLASSICAL SPIN
IN NONABELIAN GAUGE FIELDS
S.A. Pol’shin∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 27, 2011)
Using Grassmann variant of classical mechanics, we construct Lagrangian dynamics of classical spinning particle in
(possibly non-abelian) gauge fields. Quantization of this model is briefly discussed.
PACS 2010: 11.15.Kc, 11.30.Pb
In [1] the Lagrangian theory of pseudoclassical
particle moving in electromagnetic field was con-
structed and its quantization leading to Dirac equa-
tion was considered. In the present note, we gen-
eralize this theory onto the case of arbitrary gauge
group. Quantization and the case of nonzero anom-
alous magnetic momentum are discussed in last two
paragraphs.
Let xμ be even ”space-time” coordinated of our
particle, ξμ odd spin variables, Aμ
a even gauge poten-
tial, and Ja generators of certain finite-dimensional
anti-hermitian representation of our gauge group (e.g.
spinor representation of SO(3)), so [Ja, Jb] = if c
ab Jc.
Let q, q̄ be even coordinates of internal gauge degrees
of freedom of our particle (their indexes will not be
written explicitly). Define their covariant derivatives
as:
Dq
Ds
=
dq
ds
+ ieAa
μẋμJaq,
Dq̄
Ds
=
dq̄
ds
− ieAa
μẋμq̄Ja,
(1)
where s is even coordinate of a worldline and overdot
means derivative w.r.t. s. Note that ẋμẋμ �= 1 in
general, see [2] for discussion. Define gauge charge as
Qa = q̄Jaq, so
DQa
Ds
=
dQa
ds
+ eAb
μẋμf c
ab Qc. (2)
Consider the following lagrangian
L =
m
2
ẋμẋμ − 1
4
ξμξ̇μ+
+
i
2
(
−Dq̄
Ds
q + q̄
Dq
Ds
)
+
μ′
2m
F a
μνSμνQa,
(3)
where μ′ = e is the magnetic moment of a particle
(see last paragraph however), Sμν = 1
2ξμξν is spin
tensor and
F a
μν = ∂μAa
ν − ∂νAa
μ + eAb
μAc
νf a
bc (4)
is gauge field tensor. Note that unfolding the brack-
ets in r.h.s. of (3) we obtain usual interaction term
−eAa
μẋμQa due to (1). Varying the action
∫
L ds
w.r.t. q and q̄ we obtain:
iq̇ =
(
−eAa
μẋμ +
μ′
2m
F a
μνSμν
)
Jaq,
−i ˙̄q =
(
−eAa
μẋμ +
μ′
2m
F a
μνSμν
)
q̄Ja,
so using (2) we obtain equations of motion of gauge
charge
DQa
Ds
=
μ′
2m
F b
μνSμνf c
ab Qc. (5)
Varying w.r.t. ξμ we obtain spin equations of motion
mṠμν = μ′F aρ[νS μ]
ρ Qa. (6)
Finally, varying w.r.t. xμ we have to consider Qa
as geodesically constant: DQa/Ds = 0 but not
Q̇a = 0 contrary to ordinary variational calcu-
lus. Then using (2), (4) we obtain:
mẍμ = eF aμν ẋνQa +
μ′
2m
(DμF ρσ)aSρσQa, (7)
where Dμ is ordinary covariant derivative w.r.t. Aa
μ,
so d(F a
μνQa)/ds = (DρFμν)aQaẋρ + F a
μνDQa/Ds.
Eqs. (5)-(7) are just the ones obtained by Heinz [3]
by classicalizing the ordinary QCD hamiltonian (cf.
also [4]).
Due to nonabelian Bianchi identity
(D[μF ρσ])a = 0
we see that the following quantities are conserved:
C1 = ξμẋμ,
C2 =
m
2
ẋμẋμ +
μ′
2m
F a
μνSμνQa, (8)
C3 = q̄q.
Since orbits of coadjoint representation are distin-
guished by C3 values, we see that different types of
gauge charges arise in the sense of [5].
∗E-mail address: polshin.s@gmail.com
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 37-38.
37
To quantize our theory, we turn q, q̄ into bosonic
creation-destruction operators, then different values
of occupation number operator Ĉ3 correspond to dif-
ferent representations of gauge group, ξμ turn into
γ-matrices, and canonical momentum pμ = ημν ẋν −
eAa
μQa [6] turns into −i∂μ. Thus Ĉ1 becomes Dirac
operator (cf. [1] in the abelian case) and Ĉ2 becomes
Hamiltonian. If coadjoint representation of gauge
group is chosen, then Q̂a turn into Gell-Mann ma-
trices of ordinary QCD.
If μ′ �= e, we obtain theory with anomalous mag-
netic momentum. Considering C1 as a Lagrangian
constraint, we see that all the above considerations
go without substantial change, so we obtain BMT-
type [7] equations (cf. [2] for abelian case):
m(ẋρẋρ)Ṡμν = μ′(ẋκẋκ)F aρ[νS μ]
ρ Qa+
+(μ′ − e)F aρσẋ ρS
[μ
σ ẋν]Qa,
and some additional terms in the r.h.s. of eq. (7)
arise. For the case of U(1) gauge group, the quan-
tized version of C2 for an arbitrary value of μ′ was
considered in [8].
References
1. F. Ravndal. Supersymmetric Dirac Particles In
External Fields // Phys. Rev. 1980, v.D21,
p. 2823-2832.
2. S.A. Pol’shin. A simple variational principle for
classical spinning particle with anomalous mag-
netic momentum // Mod. Phys. Lett. 2009,
v. A24, p. 331-333.
3. U. Heinz. A Relativistic Colored Spinning Par-
ticle In An External Color Field // Phys. Lett.
1984, v. B144, p. 228-230.
4. N. Linden, A.J. Macfarlane and J.W. van Holten.
Particle motion in a Yang-Mills field: Wong’s
equations and spin 1/2 analogs // Czech. J. Phys.
1996, v. 46, p. 209-215.
5. A. Weinstein. A Universal Phase Space for Par-
ticles in Yang-Mills Field // Lett. Math. Phys.
1978, v. 2, p. 417-420.
6. R. Montgomery. Canonical Formulations Of A
Classical Particle In A Yang-Mills Field And
Wong’s Equations // Lett. Math. Phys. 1984,
v. 8, p. 59-67.
7. V. Bargmann, L. Michel and V. Telegdi. Preces-
sion of the polarization of particles moving in a
homogeneous electromagnetic field // Phys. Rev.
Lett. 1959, v. 2, p. 435-437.
8. A. Andreev. Atomic Spectroscopy: An Introduc-
tion to the Theory of Hyperfine Interactions.
Berlin: Springer, 2006, 274 p.
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| id | nasplib_isofts_kiev_ua-123456789-106977 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T16:00:30Z |
| publishDate | 2012 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pol’shin, S.A. 2016-10-10T07:49:09Z 2016-10-10T07:49:09Z 2012 Grassmann dynamics of classical spin in nonabelian gauge fields / S.A. Pol'shin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 37-38. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS 2010: 11.15.Kc, 11.30.Pb https://nasplib.isofts.kiev.ua/handle/123456789/106977 Using Grassmann variant of classical mechanics, we construct Lagrangian dynamics of classical spinning particle in (possibly non-abelian) gauge fields. Quantization of this model is briefly discussed. На основе грассманова варианта классической механики построена лагранжева динамика классической частицы со спином в калибровочных полях (в т.ч. неабелевых). Кратко рассмотрено квантование предложенной модели. На основi грасманова варiанта класичної механiки побудовано лагранжеву динамiку класичної частинки зi спiном у калiбрувальних полях (в т.ч. неабелевих). Коротко розглянуто квантування запропонованої моделi. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section A. Quantum Field Theory Grassmann dynamics of classical spin in nonabelian gauge fields Грассманова динамика классического спина в неабелевых калибровочных полях Грасманова динамiка класичного спiну у неабелевих калiбрувальних полях Article published earlier |
| spellingShingle | Grassmann dynamics of classical spin in nonabelian gauge fields Pol’shin, S.A. Section A. Quantum Field Theory |
| title | Grassmann dynamics of classical spin in nonabelian gauge fields |
| title_alt | Грассманова динамика классического спина в неабелевых калибровочных полях Грасманова динамiка класичного спiну у неабелевих калiбрувальних полях |
| title_full | Grassmann dynamics of classical spin in nonabelian gauge fields |
| title_fullStr | Grassmann dynamics of classical spin in nonabelian gauge fields |
| title_full_unstemmed | Grassmann dynamics of classical spin in nonabelian gauge fields |
| title_short | Grassmann dynamics of classical spin in nonabelian gauge fields |
| title_sort | grassmann dynamics of classical spin in nonabelian gauge fields |
| topic | Section A. Quantum Field Theory |
| topic_facet | Section A. Quantum Field Theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106977 |
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