Magnetic monopole and CP symmetry violation at finite temperature
An ideal gas of relativistic massive electrons in the background of a Dirac magnetic monopole is considered. We find that in the case of CP symmetry violation this system acquires, in addition to charge, also squared orbital angular momentum, squared spin, and squared total angular momentum. The fun...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Cite this: | Magnetic monopole and CP symmetry violation at finite temperature / Yu.A. Sitenko, A.V. Solovyov, and N.D. Vlasii // Вопросы атомной науки и техники. — 2007. — № 3. — С. 71-75. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859887896384765952 |
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| author | Sitenko, Yu.A. Solovyov, A.V. Vlasii, N.D. |
| author_facet | Sitenko, Yu.A. Solovyov, A.V. Vlasii, N.D. |
| citation_txt | Magnetic monopole and CP symmetry violation at finite temperature / Yu.A. Sitenko, A.V. Solovyov, and N.D. Vlasii // Вопросы атомной науки и техники. — 2007. — № 3. — С. 71-75. — Бібліогр.: 14 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | An ideal gas of relativistic massive electrons in the background of a Dirac magnetic monopole is considered. We find that in the case of CP symmetry violation this system acquires, in addition to charge, also squared orbital angular momentum, squared spin, and squared total angular momentum. The functional dependence of these quantities on the temperature and the CP-violating vacuum angle is determined. Thermal quadratic fluctuations of conserved quantities are examined, and we analyze, when charge and squared total angular momentum become sharp quantum observables rather than mere expected averages of many quantum measurements.
Розглядається ідеальний газ релятивістських масивних електронів у присутності магнітного монополя Дірака. Знайдено, що при порушенні CP-симетрії в цій системі поряд із зарядом виникають також квадрат орбітального кутового моменту, квадрат спіну та квадрат повного кутового моменту. Визначена функціональна залежність цих величин від температури та вакуумного кута, що порушує CP-симетрію. Досліджені температурні квадратичні флуктуації величин, що зберігаються, і з’ясовано, коли заряд та квадрат повного кутового моменту стають точними квантовими спостережуваними, а не просто очікуваними середніми по багатьом квантовим вимірюванням.
Рассматривается идеальный газ релятивистских массивных электронов в присутствии магнитного монополя Дирака. Обнаружено, что при нарушении CP-симметрии в этой системе наряду с зарядом возникают также квадрат орбитального углового момента, квадрат спина и квадрат полного углового момента. Определена функциональная зависимость этих величин от температуры и вакуумного угла, нарушающего CP-симметрию. Исследованы температурные квадратичные флуктуации сохраняющихся величин, и установлено, когда заряд и квадрат полного углового момента становятся точными квантовыми наблюдаемыми, а не просто ожидаемыми средними по многим квантовым измерениям.
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| first_indexed | 2025-12-07T15:53:46Z |
| format | Article |
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MAGNETIC MONOPOLE AND CP SYMMETRY VIOLATION AT FINITE
TEMPERATURE
Yu.A. Sitenko1,2, A.V. Solovyov1,3, and N.D. Vlasii1,2
1Bogolyubov Institute for Theoretical Physics
National Academy of Sciences, 03680, Kyiv, 143, Ukraine;
e-mail: yusitenko@bitp.kiev.ua;
2Physics Department, National Taras Shevchenko University of Kyiv,
03127, Kyiv, 127, Ukraine;
3Department of Physics, Princeton University,
Princeton, New Jersey 08544, USA
An ideal gas of relativistic massive electrons in the background of a Dirac magnetic monopole is considered. We
find that in the case of CP symmetry violation this system acquires, in addition to charge, also squared orbital angu-
lar momentum, squared spin, and squared total angular momentum. The functional dependence of these quantities
on the temperature and the CP-violating vacuum angle is determined. Thermal quadratic fluctuations of conserved
quantities are examined, and we analyze, when charge and squared total angular momentum become sharp quantum
observables rather than mere expected averages of many quantum measurements.
PACS: 14.80.Hv, 11.10.Wx, 11.30Er
1. INTRODUCTION
The interaction of quantized Dirac fermion fields
with classical background fields of nontrivial topology
can give rise to quantum states with rather unusual ei-
genvalues [1-3]. In particular, the ground state of a
Dirac electron in the background of a Dirac magnetic
monopole acquires nonzero electric charge, and this
results in the monopole becoming a CP symmetry vio-
lating dyon [4-6]. The effect persists when thermal fluc-
tuations of the quantized Dirac electron field are taken
into account, and this yields temperature dependence of
the induced charge [7-9].
The aim of the present paper is to show that, in addi-
tion to charge, also other quantum numbers are induced
in the magnetic monopole background both at zero and
nonzero temperatures. We find relationships between all
these quantum numbers and discuss, which of them
become sharp quantum observables rather than quantum
averages and also when this happens. At nonzero tem-
perature all quantum numbers are not sharp observ-
ables, but, instead, are thermal averages; and, appropri-
ately, the thermal quadratic fluctuations are nonvanish-
ing. If a quadratic fluctuation vanishes at zero tempera-
ture, then a corresponding quantum number at zero
temperature becomes a sharp observable. We find out,
in particular, that induced charge and squared total an-
gular momentum at zero temperature are sharp observ-
ables for almost all values of the vacuum angle with the
exception of the one corresponding to zero energy of
the bound state in the one-particle electron spectrum.
2. OPERATORS OF PHYSICAL
OBSERVABLES AND THEIR VACUUM
AND THERMAL EXPECTATION VALUES
In the second-quantized theory, the operator of a
dynamical variable (physical observable) is given by the
integrated commutator of the electron field operators,
3 +1ˆ tr ( , ), ( , ) ,
2
O d r r t r tϒ −
= Ψ ϒΨ∫
(1)
where is the appropriate one-particle operator in the
first-quantized theory, and denotes the trace over
spinor indices; in particular, O is the operator of en-
ergy, where is the one-particle Hamiltonian, and
is the operator of charge, where is the unity ma-
trix in spinor indices, and is the electron charge. The
vacuum expectation value of the observable
corresponding to Eq. (1) can be presented as
ϒ
tr
ˆH
H
ˆeIO I
e
1ˆvac vac T r sgn ( ),
2
O Hϒ = − ϒ (2)
where is the trace of an integro-differential operator
in the functional space:
Tr
3Tr tr | |U d r r U r= ∫ . The
thermal expectation value of the observable is conven-
tionally defined as (see, e.g., Ref. [10])
1
ˆ ˆSp exp( )ˆ( ) ,ˆSpexp( )
( ) ,
H
H
B
O OO T O
O
k T
β
β
β
β
ϒ
ϒ ϒ
−
−
= ≡
−
=
(3)
where is the equilibrium temperature, is the
Boltzmann constant, and is the trace or the sum
over the expectation values in the Fock state basis in the
second-quantized theory. Evidently, the zero-
temperature limit of Eq. (3) coincides with Eq. (2):
T Bk
Sp
ˆ(0) vac | | vac .O Oϒ ϒ= (4)
Thus, Eq. (3) can be presented in a way similar to that
of Eq. (2), i.e., through the functional trace of operators
in the first-quantized theory, see, e.g., Ref. [11],
1 1( ) Tr tanh .
2 2
O T Hβϒ
= − ϒ
(5)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 71-75. 71
The self-adjointness of the one-particle Hamiltonian
ensures the conservation of energy in the second-
quantized theory, and the corresponding operator is
diagonal in creation and destruction operators; the op-
erator of any other conserved observable is diagonal as
well.
If an observable is conserved, then its thermal quad-
ratic fluctuation,
( )
2
2ˆ ˆ ˆ;T O O O
ββϒ ϒ ϒ
∆ = −
,
takes form
( ) 2 21 1ˆ; Tr sech .
4 2
T O Hβϒ
∆ = ϒ
(6)
Eqs. (5) and (6) are transformed into integrals over the
energy spectrum
1( ) ( ) tanh
2 2
O T dE E Eτ
∞
ϒ ϒ
−∞
= −
∫
1 β
(7) Ψ =
and
( ) 2
21ˆ; ( )sech ,
4 2
T O dE E Eτ
∞
ϒ ϒ
−∞
∆ =
∫
1 β
E
E
(8)
where
( ) Tr ( )E Hτ δϒ = ϒ − (9)
and
2
2( ) Tr ( )E Hτ δϒ = ϒ − (10)
are the appropriate spectral densities.
3. DIRAC ELECTRON IN THE DIRAC
MONOPOLE BACKGROUND
Since the monopole mass is estimated to be much
heavier (by two orders) than the electron mass, it is rea-
sonable to adopt a viewpoint according to which the
monopole only provides a background where the elec-
tron moves. Within this approach we consider quantiza-
tion of the electron field in the background of a classical
monopole field configuration.
3.1. ONE-PARTICLE OPERATORS
A configuration of a pointlike monopole with mag-
netic charge at the origin is given by the field
strength in the form
g
3
3( ) , ( ) 4 ( ).rB r g B r g r
r
π δ= ∂ ⋅ = (11)
Following Wu and Yang [12], one can divide space into
two overlapping regions, : 0 ,
2aR πϑ< < +δ and
:
2bR π δ ϑ π− < <
sin siny r ϑ φ=
(here stands for the azi-
muthal angle in spherical coordinates, ,
, , and
0 ϑ π≤ ≤
sϑ
sin cosx r ϑ φ=
coz r= 0 ), and de-
fine the patched vector potential:
2
πδ< <
(1 cos ) , ,
( )
(1 cos ) , ,
a
b
g d r
A r dr
g d r
ϑ φ
ϑ φ
− ∈
= − + ∈
R
R
(12)
then , where is given by Eq. (11). In the
overlap
A B∂ × = B
:
2 2abR π πδ ϑ δ− < < + , the two potentials are re-
lated by gauge transformation
1,ab aba b
iA A S S
e
−= + ∂ (13)
with
2 ,ieg
abS e φ= (14)
Therefore, the vector potential serves as a connection on
a nontrivial U bundle, and the electron wave func-
tion is a section of this bundle, i.e. wave function
is two-valued with its values in the overlap
related by gauge transformation
(1)
( , )r tΨ
abR
.aba S Ψ (15) b
Generating function (14) is existing (i.e. single-
valued) only when
abS
1 ,
2
eg n n= ∈ ,
0 ,M
(16)
which is the celebrated Dirac quantization condition
[13] that has already attained its 75-year anniversary.
The Dirac Hamiltonian in the background of a static
magnetic monopole takes form
0 ( )H i eAγ γ γ= − ⋅ ∂ + + (17)
where and are the Dirac matrices, is the elec-
tron mass, and is given by Eq. (12). The magnetic
monopole background is rotationally invariant and three
generators of rotations are identified with the compo-
nents of vector – the operator of total angular mo-
mentum in the first-quantized theory,
0γ γ
A
J
M
,J = Λ +Σ (18)
where
( ) rr i eA eg
r
Λ = − × ∂ + − (19)
is its orbital part, and
4
i γ γΣ = × (20)
is its spin part; note that the last term in Eq.(19) is nec-
essary in order to ensure the correct commutation rela-
tions:
[ , ]j k jkl lJ J i Jε− = .
3.2. SELF-ADJOINT EXTENSION
AND CP SYMMETRY VIOLATION
A solution to the stationary Dirac equation
| , , | , ,H r E j m E r E j m= (21)
for the lowest partial wave with 1| |
2
j eg= − cannot be
chosen to be regular at the location of the monopole.
The procedure of the self-adjoint extension is imple-
72
mented for the corresponding partial Hamiltonian,
yielding boundary condition [14, 15]
0
cos lim ( )
2 4
sgn( )sin lim ( ),
2 4
r
rf r
i eg rg r
π
π
→
Θ +
Θ = +
(22)
where and are the radial functions of the
upper and lower components of the wave function,
( )f r ( )g r
( ) ( , )1| , | | , ,
( ) ( , )2
m
m
f r
r E eg m
g r
η ϑ ϕ
η ϑ ϕ
〈 − 〉 =
(23)
, 1/ 2
, 1/ 2
| | 1/ 2 ( , )
2 | | 1
( , ) ,
| | 1/ 2 ( , )
2 | | 1
eg m
m
eg m
eg m Y
eg
eg m Y
eg
ϑ ϕ
η ϑ ϕ
ϑ ϕ
−
+
− +
−
+ = + + +
(24)
, 1/ 2 ( , )eg mY ϑ ϕ∓
m e= −
nπΘ =
are the appropriate monopole harmon-
ics [16], and is the
self-adjoint extension parameter that plays the role of
the vacuum angle in Witten’s approach [4]. It should be
emphasized that CP symmetry is violated, unless
.
| |, | | 1, | |g eg eg− + … , Θ
In the case of , there exists, in addition to
continuum states with energies and ,
also a bound state with energy :
cos 0Θ<
E M>
BSE M
E M< −
sin= Θ
cos
sgn( )sin ( , )
2 4
| ,
cos ( , )
2 4
1 2 cos .
m
BS
m
Mr
i eg
r E m
M e
r
π η ϑ ϕ
π η ϑ ϕ
Θ
Θ +
〈 〉 =
Θ +
× − Θ
(25)
4. INDUCED QUANTUM NUMBERS
IN THE DIRAC MONOPOLE BACKGROUND
In the standard representation for the Dirac matrices
operators Λ (19) and Σ
(20) are of the block-diagonal
form,
0
,
0
Ω
ϒ = Ω
(26) ( )O
and contain no derivatives in : r
( ) ( , ) ( ) ( , ).h r h rξ ϑ φ ξ ϑ φΩ = Ω (27) 4
O T
It can be shown that, when conditions (26) and (27) are
satisfied, the contribution of partial waves with
1| |
2
j eg> − to spectral density (9) is even in energy,
and, thus, these waves do not contribute to thermal ex-
pectation value (7). The contribution of the lowest par-
tial wave to spectral density (9) is calculated to be
( ) ( ) ( )
| | 2 †( ) sin
| | 0 0
1cos sin
4
1 cos sgn( )( )
4 2 sin
2 2( ) .
2 2
eg
E d d m m
m eg
E M E M
EE M
E M
M E M
E M
π π
τ φ ϑ ϑ
θ δ δ
δ
π
θ
×
−
×
= Ω∑ ∫ ∫ϒ
=−
− Θ − Θ − −
Θ
+ +
− Θ
−
−
(28)
η η
Using the last relation, we get the following expression
for the thermal expectation value (7) of the observable
which corresponds to an operator satisfying conditions
(26) and (27):
( )
| | 21 †( ) sin
2 | | 0 0
1cos tanh( Msin )
2
1tanh( Mw)sin 2 w 2 .2 222 w sin1 w 1
eg
O T d d m m
m eg
d
π π
φ ϑ ϑη η
θ β
β
π
×
+
= − Ω ×∑ ∫ ∫ϒ
=−
− Θ Θ
∞ Θ
∫
− Θ−
(29)
The integration over angular variables with summation
over can be performed using the orthonormality of m
mη ’s. In particular, one gets immediately
( ) ( ) ( ) 0,O T O T O TJ= = =ΣΛ (30)
and, thus, rotational symmetry is not spontaneously
broken. In the case of , one gets induced charge
[7-9]
eIϒ =
1( ) | | ( cos )tanh( sin )
2
1tanh( Mw)sin 2 w 2 ;2 222 w sin1 w 1
O T e eg MeI
d
θ β
β
π
+
= − − Θ Θ
∞ Θ
∫
− Θ−
(31)
note that the last expression at 1
2
eg =
mod 2π
| | coincides with
the expression for charge which is induced in -
dimensional space-time at finite temperature by a point-
like magnetic vortex with flux [17].
2 1+
π
All other nonvanishing quantum numbers are related
to Eq. (31): squared orbital angular momentum
2
1| | (| | 1) ( ),eIT eg eg e O T−
Λ = + (32)
squared spin
2
13( ) ( ),eIe O T−
Σ = (33)
and squared total angular momentum
2
2 11( ) ( ) ( ).
4 eIJO T eg e O T− = −
(34)
Note that charge tends to finite value at zero tem-
perature
1(0) 2 | | arctan(tan ),
2eIO e eg
π
Θ
= − (35)
73
and vanishes in the high-temperature limit as inverse
temperature
( ) | | sin
4eI
eO T eg Mβ→∞ = − Θ.
2
(36)
Thus, one can conclude that at CP-violating values of
the vacuum angle (i.e. at Θ ≠ ) both charge and the
squares of orbital angular momentum, spin, and total
angular momentum are induced at finite (zero and non-
zero) temperatures.
nπ
5. QUANTUM EIGENVALUES
OR QUANTUM EXPECTED AVERAGE
VALUES
The squares of orbital angular momentum and spin
are nonconserved observables, so their values both at
zero and nonzero temperatures should be regarded as
expected averages of many quantum measurements. The
conserved observables are charge and squared total an-
gular momentum; note that the latter vanishes in the
case of the minimal monopole strength | | . 1/eg =
We analyze thermal correlations between conserved
and nonconserved observables and thermal quadratic
fluctuations of conserved observables, and find out that
these quantities at nonzero temperature are given by the
ideal gas expressions, and, thus, are Θ -independent and
proportional to the powers of spatial volume. For exam-
ple, we list here the expressions for the quadratic fluc-
tuations of charge,
2 2
2 2
2 3 2
( )ˆ( ; ) ,
14 cosh
2
eI
M
e V s MT O ds
sβ
β
π β
∞ −
∆ =
∫
2 1/ 2
(37)
and squared total angular momentum,
2
2 2
4 / 3 7 / 3 2 2 5 / 2
2 7 2
ˆ( ; )
2 3 ( ) .
1435 cosh
2
J
M
T O
V s Mds
sβ
β
ππ β
∞
∆ =
−
∫ (38)
Note that in the high-temperature limit Eqs. (37) and
(38) increase as and . Thus, the values of charge
and squared total angular momentum at nonzero tem-
perature should be regarded as expected averages of
many quantum measurements, since the corresponding
thermal quadratic fluctuations are nonvanishing.
3T 7T
However, interaction with the monopole background
reveals itself at zero temperature, yielding a Θ -
dependence of a specific type, which is due to a possi-
bility of appearance of a bound state with zero energy in
the one-particle electron spectrum, i.e. at c and
, see Eq. (25):
os <0Θ
sin 0Θ =
2
0, mod 2
ˆ(0; )
| |, mod 2
2
eIO e eg
π π
π π
Θ ≠
∆ =
Θ =
(39)
and
2
2
2
ˆ(0; )
0, mod 2
.1 1| | ( ) , mod 2
2 4
JO
eg eg
π
π π
∆ =
Θ ≠
− Θ =
(40) π
π
2
2
This fact has immediate consequences when we turn
to a question: whether the values of charge and squared
total angular momentum at zero temperature are ob-
served in a single quantum measurement, or whether
they are to be regarded as expected averages of many
such measurements. As it follows from Eqs. (39) and
(40), charge and squared total angular momentum are
sharp observables (quantum-mechanical eigenvalues),
unless . Thus, CP-conserving values of
the vacuum angle, , differ significantly. In the
case of , zero charge and zero squared total
angular momentum are observed in a single quantum
measurement. In the case of , only zero
squared total angular momentum at is a
sharp quantum observable, while zero charge at
and zero squared total angular momentum at | |
are expected average values of many quantum meas-
urements.
mod 2πΘ =
Θ =
2nπΘ =
nπ
(2 1)n πΘ = +
| |eg = 1/
eg
eg >
0≠
1/
ACKNOWLEDGEMENTS
This research was supported in part by the Swiss
National Science Foundation under the SCOPES project
No. IB7320-110848. Yu.A.S. acknowledges the support
of the State Foundation for Basic Research of Ukraine
(grant No. 2.7/00152) and INTAS (grant No. 05-
1000008-7865). N.D.V. acknowledges the INTAS sup-
port through the PhD Fellowship Grant for Young Sci-
entists (No. 05-109-5333).
REFERENCES
1. R. Jackiw, C. Rebbi. Solitons with fermion number ½
//Phys. Rev. D. 1976, v. 13, p. 3398-3409.
2. J. Goldstone, F. Wilczek. Fractional quantum numbers
on solitons //Phys. Rev. Lett. 1981, v. 47, p. 986-989.
3. A.J. Niemi, G.W. Semenoff. Fermion number fractioni-
zation in quantum field theory //Phys. Rep. 1986, v. 135,
p. 99-193.
4. E. Witten. Dyons of charge //Phys. Lett. B.
1979, v. 86, p. 283-187.
/ 2e πΘ
5. B. Grossman. Does a dyon leak? //Phys. Rev. Lett. 1983,
v. 50, p. 464-467.
6. H. Yamagishi. Fermion-monopole system reexamined
//Phys. Rev. D. 1983, v. 27, p. 2383-2396.
7. C. Coriano, R.R. Parwani. The electric charge of a Dirac
monopole at nonzero temperature //Phys. Lett. B. 1995,
v. 363, p. 71-75.
8. A.S. Goldhaber, R. Parwani, H. Singh. On the fractional
electric charge of a Dirac monopole at nonzero tempera-
ture //Phys. Lett. B. 1996, v. 386, p. 207-210.
9. G. Dunne, J. Feinberg. Finite temperature effective ac-
tion in monopole background //Phys. Lett. B. 2000,
v. 477, p. 474-481.
74
10. A. Das. Finite Temperature Field Theory. Singapore:
“World Scientific”, 1997.
11. A.J. Niemi. Topological solitons in a hot and dense
Fermi gas //Nucl. Phys. B. 1985, v. 251, p. 155-181.
12. T.T. Wu, C.N. Yang. Concept of nonintegrable phase
factors and global formulation of gauge fields //Phys.
Rev. D. 1975, v. 12, p. 3845-3857.
13. P.A.M. Dirac. Quantized singularities in the electromag-
netic field //Proc. Roy. Soc. London. A, 1931, v. 133,
p. 60-72.
14. A.S. Goldhaber. Dirac particle in a magnetic field: sym-
symmetries and their breaking by monopole singularities
//Phys. Rev. D. 1977, v. 16, p. 1815-1827.
15. C.J. Callias. Spectra of fermions in monopole fields –
exactly soluble models //Phys. Rev. D. 1977, v. 16,
p. 3068-3077.
15. T.T. Wu, C.N. Yang. Dirac monopole without strings:
monopole harmonics //Nucl. Phys. B. 1976, v. 107,
p. 365-380.
14. Yu.A. Sitenko, V.M. Gorkavenko. Fractional electric
charge of a magnetic vortex at nonzero temperature
//Nucl. Phys. B. 2004, v. 679, p. 597-620.
МАГНИТНЫЙ МОНОПОЛЬ И НАРУШЕНИЕ CP-СИММЕТРИИ
ПРИ КОНЕЧНОЙ ТЕМПЕРАТУРЕ
Ю.А. Ситенко, A.В. Соловьев, Н.Д. Власий
Рассматривается идеальный газ релятивистских массивных электронов в присутствии магнитного моно-
поля Дирака. Обнаружено, что при нарушении CP-симметрии в этой системе наряду с зарядом возникают
также квадрат орбитального углового момента, квадрат спина и квадрат полного углового момента. Опре-
делена функциональная зависимость этих величин от температуры и вакуумного угла, нарушающего CP-
симметрию. Исследованы температурные квадратичные флуктуации сохраняющихся величин, и установле-
но, когда заряд и квадрат полного углового момента становятся точными квантовыми наблюдаемыми, а не
просто ожидаемыми средними по многим квантовым измерениям.
МАГНІТНИЙ МОНОПОЛЬ ТА ПОРУШЕННЯ CP-СИМЕТРІЇ ПРИ СКІНЧЕННІЙ ТЕМПЕРАТУРІ
Ю.О. Ситенко, О.В. Соловйов, Н.Д. Власій
Розглядається ідеальний газ релятивістських масивних електронів у присутності магнітного монополя
Дірака. Знайдено, що при порушенні CP-симетрії в цій системі поряд із зарядом виникають також квадрат
орбітального кутового моменту, квадрат спіну та квадрат повного кутового моменту. Визначена функціона-
льна залежність цих величин від температури та вакуумного кута, що порушує CP-симетрію. Досліджені
температурні квадратичні флуктуації величин, що зберігаються, і з’ясовано, коли заряд та квадрат повного
кутового моменту стають точними квантовими спостережуваними, а не просто очікуваними середніми по
багатьом квантовим вимірюванням.
75
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| id | nasplib_isofts_kiev_ua-123456789-110948 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:53:46Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Sitenko, Yu.A. Solovyov, A.V. Vlasii, N.D. 2017-01-07T09:35:10Z 2017-01-07T09:35:10Z 2007 Magnetic monopole and CP symmetry violation at finite temperature / Yu.A. Sitenko, A.V. Solovyov, and N.D. Vlasii // Вопросы атомной науки и техники. — 2007. — № 3. — С. 71-75. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 14.80.Hv, 11.10.Wx, 11.30Er https://nasplib.isofts.kiev.ua/handle/123456789/110948 An ideal gas of relativistic massive electrons in the background of a Dirac magnetic monopole is considered. We find that in the case of CP symmetry violation this system acquires, in addition to charge, also squared orbital angular momentum, squared spin, and squared total angular momentum. The functional dependence of these quantities on the temperature and the CP-violating vacuum angle is determined. Thermal quadratic fluctuations of conserved quantities are examined, and we analyze, when charge and squared total angular momentum become sharp quantum observables rather than mere expected averages of many quantum measurements. Розглядається ідеальний газ релятивістських масивних електронів у присутності магнітного монополя Дірака. Знайдено, що при порушенні CP-симетрії в цій системі поряд із зарядом виникають також квадрат орбітального кутового моменту, квадрат спіну та квадрат повного кутового моменту. Визначена функціональна залежність цих величин від температури та вакуумного кута, що порушує CP-симетрію. Досліджені температурні квадратичні флуктуації величин, що зберігаються, і з’ясовано, коли заряд та квадрат повного кутового моменту стають точними квантовими спостережуваними, а не просто очікуваними середніми по багатьом квантовим вимірюванням. Рассматривается идеальный газ релятивистских массивных электронов в присутствии магнитного монополя Дирака. Обнаружено, что при нарушении CP-симметрии в этой системе наряду с зарядом возникают также квадрат орбитального углового момента, квадрат спина и квадрат полного углового момента. Определена функциональная зависимость этих величин от температуры и вакуумного угла, нарушающего CP-симметрию. Исследованы температурные квадратичные флуктуации сохраняющихся величин, и установлено, когда заряд и квадрат полного углового момента становятся точными квантовыми наблюдаемыми, а не просто ожидаемыми средними по многим квантовым измерениям. This research was supported in part by the Swiss National Science Foundation under the SCOPES project No. IB7320-110848. Yu.A.S. acknowledges the support of the State Foundation for Basic Research of Ukraine (grant No. 2.7/00152) and INTAS (grant No. 05-1000008-7865). N.D.V. acknowledges the INTAS support through the PhD FellowshipGrant for Young Scientists (No. 05-109-5333). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Magnetic monopole and CP symmetry violation at finite temperature Магнітний монополь та порушення CP-симетрії при скінченній температурі Магнитный монополь и нарушение CP-симметрии при конечной температуре Article published earlier |
| spellingShingle | Magnetic monopole and CP symmetry violation at finite temperature Sitenko, Yu.A. Solovyov, A.V. Vlasii, N.D. Quantum field theory |
| title | Magnetic monopole and CP symmetry violation at finite temperature |
| title_alt | Магнітний монополь та порушення CP-симетрії при скінченній температурі Магнитный монополь и нарушение CP-симметрии при конечной температуре |
| title_full | Magnetic monopole and CP symmetry violation at finite temperature |
| title_fullStr | Magnetic monopole and CP symmetry violation at finite temperature |
| title_full_unstemmed | Magnetic monopole and CP symmetry violation at finite temperature |
| title_short | Magnetic monopole and CP symmetry violation at finite temperature |
| title_sort | magnetic monopole and cp symmetry violation at finite temperature |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110948 |
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