Download Black-Body Radiation for Twist-Deformed Space

CHINESE JOURNAL OF PHYSICS
VOL. 53, NO. 7
December 2015
Black-Body Radiation for Twist-Deformed Space-Time
Marcin Daszkiewicz∗
Institute of Theoretical Physics, University of Wroclaw pl. Maxa Borna 9, 50-206 Wroclaw, Poland
(Received July 30, 2015; Revised September 28, 2015)
In this article we formally investigate the impact of twisted space-time on the black-body
radiation phenomena, i.e., we derive the θ-deformed Planck distribution function as well as
performing its numerical integration to the θ-deformed total radiation energy. In such a
way we indicate that the space-time noncommutativity very strongly damps the black-body
radiation process. Besides we provide for small parameter θ the twisted counterparts of the
Rayleigh-Jeans and Wien distributions, respectively.
DOI: 10.6122/CJP.20150930
PACS numbers: 02.40.Gh, 02.20.Uw, 44.40.+a
I. INTRODUCTION
The suggestion to use noncommutative coordinates goes back to Heisenberg and was
firstly formalized by Snyder in [1]. Recently, there were also found using formal arguments
based mainly on quantum gravity [2, 3] and string theory models [4, 5], indicating that
space-time at the Planck scale should be noncommutative, i.e., it should have a quantum
nature. Consequently, there are a number of papers dealing with noncommutative classical
and quantum mechanics (see, e.g., [6–9]), as well as with field theoretical models (see,
e.g., [10–12]) in which quantum space-time is employed.
It is well-known that a proper modification of the Poincaré and Galilei-Hopf algebras
can be realized in the framework of quantum groups [13, 14]. Hence, in accordance
with the Hopf-algebraic classification of all deformations of relativistic and nonrelativistic
symmetries (see [15, 16]), one can distinguish three types of quantum spaces [15, 16] (for
details see also [17]):
1) Canonical (θµν -deformed) type of quantum space [18–20]
[xµ , xν ] = iθµν ,
(1)
2) Lie-algebraic modification of classical space-time [20–23]
ρ
[xµ , xν ] = iθµν
xρ ,
(2)
and
∗
Electronic address: [email protected]
http://PSROC.phys.ntu.edu.tw/cjp
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c 2015 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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BLACK-BODY RADIATION FOR TWIST-DEFORMED . . .
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3) Quadratic deformation of Minkowski and Galilei space [20–23, 25]
ρτ
[xµ , xν ] = iθµν
xρ xτ ,
(3)
ρ
ρτ
with the coefficients θµν , θµν
, and θµν
being constants.
Moreover, it has been demonstrated in [17], that in the case of the so-called N-enlarged
(N )
Newton-Hooke Hopf algebras U0 (N H ± ) the twist deformation provides a new space-time
noncommutativity of the form [44] [45],
t
4)
[t, xi ] = 0, [xi , xj ] = if±
θij (x),
(4)
τ
with time-dependent functions
t
t
t
= f sinh
, cosh
,
f+
τ
τ
τ
t
t
t
f−
= f sin
, cos
,
τ
τ
τ
k x and τ denoting the time scale parameter — the
θij (x) ∼ θij = const or θij (x) ∼ θij
k
cosmological constant. Besides, it should be noted that the above mentioned quantum
spaces 1), 2), and 3) can be obtained by the proper contraction limit of the commutation
relations 4) [46].
Recently, there has been discussed the impact of different kinds of quantum spaces
on the dynamical structure of physical systems (see, e.g., [6–11] and [26–37]). Particulary,
it has been demonstrated that, in the case of a classical oscillator model [29] as well as
in the case of a nonrelativistic particle moving in constant external field force F~ [30],
there are generated by space-time noncommutativity additional force terms. Such a type
of investigation has been performed for a quantum oscillator model as well [29], i.e., it
was demonstrated that the quantum space in a nontrivial way affects the spectrum of the
energy operator. Besides, in the paper [31] there has been considered a model of a particle
moving on the κ-Galilei space-time in the presence of a gravitational field force. It has been
demonstrated that in such a case there is produced a force term, which can be identified
with the so-called Pioneer anomaly [33], and the value of the deformation parameter κ
can be fixed by a comparison of the obtained result with observational data. Moreover,
quite interesting results have been obtained in the series of papers [34–37] concerning the
Hall effect for canonically deformed space-time (1). Particularly, there has been found
the θ-dependent (Landau) energy spectrum of an electron moving in a uniform magnetic
as well as in a uniform electric field. Such results have been generalized to the case of
the twisted N-enlarged Newton-Hooke Hopf algebra in the papers [38] and [39]. Finally,
it should be mentioned that a similar investigation has been performed in the context of
the black-body radiation process as well. For example, it has been demonstrated with the
use of noncommutative electromagnetic field theory that black-body radiation for quantum
space becomes anisotropic. A direct implication of such a result on the cosmic microwave
background map has been argued in papers [40] and [41].
In this article we also investigate the impact of twisted space-time on the black-body
radiation phenomena. However we assume that a single mode of the photon field oscillates
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MARCIN DASZKIEWICZ
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with a frequency predicted by the new, first-quantized and canonically noncommutative
oscillator model. Precisely we derive the θ-deformed Planck distribution function as well
as perform its numerical integration to the θ-deformed total radiation energy. In such a
way we indicate that the space-time noncommutativity very strongly damps the black-body
radiation process. Besides we provide for small parameter θ the twisted counterparts of the
Rayleigh-Jeans and Wien distributions, respectively.
The paper is organized as follows. In Sect. II we recall basic facts concerning the
most wide class of twisted (N-enlarged Newton-Hooke) space-times [17], which includes the
canonically deformed one as well. The third section is devoted to the calculation of the
isotropic energy spectrum for the oscillator model defined on such a quantum spaces. In
Section IV the Planck distribution function is provided and its numerical integration to the
θ-deformed total radiation energy is performed. The final remarks are presented in the last
section.
II. TWISTED N-ENLARGED NEWTON-HOOKE SPACE-TIMES
In this section we recall the basic facts associated with the twisted N-enlarged Newton(N )
Hooke Hopf algebra Uα (N H± ) and with the corresponding quantum space-times [17].
Firstly, it should be noted that in accordance with the Drinfeld twist procedure, the alge(N )
braic sector of the twisted Hopf structure Uα (N H± ) remains undeformed, i.e., it takes
the form
[Mij , Mkl ] = i (δil Mjk − δjl Mik + δjk Mil − δik Mjl ) ,
h
Mij , Gk
h
i
(k)
(k−1)
Gi , H = −ikGi
,
(n)
i
(n)
(n)
= i δjk Gi − δik Gj
,
(0)
h
(n)
h
i
i (1)
(0)
H, Gi
= ± Gi ;
τ
(1)
(m)
Gi , Gj
i
[H, Mij ] = 0,
(5)
= 0,
(6)
k > 1,
(7)
(n)
where τ , Mij , H, Gi (= Pi ), Gi (= Ki ), and Gi (n > 1) can be identified with the cosmological time parameter, rotation, time translation, momentum, boost, and acceleration
operators, respectively. Besides, the coproducts and antipodes of the considered algebra
are given by [47]
∆α (a) = Fα ◦ ∆0 (a) ◦ Fα−1 , Sα (a) = uα S0 (a)u−1
(8)
α ,
P
P
with uα = f(1) S0 (f(2) ) (we use the Sweedler’s notation Fα = f(1) ⊗ f(2) ) and with the
(N )
(N )
twist factor Fα ∈ Uα (N H± ) ⊗ Uα (N H± ) satisfying the classical cocycle condition
Fα12 · (∆0 ⊗ 1) Fα = Fα23 · (1 ⊗ ∆0 ) Fα ,
(9)
and the normalization condition
( ⊗ 1)Fα = (1 ⊗ )Fα = 1,
(10)
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BLACK-BODY RADIATION FOR TWIST-DEFORMED . . .
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such that Fα12 = Fα ⊗ 1 and Fα23 = 1 ⊗ Fα .
The corresponding quantum space-times are defined as the representation spaces
(N )
(Hopf modules) for the N-enlarged Newton-Hooke Hopf algebra Uα (N H± ). Generally,
they are equipped with two the spatial directions commuting to classical time, i.e., they
take the form
[t, x̂i ] = [x̂1 , x̂3 ] = [x̂2 , x̂3 ] = 0,
[x̂1 , x̂2 ] = if (t);
i = 1, 2, 3.
(11)
However, it should be noted that this type of noncommutativity has been constructed
explicitly only in the case of the 6-enlarged Newton-Hooke Hopf algebra, with [17] [48]
t
t
2
f (t) = fκ1 (t) = f±,κ1
= κ1 C ±
,
τ
τ
t
t
t
= κ2 τ C ±
S±
,
f (t) = fκ2 (t) = f±,κ2
τ
τ
τ
·
·
(12)
·
!
t
t
1 t 4 1 t 2
11
f (t) = fκ35
= 86400κ35 τ
±C±
∓
−
∓1 ×
τ
τ
24 τ
2 τ
!
1 t 3 t
t
∓
−
,
× S±
τ
6 τ
τ
!2
t
t
1 t 4 1 t 2
12
±C±
f (t) = fκ36
= 518400κ36 τ
∓
−
∓1 ,
τ
τ
24 τ
2 τ
and
C+/−
t
t
t
t
= cosh / cos
and S+/−
= sinh / sin
.
τ
τ
τ
τ
Moreover, one can easily check that when τ is approaching the infinity limit the above
quantum spaces reproduce the canonical (1), Lie-algebraic (2), and quadratic (3) type of
space-time noncommutativity, i.e., for τ → ∞ we get
fκ1 (t) = κ1 ,
fκ2 (t) = κ2 t,
·
·
·
fκ35 (t) = κ35 t11 ,
fκ36 (t) = κ36 t12 .
(13)
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MARCIN DASZKIEWICZ
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Of course, for all deformation parameters κa going to zero the above deformations disappear.
III. QUANTUM OSCILLATOR MODEL FOR TWISTED N-ENLARGED
NEWTON-HOOKE SPACE-TIME
Let us now turn to the oscillator model defined on quantum space-times (11)–(13).
In first step of our construction, we extend the described in pervious section spaces to the
whole algebra of momentum and position operators as follows:
[x̂1 , x̂2 ] = 2ifκa (t), [p̂1 , p̂2 ] = 2igκa (t),
[x̂i , p̂j ] = ih̄δij 1 + fκa (t)gκa (t)/h̄2 ,
(14)
(15)
with the arbitrary function gκa (t). One can check that relations (14), (15) satisfy the Jacobi
identity and for deformation parameters κa approaching zero become classical.
Next, by analogy to the commutative case, we define the Hamiltonian operator
Ĥ =
1
1
p̂21 + p̂22 + mω 2 x̂21 + x̂22 .
2m
2
(16)
with m and ω denoting the mass and frequency of a particle, respectively. In order to analyze
the above system, we represent the noncommutative operators (x̂i , p̂i ) by the classical ones
(xi , pi ) as (see, e.g., [27–29])
x̂1 = x1 − fκa (t)p2 /h̄,
(17)
x̂2 = x2 + fκa (t)p1 /h̄,
(18)
p̂1 = p1 + gκa (t)x2 /h̄,
(19)
p̂2 = p2 − gκa (t)x1 /h̄,
(20)
where
[xi , xj ] = 0 = [pi , pj ],
[xi , pj ] = ih̄δij .
(21)
Then, the Hamiltonian (16) takes the form
Ĥ = Ĥ(t) =
1
1
p21 + p22 + M (t)Ω2 (t) x21 + x22 − S(t)L,
2M (t)
2
(22)
with
L = x1 p2 − x2 p1 ,
(23)
mω 2 fκ2a (t)/h̄2 ,
1/M (t) = 1/m +
q
Ω(t) =
1/m + mω 2 fκ2a (t)/h̄2 mω 2 + gκ2a (t)/(h̄2 m) ,
(24)
(25)
and
S(t) = mω 2 fκa (t)/h̄ + gκa (t)/(h̄m).
(26)
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In accordance with the scheme proposed in [29], we introduce a set of time-dependent
creation (a†A (t)) and annihilation (aA (t)) operators
"
#
p
(p2 ± ip1 )
1
p
â± (t) =
− i M (t)Ω(t)/h̄(x2 ± ix1 ) ,
(27)
2
M (t)Ω(t)h̄
satisfying the standard commutation relations
[âA , âB ] = 0,
[â†A , â†B ] = 0,
[âA , â†B ] = δAB ;
A, B = ±.
(28)
Then, it is easy to see that in terms of the objects (27) the Hamiltonian function (22) can
be written as follows
1
1
Ĥ(t) = h̄Ω+ (t) N̂+ (t) +
+ h̄Ω− (t) N̂− (t) +
,
(29)
2
2
with the coefficient Ω± (t) and the particle number operators N̂± (t) given by
Ω± (t) = Ω(t) ∓ S(t),
(30)
↱ (t)Ⱡ(t).
(31)
N̂± (t) =
Besides, one can observe that the eigenvectors of Hamiltonian (29) can be written as
1 † n+ † n−
1
p
â+ (t)
â− (t)
|0i,
(32)
|n+ , n− , ti = p
n+ ! n− !
while the corresponding eigenvalues take the form
1
1
En+ ,n− (t) = h̄Ω+ (t) n+ +
+ h̄Ω− (t) n− +
.
2
2
(33)
Let us now consider an interesting situation such that
S(t) = 0.
(34)
One can check that it appears when the functions fκa (t) and gκa (t) satisfy the following
condition
fκa (t) = −gκa (t)/(ω 2 m2 ).
(35)
Then we have
Ω− (t) = Ω+ (t) = Ω(t) = mω 1/m + mω 2 fκ2a (t)/h̄2 ,
(36)
and, consequently, the spectrum (33) provides the energy levels for a two-dimensional
isotropic oscillator model with the time-dependent frequency: Ω(t)
1
1
En+ ,n− (t) = h̄Ω(t) n+ +
+ h̄Ω(t) n− +
.
(37)
2
2
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MARCIN DASZKIEWICZ
Particularly, for the canonical deformation fκ1 (t) = κ1 = θ we get
1
1
En+ ,n− ,θ = h̄Ωθ n+ +
+ h̄Ωθ n− +
,
2
2
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(38)
with constant frequency
Ωθ = Ωθ (ω) = mω 1/m + mω 2 θ2 /h̄2 ,
(39)
such that limθ→0 Ωθ = ω.
IV. BLACK-BODY RADIATION FOR TWISTED SPACE-TIME
In this section we derive the black-body radiation for θ-deformed nonrelativistic space,
i.e., due to the results of the pervious section we take under consideration the following
energies of a single mode of the photon field [49]
En,θ (ω) = h̄Ωθ (ω)n;
n = 1, 2, 3, . . . ,
(40)
where factor Ωθ (ω) is given just by (39). Consequently, in accordance with quantum theory [42] its average energy takes the form
E θ (ω) =
exp
h̄Ω (ω)
θ h̄Ωθ (ω)
kT
,
(41)
−1
with the Boltzman constant k and temperature T . Besides, due to the fact that the number
of states with frequency from ω to ω + dω per volume unit is [42], [43]
8πω 2
dω,
c3
we get the following θ-deformed Planck distribution function [50]:
ρ(ω)dω =
fθ (ω) =
8πh̄ω 2
Ω (ω)
θ
·
.
3
θ (ω)
c
exp h̄ΩkT
−1
(42)
(43)
Of course, in the θ approaching zero limit we reproduce from (43) the well-known Planck
formula
8πh̄
ω3
f (ω) = 3 ·
,
(44)
h̄ω
c
−1
exp kT
while for small values of the deformation parameter, we have
fθ (ω) =
8πh̄
ω3
·
+
h̄ω
c3 exp kT
−1
h̄ω
h̄ω
− kT exp kT
− 1 θ2
8πh̄ m2 ω 5 h̄ω exp kT
+
− 3 ·
2
c
exp h̄ω − 1 h̄2 kT
kT
+ O(θ3 ).
(45)
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Besides, in accordance with (45) in the high-temperature (kT h̄ω) as well as in the highfrequency (h̄ω kT ) limit, we obtain the θ-deformed counterparts of the Rayleigh-Jeans
fθ (ω) =
8πm2 ω 5 kT θ2
8πω 2 h̄2 kT
−
+ O(θ3 ),
c3
c3 h̄
(46)
and Wien
h̄ω
8πh̄ 3
+
fθ (ω) = 3 · ω exp −
c
kT
8πh̄ m2 ω 5 (h̄ω − kT )
h̄ω
− 3 ·
exp −
θ2 + O(θ3 ),
c
kT
kT h̄2
(47)
distributions, respectively.
Let us now find the total energy density of the radiation by integration of the formula (43) over all frequencies:
Z ∞
uθ =
fθ (ω)dω;
(48)
0
the results of the numerical calculations are summarized in Figures 3 and 4, respectively [51].
Consequently one can notice that the value of the deformed total radiation energy for fixed
temperature T = 1 and 2 strongly decreases with increasing deformation parameter θ. This
means that the biggest value of the energy appears for the undeformed case (with θ equal
zero). Such a result (Figures 1–4 and formula (43)) formally indicates that the canonical
space-time noncommutativity effectively damps the black-body radiation process.
FIG. 1: The shape of the distribution fθ (ω) for three different values of the deformation parameter
theta: θ = 0 (continuous line), θ = 0.1 (dashed line), and θ = 0.2 (dotted line). In all three cases
we fix the temperature T = 1 and the parameter m = 1.
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MARCIN DASZKIEWICZ
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FIG. 2: The shape of the distribution fθ (ω) for three different values of the temperature: T = 1
(continuous line), T = 1.1 (dashed line), and T = 1.2 (dotted line). In all three cases we fix the
deformation parameter at θ = 1 and the parameter m = 1.
FIG. 3: The values of the total radiation energy uθ as a function of the deformation parameter
θ = 0, 1, 2, . . . , 21 for fixed temperature T = 1.
V. FINAL REMARKS
In this article we formally investigate the impact of twisted space-time on black-body
radiation phenomena. Precisely we derive the θ-deformed Planck distribution function (see
formula (43)) as well as we perform its numerical integration to the θ-deformed total radi-
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BLACK-BODY RADIATION FOR TWIST-DEFORMED . . .
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FIG. 4: The values of the total radiation energy uθ as a function of the deformation parameter
θ = 0, 1, 2, . . . , 21 for fixed temperature T = 2.
ation energy. In such a way we indicate that space-time noncommutativity very strongly
damps the black-body radiation process. Besides we provide for small θ the twisted counterparts of the Rayleigh-Jeans and Wien distributions, respectively. Obviously for the
deformation parameter θ approaching zero all obtained results become classical.
Acknowledgments
The author would like to thank J. Lukierski, W. Sobkow, and J. Miskiewicz for
valuable discussions. This paper has been financially supported by the Polish NCN grant
No. 2014/13/B/ST2/04043.
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doi: 10.1209/epl/i2006-10149-x
[42] M. Planck, Annalen der Physik 4, 553 (1901). doi: 10.1002/andp.19013090310
[43] K. Huang, Statistical Mechanics 2nd ed., (John Wiley and Sons, 1987).
[44] x0 = ct.
[45] The discussed space-times have been defined as quantum representation spaces, the so-called
Hopf modules (see e.g. [18, 19]), for the quantum N-enlarged Newton-Hooke Hopf algebras.
[46] Such a result indicates that the twisted N-enlarged Newton-Hooke Hopf algebra plays a role of
the most general type of quantum group deformation at the nonrelativistic level.
[47] ∆0 (a) = a ⊗ 1 + 1 ⊗ a, S0 (a) = −a.
[48] κa = α (a = 1, . . . , 36) denote the deformation parameters.
[49] Due to the isotropy of the spectrum (38) we consider excitations only in one direction. Besides
as a single (emitted) quanta we take E = En+1 − En = h̄Ωθ (ω).
[50] The distribution function fθ (ω) has been plotted for different values of the parameters θ and
T in Figures 1 and 2, respectively.
[51] We use the formula (43) with m = k = c = h̄ = 1 and without the factor 8π.