Minimal Length Effects on Schwinger Mechanism

0.15 MB·English

by  Benrong Mu

CTP-SCU/2015002
Minimal Length Effects on Schwinger Mechanism
Benrong Mua,b, Peng Wangb, and Haitang Yangb
∗ † ‡
aSchool of Physical Electronics, University of Electronic Science
and Technology of China ,Chengdu, 610054, China and
bCenter for Theoretical Physics, College of Physical Science and Technology,
Sichuan University, Chengdu, 610064, PR China
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Abstract
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In this paper, we investigate effects of the minimal length on the Schwinger mechanism using
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J the quantum field theory (QFT) incorporating the minimal length. We first study the Schwinger

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2 mechanism for scalar fields in both usual QFT and the deformed QFT. The same calculations are

] then performed in the case of Dirac particles. Finally, we discuss how our results imply for the
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q
corrections to the Unruh temperature and the Hawking temperature due to the minimal length.
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∗Electronic address: [email protected]
†Electronic address: [email protected]
‡Electronic address: [email protected]
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Contents
I. Introduction 2
II. Scalar Pair Production 3
A. Usual Scalar Field 5
B. Minimal Length Modified Scalar Field 6
III. Fermion Pair Production 8
A. Usual Fermion Field 9
B. Minimal Length Modified Fermion Field 10
IV. Discussion and Conclusion 11
References 13
I. INTRODUCTION
Using the proper-time method, Schwinger[1] calculated the effective action of a charged
particle in an external electromagnetic field. He found that the action has an imaginary
part for a uniform electric field, which leads to the vacuum decay through pair production.
Due to its purely non-perturbative nature, this quantum field theoretical prediction is of
fundamental importance. The Schwinger mechanism sheds lights on topics as diverse as the
string breaking rate in QCD[2, 3] and on black hole physics[4].
On the other hand, various theories of quantum gravity, such as string theory, loop
quantum gravityandquantum geometry, predict theexistence ofaminimal length[5–7]. The
generalized uncertainty principle(GUP)[8] is a simply way to realize this minimal length. An
effective model of the GUP in one dimensional quantum mechanics is given by[9, 10]
1
k(p) = tanh βp , (1)
√β
(cid:16)p (cid:17)
1
ω(E) = tanh βE , (2)
√β
(cid:16)p (cid:17)
where the generators of the translations in space and time are the wave vector k and the
frequency ω, β = β0 , m is the Planck mass and β is a dimensionless parameter marking
m2 p 0
p
2
quantum gravity effects. We set c = ~ = G = 1 in the paper. The quantization in position
representation xˆ = x leads to
k = i∂ , ω = i∂ . (3)
x t
−
Therefore, the low energy limit p m including order of p3 gives
≪ p Mf3
β
p = i∂ 1 ∂2 , (4)
− x − 3 x
(cid:18) (cid:19)
β
E = i∂ 1 ∂2 . (5)
t − 3 t
(cid:18) (cid:19)
From eqn. (1), it is noted that although one can increase p arbitrarily, k has an upper bound
which is 1 . The upper bound on k implies that that particles could not possess arbitrarily
√β
small Compton wavelengths λ = 2π/k and that there exists a minimal length √β.
∼
In this paper, we investigate scalars and fermions pair production from a static classical
electric field using the deformed QFT which incorporates the minimal length via eqn. (4)
and eqn. (5). The organization of this paper is as follows. In section II, based on the usual
and the minimal length modified field theoretic considerations, Schwinger’s mechanism is
derived in the case of spinless particles. We then show how the same calculations can be
performedinthecaseofDiracparticlesinsectionIII.SectionIVisdevotedtoourdiscussions
and conclusions.
II. SCALAR PAIR PRODUCTION
In this section, we first derive the formula for the scalar pair production rate in the
framework of QFT. We then use the formula to calculate pair creation rate in usual QFT
and the minimal length modified QFT. As shown in ref. [11], the scalar field theoretic
vacuum to vacuum amplitude can be written as
vac vac [dφ]exp i dx4 , (6)
h → i ∝ L
Z (cid:18) Z (cid:19)
where is the Lagrange density and φ is the corresponding scalar field. Assume is
L L
given by =φ+ φ where is some differential operator. Defining eigenfunctions φ with
s s n
L O O
eigenvalues λ ,
n
φ = λ φ , (7)
s n n n
O
3
we expand
φ = a φ . (8)
n n
n
X
Using orthogonality of φ , we find
n
e ξs vac vac da da exp i dx4λ a 2
− ≡ h → i ∝ n ∗n n| n|
Z (cid:18) Z (cid:19)
1 1
= = exp( Tr(ln )) = exp lnλ . (9)
s n
∝ λ det − O −
n s !
n O n
Y X
Using the integral
+ e as
∞ −
lna = ds +const., (10)
− s
Z0
one has
ξ = ξE +C, (11)
s s
where C is a constant and we define
+ ds
ξE ∞ e λns. (12)
s ≡ − s −
n Z0
X
The imaginary part of ξ is always infinite due to vacuum energy shift. Here, we are only
s
interested in the real part of ξs which gives the vacuum decay. As shown later in this section,
the electric field E only appears in λ and C is independent of E. To obtain C we consider
n
the case without the electric field. When there is no electric field, the vacuum is stable and
no pairs are produced. In this case, the vacuum to vacuum amplitude is
eiα = e ξs = exp ξ0 +C
− − s
(cid:0) (cid:1)
where α is a phase irrelevant to the vacuum decay and ξ0 is ξE with E = 0. Thus, one has
s s
C = ξ0 +iα. (13)
s
Plugging eqn. (13) into eqn. (11), we find that the real part of ξ is
s
γ Reξ = Re ξE ξ0 . (14)
s ≡ s s − s
(cid:0) (cid:1)
Squaring γ , we observe that the total pair production rate per unit volume is simply
s
prob 1 2γ
pair = 1 e 2γs s. (15)
−
VT VT − ≈ VT
(cid:0) (cid:1)
4
A. Usual Scalar Field
For the case of a scalar field φ of mass m and charge e in the presence of an external
electromagnetic interaction described by vector potential A , the Lagrangian is given by
µ
= φ+ (0)φ, (16)
Ls Os
where (0)=(∂ +ieA)2+m2. If there is a uniform electric field E = Ee , we can utilize the
s z
O
gauge A = Ete . The operator (0) then becomes
z s
− O
(0)= ∂2 ∂2 (∂ +ieEt)2 +m2, (17)
Os t − − z
⊥
where ∂2 = ∂2 + ∂2. Assume that eigenfunction of (0) takes the form as ψ =
x y Os
⊥
exp(ik r +ik z)σ(t) where r = xe +ye and σ(t) satisfies
z x y
⊥ · ⊥ ⊥
2
d
+k2 +(k +eEt)2 +m2 σ(t) = λσ(t). (18)
z
dt
⊥ !
(cid:18) (cid:19)
Performing the Wick Rotation t iτ and E iE˜, we have
→ − → −
2 2
d k
+e2E˜2 τ z σ(t) = λ k2 m2 σ(t). (19)
− dτ − eE˜ ! − ⊥ −
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Obviously, eqn. (19) describes a one-dimensional harmonic oscillator with its well centered
at kz and a resonant frequency 2eE˜. One can express d and τ in terms of ladder operators,
eE˜ dτ
a and a+, as
k 1
τ z = a+ +a , (20)
− eE˜ 2eE˜
(cid:0) (cid:1)
d p eE˜
= a+ a . (21)
dτ −s 2 −
(cid:0) (cid:1)
Thus, the energy levels are quantized as
λs(0) = k2 +m2 +(2n+1)eE˜ (22)
n,k⊥
⊥
where n = 0,1,2 . Note that k and k range over all values from to , but k
x y z
··· −∞ ∞
is constrained to be in the range 0 < k < eE˜iT in order that the entire range of time
z
is included as k is varied, where T is a total interaction time. Thus, the corresponding
z
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degeneracy is eETVdk⊥ where V is the volume. After switching back to t and E, eqn. (12)
3
(2π)
and eqn. (22) yields
ξE = eEVT dk +∞ ds exp k2 +m2 s ∞ exp[ i(2n+1)eEs]. (23)
s − (2π)3 Z ⊥Z0 s − ⊥ n=0 −
(cid:2) (cid:0) (cid:1) (cid:3)X
With the help of Dirac comb
∞ ∞
π δ(t kπ) = exp( i2nt), (24)
− −
k= n=
X−∞ X−∞
one easily gets
ReξE = eEVT dk +∞ ds exp k2 +m2 s exp( ieEs)π ∞ δ(eEs kπ).
s −2(2π)3 Z ⊥Z0 s − ⊥ − k= −
(cid:2) (cid:0) (cid:1) (cid:3) X−∞
(25)
When E approaches zero, we have for Reξ0
s
VT + ds
Reξ0 = dk ∞ exp k2 +m2 s δ(s). (26)
s −16π2 Z ⊥Z0 s − ⊥
(cid:2) (cid:0) (cid:1) (cid:3)
Thus, we find
γ(0) = Re ξE ξ0
s s − s
= V(cid:0)T dk(cid:1) +∞ ds exp k2 +m2 s exp( ieEs) ∞ δ s kπ δ(s)
−16π2 Z ⊥Z0 s − ⊥ − "k= (cid:18) − eE(cid:19)− #
(cid:2) (cid:0) (cid:1) (cid:3) X−∞
e2E2VT ∞ ( 1)k+1 kπ
= − exp m2 . (27)
16π3 k2 − eE
k=1 (cid:18) (cid:19)
X
B. Minimal Length Modified Scalar Field
In the presence of an external electromagnetic potential A , eqn. (4) and eqn. (5) can
µ
be generalized to
β
p = iD 1 D2 , (28)
i − i − 3 i
(cid:18) (cid:19)
β
E = iD 1 D2 , (29)
t − 3 t
(cid:18) (cid:19)
where D = ∂ +ieA . For a scalar field φ of mass m and charge e in the external electro-
µ µ µ
magnetic potential A , the Lagrangian incorporating eqn. (28) and eqn. (29) can be written
µ
as
= ηµν (p φ)+(p φ)+m2, (30)
s µ ν
L
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where p = (E,p ). After integrating by part, we have
µ i
= φ+ (0) +δ φ, (31)
Ls Os Os
(cid:0) (cid:1)
where we define
2
δ = β D4 D4 +D4 +D4 + β2 . (32)
Os −3 t − x y z O
(cid:2) (cid:0) (0) (cid:1)(cid:3) (cid:0) (cid:1)
In order to get eigenfunctions and eigenvalues of +δ , we write the eigenfunctions in
s s
O O
the form φ = exp(ik r +ik z)σ(t). For σ(t), δ becomes
z s
⊥ · ⊥ O
2
δ = β ∂4 k4 (k +eEt)4 + β2 . (33)
Os −3 t − − z O
⊥
(cid:2) (cid:3) (cid:0) (cid:1)
Rotating to imaginary time τ and E˜, we can use eqn. (20) and eqn. (21) to write δ in
s
O
terms of ladder operators
1 4k4
δ = βe2E˜2 a+ a 4 a+ +a 4 + β2 . (34)
Os −6 − − e2E˜⊥2 − O
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
s(0)
Treating δ as perturbations, we find the first-order correction to λ
Os n,k⊥
2βk4
δλs = n δ n = , (35)
n,k⊥ h | Os| i 3 ⊥
where n is nth eigenstate for the one-dimensional harmonic oscillator. The corresponding
| i
degeneracy stays same, namely eETVdk⊥. Therefore, eqn. (12) gives to (β)
3
(2π) O
ξE eEVT dk +∞ ds exp k2 +m2 + 2βk4 s ∞ exp[ i(2n+1)eEs].
s ≈ − (2π)3 Z ⊥Z0 s (cid:20)−(cid:18) ⊥ 3 ⊥(cid:19) (cid:21)n=0 −
X
(36)
Note that the above equation is the same as eqn. (23) except the prefactor exp 2βk⊥4 s .
− 3
Following calculations for γ(0), one obtains to (β) (cid:16) (cid:17)
s
O
γ = Re ξE ξ0
s s − s
eE(cid:0)VT (cid:1) ∞ 1 2βk4 kπ
dk ( 1)k+1 exp k2 +m2 +
⊥
≈ 16π3 ⊥ − k − ⊥ 3 eE
Z k=1 (cid:20) (cid:18) (cid:19) (cid:21)
X
e2E2VT ∞ ( 1)k+1 m2πk 4βeE
− exp . (37)
≈ 16π3 k2 − eE − 3πk
k=1 (cid:18) (cid:19)
X
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III. FERMION PAIR PRODUCTION
The procedure in Section II can also be applied to calculations for a fermion field. How-
ever, two differences should be noted. First, the Grassmann numbers are used in fermion
case. Second, instead of , we usually calculate eigenvalues of ˜ , where ˜ is defined
f f f f
O O O O
as follows. In even dimension, there exist two charge conjugate operators C and C such
+
−
that
C γµC 1 = γµT, (38)
−
± ± ±
where γµ are Gamma matrices. We then define
O˜f ≡ C−−1 γ0 C+OfC+−1 ∗γ0 T C−. (39)
˜ (cid:0) (cid:0) (cid:1) (cid:1)
Since is Hermitian, one finds
f
O
det ˜ =det = det + = det . (40)
Of Of∗ Of Of
Assume the Lagrangian for a fermion field ψ is give in the form of =ψ ψ. Defining
f
L O
eigenfunctions ψ with eigenvalues λ ,
n ′n
ψ = λ ψ , (41)
Of n ′n n
we expand
ψ = ξ ψ . (42)
n n
n
X
Thus, the vacuum to vacuum amplitude for ψ is
e−ξf ≡ hvac → vaci ∝ dξndξn∗exp i dx4λ′n|ξn|2
Z (cid:18) Z (cid:19)
1 1 1
λ = det = det ˜ 2 = exp Tr ln ˜ = exp lnλ (43)
∝ ′n Of OfOf 2 OfOf 2 n
!
Yn (cid:16) (cid:17) (cid:20) (cid:16) (cid:17)(cid:21) Xn
where λ are eigenvalues of ˜ . The real part of ξ is given by
n f f f
O O
γ = Reξ = Re ξE ξ0 (44)
f f f − f
(cid:0) (cid:1)
where ξ0 is ξE with E = 0 and we define
f f
1 + ds
ξE ∞ e λns. (45)
f ≡ 2 s −
n Z0
X
Squaring γ , we observe that the total pair production rate per unit volume is simply
f
prob 1 2γ
pair = 1 e−2γf f. (46)
VT VT − ≈ VT
(cid:0) (cid:1)
8
A. Usual Fermion Field
The Lagrangian for a charged spinor of mass m and charge e is
= iψDµγ ψ mψψ = ψ (0)ψ, (47)
L µ − Of
where (0)=iDµγ m. Using eqn. (39), one finds
Of µ −
(0) ˜(0) = κ +κ , (48)
Of Of 1 2
where σµν = i [γµ,γν], [D ,D ] = ieF , κ D2 + m2 and κ = eF σµν. Here the
2 µ ν µν 1 ≡ 2 2 µν
eigenvalues of κ in the case of a constant electric field were determined in Section II and
1
are given by
λ = k2 +m2 +(2n+1)eE˜. (49)
1
⊥
In chiral representation, we have
σ3 0
κ = ieE − , (50)
2
 0 σ3
 
where A = Ete for a uniform electric field E = Ee . Temporarily rotating to imaginary
z z
−
time as before, we find that the eigenvalues of (0) ˜(0) are
Of Of
λf(0) = k2 +m2 +2neE˜ for n = 0,1,2 , (51)
n,k⊥ ···
⊥
and the corresponding degeneracies are
2g eETVdk
n
⊥, (52)
(2π)3
where g = 1 and g = 2. Thus one finds
0 n>0
ξE = eEVT dk +∞ ds exp k2 +m2 s ∞ g exp( 2ineEs). (53)
f (2π)3 Z ⊥Z0 s − ⊥ n=0 n −
(cid:2) (cid:0) (cid:1) (cid:3)X
Using Dirac comb, we get
ReξE = eEVT dk +∞ ds exp k2 +m2 s π ∞ δ(eEs kπ), (54)
f (2π)3 Z ⊥Z0 s − ⊥ k= −
(cid:2) (cid:0) (cid:1) (cid:3) X−∞
VT + ds
Reξ0 = dk ∞ exp k2 +m2 s πδ(s). (55)
f (2π)3 Z ⊥Z0 s − ⊥
(cid:2) (cid:0) (cid:1) (cid:3)
Therefore, we have
γ(0) = Re ξE ξ0
f f − f
e2E(cid:0)2VT ∞(cid:1) 1 kπ
= exp m2 . (56)
8π3 k2 − eE
k=1 (cid:18) (cid:19)
X
9
B. Minimal Length Modified Fermion Field
For this case, the Lagrangian incorporating eqn. (28) and eqn. (29) for a charged spinor
ψ of mass m and charge e can be written as
= ψ(ip γµ m)ψ, (57)
f µ
L −
where p = (E,p ). We then find
µ i
(0)
= ψ ψ = ψ +δ ψ, (58)
Lf Of Of Of′
(cid:16) (cid:17)
where we define
i
δ = βD3γµ + β2 . (59)
Of′ −3 µ O
Eqn. (39) and eqn. (59) gives the product of and (cid:0)˜ (cid:1)to (β)
f f
O O O
˜ (0) ˜(0)
= +δ , (60)
OfOf Of Of Of
where we find
δ = δ eβF D2σµν. (61)
Of Os − µν µ
The first term in eqn. (61) is just δ given in eqn. (33), while one can express the second
s
O
term in terms of ladder operators after rotating to imaginary time. In fact, using eqn. (20)
and eqn. (21) gives
σ3 0
F D2σµν = E2 − a+2 +a2 , (62)
µν µ  0 σ3
(cid:0) (cid:1)
 
f(0)
which doesn’t contribute to the first-order correction to eigenvalues λ of the leading
n,k⊥
operator (0) ˜(0) in eqn. (60) since n F D2σµν n is zero. Thus, the eigenvalues of
Of Of µν µ
˜
f f to (β) are (cid:10) (cid:12) (cid:12) (cid:11)
O O O (cid:12) (cid:12)
2βk4
λf k2 +m2 +2neE˜ + for n = 0,1,2 , (63)
n,k⊥ ≈ ⊥ 3 ⊥ ···
where we use eqn. (35). The corresponding degeneracies are also
2g eETVdk
n
⊥, (64)
(2π)3
where g = 1 and g = 2. Therefore, one finds for ξE
0 n>0 f
ξE eEVT dk +∞ ds exp k2 +m2 + 2βk4 s ∞ g exp 2neE˜s . (65)
f ≈ (2π)3 Z ⊥Z0 s (cid:20)−(cid:18) ⊥ 3 ⊥(cid:19) (cid:21)Xn=0 n (cid:16)− (cid:17)
10