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Incorporation of generalized uncertainty principle into Lifshitz field theories
Annals of Physics 357 (2015) 49–58
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Incorporation of generalized uncertainty principle into Lifshitz field theories Mir Faizal a,∗ , Barun Majumder b a
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
b
Indian Institute of Technology Gandhinagar, Ahmedabad, 382424, India
article
info
Article history: Received 17 February 2015 Accepted 18 March 2015 Available online 25 March 2015
abstract In this paper, we will incorporate the generalized uncertainty principle into field theories with Lifshitz scaling. We will first construct both bosonic and fermionic theories with Lifshitz scaling based on generalized uncertainty principle. After that we will incorporate the generalized uncertainty principle into a non-abelian gauge theory with Lifshitz scaling. We will observe that even though the action for this theory is non-local, it is invariant under local gauge transformations. We will also perform the stochastic quantization of this Lifshitz fermionic theory based generalized uncertainty principle. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The classical picture of spacetime breaks down in most approaches to quantum gravity. This is due to the fluctuations in the geometry being of order one at Planck scale. Thus, the picture of spacetime as a continuous differential manifold cannot be valid below Planck length. Furthermore, the existence of a minimum length scale is also a feature of string theory [1–5]. In fact, in loop quantum gravity the existence of minimum length turns big bang into a big bounce [6]. However, the existence of minimum length is not consistent with conventional uncertainty principle, which states that one can measure length with arbitrary accuracy, if one takes no measurement of momentum [1,2,7–19]. Thus, the uncertainty principle has to be modified if one wants to incorporate the existence of minimum length scale. These considerations have led to a modification of the Heisenberg uncertainty principle,
∗
Corresponding author. E-mail addresses: [email protected] (M. Faizal), [email protected] (B. Majumder).
http://dx.doi.org/10.1016/j.aop.2015.03.022 0003-4916/© 2015 Elsevier Inc. All rights reserved.
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which in turn has led to a modification of the Heisenberg algebra. It may be noted that the implications of this modified uncertainty principle for quantum field theory have also been studied [20–22]. In this paper, we analyse a quantum field theory based on generalized uncertainty with Lifshitz scaling. Lifshitz field theories are quantum field theories based on an anisotropic scaling between space and time. Lifshitz theories were first introduced in condensed matter physics to model quantum criticality [23–26]. In fact, a Fermi-surface-changing Lifshitz transition occurs for some heavy fermion compounds [27]. The location of this Fermi-surface-changing Lifshitz transition is influenced by carrier doping. Due to strong correlations, a heavy band does not shift rigidly with the chemical potential and the actual shift is determined by the interplay of heavy and additional light bands crossing the Fermi level. Furthermore, meta-magnetic transitions in models for heavy fermions have also been analysed using doped Kondo lattice model in two dimensions [28]. Some heavy fermion metals display a fielddriven quantum phase transition due to a breakdown of the Kondo effect [29,30]. Many of the properties have been described by a Zeeman-driven Lifshitz transition of narrow heavy fermion bands [31]. Materials that cannot be described with the local dielectric response have been described by a generalization of the usual Lifshitz theory [32]. In fact, the temperature correction to the Casimir–Lifshitz free energy between two parallel plates made of dielectric material, possessing a constant conductivity at low temperatures, has been calculated [33]. Lifshitz theory has also been used for calculating the van der Waals and Casimir interaction between graphene and a material plate, graphene and an atom or a molecule, and between a single-wall carbon nanotube and a plate [34]. In this model the reflection properties of electromagnetic oscillations on graphene are governed by the specific boundary conditions imposed on the infinitely thin positively charged plasma sheet, carrying a continuous fluid with some mass and charge density. Fermionic retarded Green’s function with z = 2 has been studied at finite temperature and finite chemical potential [35]. Here the usual Lifshitz geometry was replaced by a Lifshitz black hole. Hawking radiation for Lifshitz fermions has also been studied [36]. Fermionic theories with z = 2 Lifshitz scaling have also been constructed using a non-local differential operator [37]. This non-local differential operator is defined using harmonic extension of a compactly supported function [38–42]. It appears as a map from the Dirichlet-type problem to the Neumann type problem. It may be noted that fermionic theories with z = 3 have also been studied [43,44]. It has been demonstrated that Nambu–Jona-Lasinio type four-fermion coupling at the z = 3 Lifshitz fixed point in four dimensions is asymptotically free and generates a mass scale [45]. In this paper, we will study both bosonic and fermionic Lifshitz field theory, consistent with generalized uncertainty principle. We will also study the gauge symmetry for these theories. It may also be noted that another interesting deformation of quantum mechanics comes from stochastic quantization [46–49]. Stochastic quantization has provided a powerful framework for analysing bosonic theories with Lifshitz scaling [50,51]. In fact, effect of ohmic noise on the non-Markovian spin dynamics resulting in Kondo-type correlations has been studied using stochastic quantization [52] In this paper, we will analyse the stochastic quantization of Lifshitz Dirac equation with minimum length. 2. Generalized Uncertainty Principle In the Lifshitz field theories the scaling is usually taken as x → bx and t → bz t, where b is called the scaling factor and z is called the degree of anisotropy. For z = 1, this reduces to the usual conformal transformation. In this paper, we will analyse the Lifshitz theories with z = 2. The Lifshitz action for a bosonic field with z = 2, can be written as [37] Sb =
1 2
dd+1 x (φ∂ 0 ∂0 φ − κ 2 φ(∂ i ∂i )2 φ).
(1)
The Lifshitz theories are unitarity because they contain no higher order temporal derivatives. So, we will leave the temporal part of the Lifshitz action for a bosonic field undeformed. However, we will deform its spatial part, to make it consistent with the existence of a minimum measurable length [19,20]. The Heisenberg uncertainty principle is not consistent with the existence of a
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minimum measurable length, as according to it, we can measure length up to arbitrary accuracy, if we do not measure the momentum. So, to accommodate the existence of a minimum measurable length scale, the Heisenberg uncertainty principle has to be modified to the generalized uncertainty principle. The generalized uncertainty principle can be derived from a deformed Heisenberg algebra. The deformation of the Heisenberg algebra in turn deforms the coordinate representation of the momentum operator, and this deforms the Laplacian to ∂ i ∂i → ∂ i ∂i (1 − β∂ j ∂j ) [20]. Now using this definition of the deforms of the Laplacian, the deformed Lifshitz action can be written as Sb =
1
2
dd+1 x φ∂ 0 ∂0 φ − κ 2 φ[∂ i ∂i (1 − β∂ j ∂j )]2 φ .
(2)
Here we have to promote that parameter β to a background field, such that it scales as β → b2 β . This ensures that theory still has Lifshitz scaling after it has been deformed by the generalized uncertainty principle. It may be noted that it is common to promote parameters in conformal field theories to background field in this way [53,54]. These background fields have scaling properties that ensure the conformal invariance of the deformed theory. Now we can write this action as Sb =
1
2
dd+1 x φ∂ 0 ∂0 φ − κ 2 ∂ i φ T∂2 (1 − β∂ j ∂j )2 ∂i φ ,
(3)
where T∂ = −∂ i ∂i . It may be noted that the non-local differential operator T∂ is crucial in constructing the fermionic action with Lifshitz scaling. Even though this operator is non-local it can be effectively viewed as a local operator, by using the theory of harmonic extension of functions from Rd to Rd × (0, ∞) [37–42]. Thus, we can define T∂ by its action on functions f : Rd → R, such that its harmonic extension u : Rd × (0, ∞) → R satisfies, T∂ f (x) = −∂y u(x, y)|y=0 . This is because if we start with a function f : Rd → R, and find a harmonic function u : Rd × (0, ∞) → R, such that its restriction to Rd coincides with the original function f : Rd → R, then it is possible to find u by solving a Dirichlet problem. This Dirichlet problem can be expressed in terms of the Laplacian in Rd+1 , which is denoted by ∂d2+1 . So, for x ∈ Rd and y ∈ R, we have, u(x, 0) = φ(x) and ∂d2+1 u(x, y) = 0. In fact, for a smooth function C0∞ (Rd ), there is a unique harmonic extension u ∈ C ∞ (Rd × (0, ∞)). Now as T∂ φ(x) also has a harmonic extension to Rd × (0, ∞), we can obtain the following result, T∂2 φ(x) = ∂y2 u(x, y)|y=0 = −∂ i ∂i u(x, y)|y=0 . Thus, it is possible to
define T∂ = −∂ i ∂i , because T∂2 φ(x) = −∂ i ∂i φ(x). So, we can write T∂ exp ikx = |k| exp ikx, because T∂2 exp ikx = |k|2 exp ikx. Furthermore, if we start with two fields φ1 (x) and φ2 (x), such that u1 (x, y) and u2 (x, y) are their harmonic extensions to C = Rd × (0, ∞), and both these harmonic extensions vanish for |x| → ∞ and |y| → ∞, then we can write [55]
dd xdy u1 (x, y)∂n2+1 u2 (x, y) −
C
dd xdy u2 (x, y)∂n2+1 u1 (x, y) = 0.
(4)
C
Thus, we get the following expression
dd x u1 (x, y)∂y u2 (x, y) − u2 (x, y)∂x u1 (x, y) |y=0 = 0.
Rd
(5)
This can now be written in terms of φ1 (x) and φ2 (x) as
dd x φ1 (x)∂y φ2 (x) − φ2 (x)∂x φ1 (x) = 0.
Rd
(6)
So, the operator T∂ can be moved from φ2 (x) to φ1 (x),
Rd
dd x φ1 (x)T∂ f φ2 (x) =
Rd
dd x φ2 (x)T∂ φ1 (x).
(7)
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Now the Lifshitz bosonic action, consistent with generalized uncertainty principle, can also be written as Sb =
1 2
dd+1 x ∂ µ φ Gµν ∂ ν φ,
(8)
where Gµν can be written as
Gµν =
I1×1 0d×1
01×d
−κ 2 T∂2 (1 − β∂ j ∂j )2 Id×d
.
This equation can now be regarded as defining a scalar product for vector fields, such that for any two vectors V and W , we have
(V (x), W (x)) =
dd+1 x V0 W0 − κ 2 Vi T∂2 (1 − β∂ j ∂j )Wi .
(9)
The group of isometries under this inner product remains invariant. So, we can write, (Λ(V ), Λ(W )) = (V , W ). Thus, we can write Λ0µ Λ0ν − κ 2 Λiµ Λiν T∂ (1 − β∂ j ∂j ) = Gµν . From this we can infer that Λ00 = Λi0 = Λ0i = 0 and Λki Λkj = δij . Now a set of local gamma matrices can be defined, such that {Γµ , Γν } = 2Gµν . Furthermore, an appropriate choice for these local gamma matrices is Γ0 = γ0 and Γi = κ T∂ (1 − β∂ j ∂j )γi , where {γa , γb } = 2ηab . We can thus define a fermionic Lifshitz operator as Γ µ ∂µ = γ 0 ∂0 + κγ i T∂ (1 − β∂ j ∂j )∂i . We observe that Γ µ ∂µ Γ ν ∂ν = ∂ 0 ∂0 − κ 2 [∂ i ∂i (1 − 2β∂ j p∂j )]2 . The Lifshitz action for a massless fermionic field can be written as 1 ¯ Γ µ ∂µ ψ dd+1 x ψ Sf = 2
=
1
2
¯ γ 0 ∂0 + γ i κ T∂ (1 − β∂ j ∂j )∂i ψ. dd+1 x ψ
(10)
3. Gauge symmetry In this section, we will analyse gauge theories with Lifshitz corresponding to generalized uncertainty principle. We note that if the covariant derivative is gauge covariant, then so, is any function of the covariant derivative. We will construct a covariant derivative from using a non-abelian gauge C TC . Now if ψ → U ψ , then we should have Dµ ψ → UDµ ψ . We field Aµ = AAµ TA , where [TA , TB ] = ifAB can construct a covariant derivative with this transformation property if, we assume that the gauge field transforms as Aµ → iUDµ U −1 and define the gauge covariant derivative as, Dµ = ∂µ + iAµ . This is because now the covariant derivative will transform as Dµ → UDµ U −1 ,
(11)
and so, Dµ ψ → UDµ ψ , if ψ → U ψ . Now any function of the covariant derivative is also gauge covariant. So, if we take a general function of Dµ , f (Dν Dν )Dµ , then it transforms as f (Dν Dν )Dµ → Uf (Dν Dν )Dµ U −1 , ν
(12)
ν
such that, f (D Dν )Dµ ψ → Uf (D Dν )Dµ ψ . We can now use different f (Dν Dν ), for the spatial and the temporal part of the covariant derivative. Now we define f1 (Dν Dν ) to be the function for the temporal part of the derivative and f2 (Dν Dν ) to be the function for the spatial part of the derivative. The theory has Lifshitz scaling, if we choose f1 (Dν Dν )D0 = D0 , f2 (Dν Dν )Di = κ TD Di , where TD =
−
Di D
D0 → UD0 U
i.
−1
(13)
The covariant derivative will still transform as
,
κ TD Di → U κ TD Di U −1 .
(14)
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Here we have to again assume that β is background field which scales like β → b2 β . [53,54]. This ensures that the theory constructed also has Lifshitz scaling. However, we also want a theory that will correspond to the generalized uncertainty principle. In particular the matter part of the Lagrangian should reduce to the Lagrangian derived in the previous section, if we set all the gauge fields to zero. Thus, we re-define f1 (Dν Dν ) and f2 (Dν Dν ) as f1 (Dν Dν )D0 = D0 , f2 (Dν Dν )Di = κ TD (1 − β Dj Dj )Di .
(15)
It may be noted that the covariant derivative will still transform as D0 → UD0 U −1 ,
κ TD (1 − β Dj Dj )Di → U κ TD (1 − β Dj Dj )Di U −1 .
(16)
So, we can now write the final action as S=
1
¯ 0 D0 + γ i κ TD (1 − β Dj Dj )Di )ψ]. dd+1 x Tr [ψ(γ
2
(17)
¯ 0 D0 )ψ] Now the temporal part of this action is invariant under local gauge transforms because, Tr [ψ(γ −1 0 −1 0 ¯ ¯ → Tr [ψ U U (γ D0 )U U ψ] = Tr [ψ(γ D0 )ψ], and the spatial part of this action is also invariant ¯ i κ TD (1 − β Dj Dj )Di )ψ] → Tr [ψ¯ U −1 U (γ i κ TD (1 − under local gauge transforms because, Tr [ψ(γ j −1 i j ¯ κ TD (1 − β D Dj )Di )ψ]. So, even though this action is non-local, it is β D Dj )Di )U U ψ] = Tr [ψ(γ invariant under local gauge transformations, Aµ → iUDµ U −1 . It may be noted that we can now define a gauge field tensor for this theory as Fi0 = −i[D0 , κ TD (1 − β Dj Dj )Di ], Fij = −i[κ TD (1 − β Dk Dk )Di , κ TD (1 − β Dl Dl )Dj ].
(18)
It transforms as Fi0 → −i[UD0 U −1 , U κ TD (1 − β Dj Dj )Di U −1 ],
= UFi0 U −1 , Fij → −i[U κ TD U −1 (1 − β UDk U −1 UDk U − 1)UDi U −1 , U κ TD U −1 (1 − β UDl U −1 UDl )U −1 UDj U −1 ]
= UFij U −1 .
(19)
Now we can write the action for the gauge part of the action as follows, Sg = −
1 4
dd+1 x Tr [F µν Fµν ].
(20)
It may be noted that even though this action is non-local, it is invariant under local gauge transformations, Aµ → iUDµ U −1 , because, Tr [F µν Fµν ] → Tr [UF µν U −1 UFµν U −1 ] = Tr [F µν Fµν ]. Now we can write the gauge fixing term for this theory,
Sgh =
dd+1 x Tr [b∂ 0 A0 − bκ∂ i T∂ (1 − β∂ j ∂j )Ai ].
(21)
The ghost term for corresponding to this gauge fixing term, can be written as
Sgf =
dd+1 x Tr [¯c ∂ 0 D0 c − κ 2 c¯ ∂ i T∂ (1 − β∂ j ∂j )Di (1 − β Dk Dk )TD c ].
(22)
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4. Stochastic quantization In this section, the stochastic quantization of the Lifshitz fermionic theory based generalized uncertainty principle will be analysed. To perform this analysis an extra fictitious time variable τ ¯ x) → ψ( ¯ x, t ) [46–49]. We will also use an will be introduced, such that ψ(x) → ψ(x, t ), and ψ( appropriate Kernel K (x, y) to ensure the relaxation process is such that the systems will approach equilibrium as τ → 0. The anticommuting fermionic Gaussian noise, η(x, τ ) and η(x, τ ), will satisfy
⟨η(x, τ )⟩ = 0 ⟨η( ¯ x, τ )⟩ = 0, ⟨η(x′ , τ ′ )η( ¯ x, τ )⟩ = 2δ(τ − τ ′ )K (x, x′ )δ 4 (x − x′ ).
(23)
The Langevin equations for the Lifshitz fermionic theory based on the generalized uncertainty principle can be written as
¯ δ S [ψ, ψ] ∂ψ(x, τ ) = − d4 yK (x, y) + η(x, τ ), ∂τ δ ψ¯ ¯ x, τ ) 4 δ S [ψ, ψ] ¯ ∂ ψ( = d y K (x, y) + η( ¯ x, τ ). ∂τ δψ
(24)
Here the action for the Lifshitz fermionic theory based generalized uncertainty principle is given by S=
1
2
¯ 0 D0 + γ i κ TD (1 − β Dj Dj )Di )ψ]. dd+1 x Tr [ψ(γ
(25)
¯ are the dynamical variables, as we are analysing the system It may be noted that only ψ and ψ on a fixed background. The partition function for the Lifshitz fermionic theory based generalized uncertainty principle, can be expressed as
Dη¯ Dη exp −
Z =
1
2
d xd ydτ η( ¯ x, τ )K 4
4
−1
(x, y)η(y, τ ) .
(26)
This partition function can now be expressed as
Z =
¯ Dψ det Dψ
δ η¯ δ ψ¯
−1
δη det δψ
−1
1 × exp − d4 xd4 ydτ η( ¯ x, τ )K −1 (x, y)η(y, τ ) . 2
(27)
Finally, using Langevin equations, we obtain,
¯ ψ] −1 ¯ ψ] −1 ∂ δ 2 S [ψ, δ 2 S [ψ, ∂ − det K −1 + ¯ ∂τ ∂τ δ ψδψ δψδ ψ¯ ¯ ψ] ¯ ψ] 1 ∂ ψ¯ −1 δ S [ψ, ∂ψ δ S [ψ, × exp − . d4 xd4 ydτ K − +K 2 ∂τ δψ ∂τ δ ψ¯
Z =
¯ Dψ det K −1 Dψ
(28)
Here the determinant is defined as the regularized product of eigenvalues. This is done by using ghost fields (c1 , c¯1 , c2 , c¯2 ), and writing [56],
∂ δ2S − c1 = λ¯c1 c1 , ∂τ δψδ ψ¯ δ2S −1 ∂ c¯2 det K − c2 = λ¯c2 c2 . ∂τ δψδ ψ¯
c¯1 det K −1
(29)
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So, we get
¯ −1 ∂ψ ¯ ψ] δ S [ψ, ¯ ψ] ¯ ψ] ∂ 1 ∂ψ 1 δ S [ψ, δ 2 S [ψ, + c¯1 K −1 c1 K − K − ¯ 2 ∂τ ∂τ 2 δψ ∂τ δ ψ¯ δ ψδψ
Leff =
+ c¯2 K
−1
¯ ψ] ∂ δ 2 S [ψ, + c2 . ∂τ δψδ ψ¯
(30)
The auxiliary fields F¯ and F are introduced to write the partition function as [51,57]
Z =
4 ¯ ¯ Dψ Dψ Dc1 Dc¯1 Dc2 Dc¯2 DFDF exp − d xeLeff ,
(31)
where
¯ ψ] ¯ ψ] ∂ ψ¯ −1 δ S [ψ, ∂ψ δ S [ψ, K − F + iF¯ +K ∂τ δψ ∂τ δ ψ¯ 2 2 ¯ ψ] ¯ ψ] ∂ δ S [ψ, δ S [ψ, ∂ + c¯1 K −1 − + c1 + c¯2 K −1 c2 . ¯ ∂τ ∂τ δ ψδψ δψδ ψ¯
Leff = 2F¯ K −1 F + i
(32)
This action can be written using the superfield formalism. Thus, complex superfields superfields
Ω (x, τ , θ , θ¯ ) and Ω (x, τ , θ , θ¯ ), are defined as Ω (x, τ , θ , θ¯ ) = ψ(x, τ ) + θ¯ c1 (x, τ ) + c¯2 (x, τ )θ + iθ θ¯ F (x, τ ),
¯ x, τ ) + θ¯ c2 (x, τ ) + c¯1 (x, τ )θ + iθ¯ θ F¯ (x, τ ). ¯ (x, τ , θ , θ¯ ) = ψ( Ω
(33)
The following superderivatives and supercharges are also defined,
∂ ∂ −θ , ¯ ∂τ ∂θ ∂ ∂ Q¯ = + θ¯ , ∂θ ∂τ
∂ , ∂ θ¯ ¯ = ∂ . D ∂θ
D=
Q =
(34)
The commutators of these superderivatives and supercharges are given by
{D, D¯ } = −
∂ , ∂τ
{Q , Q¯ } =
∂ . ∂τ
(35)
Now we can define a superspace action S = S1 + S2 + S3 , where
S1 =
= S2 =
= S3 =
=
¯Ω ¯ DΩ d4 xedθ¯ dθ D
d4 xe c¯1
∂ c1 ¯ ∂ψ + F F + iF¯ , ∂τ ∂τ
(36)
¯ Ω DΩ ¯ d4 xedθ¯ dθ D
d4 xe c¯2
∂ c2 ∂ ψ¯ + F F¯ + iF¯ , ∂τ ∂τ
¯ , Ω] d4 xedθ¯ dθ S [Ω
d4 xe F¯
δ S3 δ S3 δ 2 S3 δ 2 S3 F + c¯1 + c1 + c¯2 c2 . δψ δ ψ¯ δψδ ψ¯ δψδ ψ¯
It may now be noted that this action S coincides with the component action, if we make the following ¯ K −1 → ψ, ¯ c¯1 K −1 → c¯1 , c¯2 K −1 → c¯2 , and F → F , ψ → ψ, c1 → changes of variables F¯ K −1 → F¯ , ψ c1 , c2 → c2 [37]. Thus, we have performed stochastic quantization of the Lifshitz fermionic theory based generalized uncertainty principle using superspace formalism. It may be noted that stochastic quantization of this system induces a supersymmetry corresponding to the extra fictitious time variable.
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5. Conclusion In this paper we deformed the Lifshitz field theories to make them consistent with the existence of a minimum measurable length. This was done by incorporating generalized uncertainty principle in them. We had to promote a parameter used in the theory to a background field with interesting scaling properties, to preserve the Lifshitz scaling of the deformed theory. We also analysed a deformed Lifshitz theory gauge theory based on the generalized uncertainty principle. We observed that even though this theory is non-local, it is invariant under local gauge transformations. We expect to obtain similar results, if we generalize this work by incorporating terms linear in the momentum, in the deformed Heisenberg algebra [58–62]. It would also be interesting to analyse the BRST symmetry for this theory. We also performed the stochastic quantization of the deformed Dirac equation semiclassically. It may be noted that the deformed version of the general relativity has been obtained based on generalized uncertainty principle [20]. It would be interesting to analyse the effect a combination of Lipschitz scaling and generalized uncertainty principle can have on general relativity. The holographic dual to the Lifshitz field theory has also been studied [63–67]. The dual of the field theory vacuum has a bulk metric, ds2 = −r 2z dt 2 + r 2 dx2 + L2 r −2 dr 2 ,
(37)
where L2 represents the overall curvature scale. It is obvious that for z = 1, this metric reduces to the usual AdS metric. In these Lifshitz theories, the renormalization group flow at finite temperature is used for evaluating the dependence of physical quantities such as the energy density on the momentum scale [68]. Furthermore, the holographic renormalization of gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling [69]. The holographic counter-terms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate value of the dynamical critical exponent. The holographic renormalization of Horava–Lifshitz gravity reproduces the full structure of the z = 2 anisotropic Weyl anomaly in dual field theories in three dimensions [70]. In fact, Lifshitz theories have also become important because of the development of Horava–Lifshitz gravity [71–75]. Horava–Lifshitz gravity is a renormalizable theory of gravity, in which unitarity is not spoiled. Even though gravity is not renormalizable, it can be made renormalizable by adding higher order curvature terms to it. However, the addition of higher order temporal derivatives spoils the unitarity of the theory. A way out of this problem is to add higher order spatial derivatives without adding any higher order temporal derivatives. Even though this breaks Lorentz symmetry, the Horava–Lifshitz theory of gravity reproduces General Relativity in the infrared limit. It may be noted that the string theory comes naturally equipped with a minimum measurable length scale, which is the string length scale. This is because the spacetime cannot be probed below this scale [1]. Furthermore, the existence of a minimum measurable length scale in a theory produces higher derivative correction terms, due to the existence of generalized uncertainty principle [19]. The CFT dual to a massive free scalar field theory with such higher derivative corrections has been analysed using the AdS /CFT correspondence [76]. So, it will be interesting to analyse the Lifshitz deformation of AdS /CFT correspondence, consistent with generalized uncertainty principle. As we have both the fermionic and bosonic actions, it will be interesting to analyse supersymmetric theories based on such deformations. This can be done by constructing various supersymmetric theories based on generalized uncertainty principle with Lifshitz scaling. It may be noted that Lifshitz supersymmetric theories have already been constructed [77]. In fact, according to AdS /CFT correspondence type IIB superstring on AdS 5 × S 5 is dual to the maximally supersymmetric N = 4 super-Yang–Mills theory in four dimensions [78–81], so, this result will can be used to study the gravity dual to such a theory. Furthermore, as AdS 5 × S 5 ∼ SO(2, 4)/SO(1, 4) × SO(6)/SO(5) ⊂ SU (2, 2|4)/SO(1, 4) × SO(5), so, the superisometries of this background are generated by the supergroup SU (2, 2|4), which also generates the superconformal invariance of N = 4 super-Yang–Mills theory in four dimensions. Here the four dimensional superconformal transformations are generated by SO(2, 4) and the R-symmetry is generated by SO(6) ∼ SU (4). Furthermore, N = 4 superYang–Mills theory, with U (N ) as the gauge group, is the low-energy limit for a stack of multiple coincident D3-branes on AdS 5 × S 5 . Here the transverse D3-brane coordinates give rise to six scalar
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57
fields in the N = 4 super-Yang–Mills theory. Apart from these six bosons, there are also sixteen fermions. Thus, a Lifshitz deformation of bulk theory, consistent with generalized uncertainty, may produce interesting deformation of the N = 4 super-Yang–Mills theory. It will be interesting to analyse such deformed super-Yang–Mills theories. Acknowledgements We would like to thank Ali Nassar for pointing out to us an interesting technique used in conformal field theories, i.e., the parameters in a conformal field theory can be promoted to background fields. These background fields can have interesting scaling properties. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
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Incorporation of generalized uncertainty principle into Lifshitz field theories Mir Faizal a,∗ , Barun Majumder b a
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
b
Indian Institute of Technology Gandhinagar, Ahmedabad, 382424, India
article
info
Article history: Received 17 February 2015 Accepted 18 March 2015 Available online 25 March 2015
abstract In this paper, we will incorporate the generalized uncertainty principle into field theories with Lifshitz scaling. We will first construct both bosonic and fermionic theories with Lifshitz scaling based on generalized uncertainty principle. After that we will incorporate the generalized uncertainty principle into a non-abelian gauge theory with Lifshitz scaling. We will observe that even though the action for this theory is non-local, it is invariant under local gauge transformations. We will also perform the stochastic quantization of this Lifshitz fermionic theory based generalized uncertainty principle. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The classical picture of spacetime breaks down in most approaches to quantum gravity. This is due to the fluctuations in the geometry being of order one at Planck scale. Thus, the picture of spacetime as a continuous differential manifold cannot be valid below Planck length. Furthermore, the existence of a minimum length scale is also a feature of string theory [1–5]. In fact, in loop quantum gravity the existence of minimum length turns big bang into a big bounce [6]. However, the existence of minimum length is not consistent with conventional uncertainty principle, which states that one can measure length with arbitrary accuracy, if one takes no measurement of momentum [1,2,7–19]. Thus, the uncertainty principle has to be modified if one wants to incorporate the existence of minimum length scale. These considerations have led to a modification of the Heisenberg uncertainty principle,
∗
Corresponding author. E-mail addresses: [email protected] (M. Faizal), [email protected] (B. Majumder).
http://dx.doi.org/10.1016/j.aop.2015.03.022 0003-4916/© 2015 Elsevier Inc. All rights reserved.
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which in turn has led to a modification of the Heisenberg algebra. It may be noted that the implications of this modified uncertainty principle for quantum field theory have also been studied [20–22]. In this paper, we analyse a quantum field theory based on generalized uncertainty with Lifshitz scaling. Lifshitz field theories are quantum field theories based on an anisotropic scaling between space and time. Lifshitz theories were first introduced in condensed matter physics to model quantum criticality [23–26]. In fact, a Fermi-surface-changing Lifshitz transition occurs for some heavy fermion compounds [27]. The location of this Fermi-surface-changing Lifshitz transition is influenced by carrier doping. Due to strong correlations, a heavy band does not shift rigidly with the chemical potential and the actual shift is determined by the interplay of heavy and additional light bands crossing the Fermi level. Furthermore, meta-magnetic transitions in models for heavy fermions have also been analysed using doped Kondo lattice model in two dimensions [28]. Some heavy fermion metals display a fielddriven quantum phase transition due to a breakdown of the Kondo effect [29,30]. Many of the properties have been described by a Zeeman-driven Lifshitz transition of narrow heavy fermion bands [31]. Materials that cannot be described with the local dielectric response have been described by a generalization of the usual Lifshitz theory [32]. In fact, the temperature correction to the Casimir–Lifshitz free energy between two parallel plates made of dielectric material, possessing a constant conductivity at low temperatures, has been calculated [33]. Lifshitz theory has also been used for calculating the van der Waals and Casimir interaction between graphene and a material plate, graphene and an atom or a molecule, and between a single-wall carbon nanotube and a plate [34]. In this model the reflection properties of electromagnetic oscillations on graphene are governed by the specific boundary conditions imposed on the infinitely thin positively charged plasma sheet, carrying a continuous fluid with some mass and charge density. Fermionic retarded Green’s function with z = 2 has been studied at finite temperature and finite chemical potential [35]. Here the usual Lifshitz geometry was replaced by a Lifshitz black hole. Hawking radiation for Lifshitz fermions has also been studied [36]. Fermionic theories with z = 2 Lifshitz scaling have also been constructed using a non-local differential operator [37]. This non-local differential operator is defined using harmonic extension of a compactly supported function [38–42]. It appears as a map from the Dirichlet-type problem to the Neumann type problem. It may be noted that fermionic theories with z = 3 have also been studied [43,44]. It has been demonstrated that Nambu–Jona-Lasinio type four-fermion coupling at the z = 3 Lifshitz fixed point in four dimensions is asymptotically free and generates a mass scale [45]. In this paper, we will study both bosonic and fermionic Lifshitz field theory, consistent with generalized uncertainty principle. We will also study the gauge symmetry for these theories. It may also be noted that another interesting deformation of quantum mechanics comes from stochastic quantization [46–49]. Stochastic quantization has provided a powerful framework for analysing bosonic theories with Lifshitz scaling [50,51]. In fact, effect of ohmic noise on the non-Markovian spin dynamics resulting in Kondo-type correlations has been studied using stochastic quantization [52] In this paper, we will analyse the stochastic quantization of Lifshitz Dirac equation with minimum length. 2. Generalized Uncertainty Principle In the Lifshitz field theories the scaling is usually taken as x → bx and t → bz t, where b is called the scaling factor and z is called the degree of anisotropy. For z = 1, this reduces to the usual conformal transformation. In this paper, we will analyse the Lifshitz theories with z = 2. The Lifshitz action for a bosonic field with z = 2, can be written as [37] Sb =
1 2
dd+1 x (φ∂ 0 ∂0 φ − κ 2 φ(∂ i ∂i )2 φ).
(1)
The Lifshitz theories are unitarity because they contain no higher order temporal derivatives. So, we will leave the temporal part of the Lifshitz action for a bosonic field undeformed. However, we will deform its spatial part, to make it consistent with the existence of a minimum measurable length [19,20]. The Heisenberg uncertainty principle is not consistent with the existence of a
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51
minimum measurable length, as according to it, we can measure length up to arbitrary accuracy, if we do not measure the momentum. So, to accommodate the existence of a minimum measurable length scale, the Heisenberg uncertainty principle has to be modified to the generalized uncertainty principle. The generalized uncertainty principle can be derived from a deformed Heisenberg algebra. The deformation of the Heisenberg algebra in turn deforms the coordinate representation of the momentum operator, and this deforms the Laplacian to ∂ i ∂i → ∂ i ∂i (1 − β∂ j ∂j ) [20]. Now using this definition of the deforms of the Laplacian, the deformed Lifshitz action can be written as Sb =
1
2
dd+1 x φ∂ 0 ∂0 φ − κ 2 φ[∂ i ∂i (1 − β∂ j ∂j )]2 φ .
(2)
Here we have to promote that parameter β to a background field, such that it scales as β → b2 β . This ensures that theory still has Lifshitz scaling after it has been deformed by the generalized uncertainty principle. It may be noted that it is common to promote parameters in conformal field theories to background field in this way [53,54]. These background fields have scaling properties that ensure the conformal invariance of the deformed theory. Now we can write this action as Sb =
1
2
dd+1 x φ∂ 0 ∂0 φ − κ 2 ∂ i φ T∂2 (1 − β∂ j ∂j )2 ∂i φ ,
(3)
where T∂ = −∂ i ∂i . It may be noted that the non-local differential operator T∂ is crucial in constructing the fermionic action with Lifshitz scaling. Even though this operator is non-local it can be effectively viewed as a local operator, by using the theory of harmonic extension of functions from Rd to Rd × (0, ∞) [37–42]. Thus, we can define T∂ by its action on functions f : Rd → R, such that its harmonic extension u : Rd × (0, ∞) → R satisfies, T∂ f (x) = −∂y u(x, y)|y=0 . This is because if we start with a function f : Rd → R, and find a harmonic function u : Rd × (0, ∞) → R, such that its restriction to Rd coincides with the original function f : Rd → R, then it is possible to find u by solving a Dirichlet problem. This Dirichlet problem can be expressed in terms of the Laplacian in Rd+1 , which is denoted by ∂d2+1 . So, for x ∈ Rd and y ∈ R, we have, u(x, 0) = φ(x) and ∂d2+1 u(x, y) = 0. In fact, for a smooth function C0∞ (Rd ), there is a unique harmonic extension u ∈ C ∞ (Rd × (0, ∞)). Now as T∂ φ(x) also has a harmonic extension to Rd × (0, ∞), we can obtain the following result, T∂2 φ(x) = ∂y2 u(x, y)|y=0 = −∂ i ∂i u(x, y)|y=0 . Thus, it is possible to
define T∂ = −∂ i ∂i , because T∂2 φ(x) = −∂ i ∂i φ(x). So, we can write T∂ exp ikx = |k| exp ikx, because T∂2 exp ikx = |k|2 exp ikx. Furthermore, if we start with two fields φ1 (x) and φ2 (x), such that u1 (x, y) and u2 (x, y) are their harmonic extensions to C = Rd × (0, ∞), and both these harmonic extensions vanish for |x| → ∞ and |y| → ∞, then we can write [55]
dd xdy u1 (x, y)∂n2+1 u2 (x, y) −
C
dd xdy u2 (x, y)∂n2+1 u1 (x, y) = 0.
(4)
C
Thus, we get the following expression
dd x u1 (x, y)∂y u2 (x, y) − u2 (x, y)∂x u1 (x, y) |y=0 = 0.
Rd
(5)
This can now be written in terms of φ1 (x) and φ2 (x) as
dd x φ1 (x)∂y φ2 (x) − φ2 (x)∂x φ1 (x) = 0.
Rd
(6)
So, the operator T∂ can be moved from φ2 (x) to φ1 (x),
Rd
dd x φ1 (x)T∂ f φ2 (x) =
Rd
dd x φ2 (x)T∂ φ1 (x).
(7)
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Now the Lifshitz bosonic action, consistent with generalized uncertainty principle, can also be written as Sb =
1 2
dd+1 x ∂ µ φ Gµν ∂ ν φ,
(8)
where Gµν can be written as
Gµν =
I1×1 0d×1
01×d
−κ 2 T∂2 (1 − β∂ j ∂j )2 Id×d
.
This equation can now be regarded as defining a scalar product for vector fields, such that for any two vectors V and W , we have
(V (x), W (x)) =
dd+1 x V0 W0 − κ 2 Vi T∂2 (1 − β∂ j ∂j )Wi .
(9)
The group of isometries under this inner product remains invariant. So, we can write, (Λ(V ), Λ(W )) = (V , W ). Thus, we can write Λ0µ Λ0ν − κ 2 Λiµ Λiν T∂ (1 − β∂ j ∂j ) = Gµν . From this we can infer that Λ00 = Λi0 = Λ0i = 0 and Λki Λkj = δij . Now a set of local gamma matrices can be defined, such that {Γµ , Γν } = 2Gµν . Furthermore, an appropriate choice for these local gamma matrices is Γ0 = γ0 and Γi = κ T∂ (1 − β∂ j ∂j )γi , where {γa , γb } = 2ηab . We can thus define a fermionic Lifshitz operator as Γ µ ∂µ = γ 0 ∂0 + κγ i T∂ (1 − β∂ j ∂j )∂i . We observe that Γ µ ∂µ Γ ν ∂ν = ∂ 0 ∂0 − κ 2 [∂ i ∂i (1 − 2β∂ j p∂j )]2 . The Lifshitz action for a massless fermionic field can be written as 1 ¯ Γ µ ∂µ ψ dd+1 x ψ Sf = 2
=
1
2
¯ γ 0 ∂0 + γ i κ T∂ (1 − β∂ j ∂j )∂i ψ. dd+1 x ψ
(10)
3. Gauge symmetry In this section, we will analyse gauge theories with Lifshitz corresponding to generalized uncertainty principle. We note that if the covariant derivative is gauge covariant, then so, is any function of the covariant derivative. We will construct a covariant derivative from using a non-abelian gauge C TC . Now if ψ → U ψ , then we should have Dµ ψ → UDµ ψ . We field Aµ = AAµ TA , where [TA , TB ] = ifAB can construct a covariant derivative with this transformation property if, we assume that the gauge field transforms as Aµ → iUDµ U −1 and define the gauge covariant derivative as, Dµ = ∂µ + iAµ . This is because now the covariant derivative will transform as Dµ → UDµ U −1 ,
(11)
and so, Dµ ψ → UDµ ψ , if ψ → U ψ . Now any function of the covariant derivative is also gauge covariant. So, if we take a general function of Dµ , f (Dν Dν )Dµ , then it transforms as f (Dν Dν )Dµ → Uf (Dν Dν )Dµ U −1 , ν
(12)
ν
such that, f (D Dν )Dµ ψ → Uf (D Dν )Dµ ψ . We can now use different f (Dν Dν ), for the spatial and the temporal part of the covariant derivative. Now we define f1 (Dν Dν ) to be the function for the temporal part of the derivative and f2 (Dν Dν ) to be the function for the spatial part of the derivative. The theory has Lifshitz scaling, if we choose f1 (Dν Dν )D0 = D0 , f2 (Dν Dν )Di = κ TD Di , where TD =
−
Di D
D0 → UD0 U
i.
−1
(13)
The covariant derivative will still transform as
,
κ TD Di → U κ TD Di U −1 .
(14)
M. Faizal, B. Majumder / Annals of Physics 357 (2015) 49–58
53
Here we have to again assume that β is background field which scales like β → b2 β . [53,54]. This ensures that the theory constructed also has Lifshitz scaling. However, we also want a theory that will correspond to the generalized uncertainty principle. In particular the matter part of the Lagrangian should reduce to the Lagrangian derived in the previous section, if we set all the gauge fields to zero. Thus, we re-define f1 (Dν Dν ) and f2 (Dν Dν ) as f1 (Dν Dν )D0 = D0 , f2 (Dν Dν )Di = κ TD (1 − β Dj Dj )Di .
(15)
It may be noted that the covariant derivative will still transform as D0 → UD0 U −1 ,
κ TD (1 − β Dj Dj )Di → U κ TD (1 − β Dj Dj )Di U −1 .
(16)
So, we can now write the final action as S=
1
¯ 0 D0 + γ i κ TD (1 − β Dj Dj )Di )ψ]. dd+1 x Tr [ψ(γ
2
(17)
¯ 0 D0 )ψ] Now the temporal part of this action is invariant under local gauge transforms because, Tr [ψ(γ −1 0 −1 0 ¯ ¯ → Tr [ψ U U (γ D0 )U U ψ] = Tr [ψ(γ D0 )ψ], and the spatial part of this action is also invariant ¯ i κ TD (1 − β Dj Dj )Di )ψ] → Tr [ψ¯ U −1 U (γ i κ TD (1 − under local gauge transforms because, Tr [ψ(γ j −1 i j ¯ κ TD (1 − β D Dj )Di )ψ]. So, even though this action is non-local, it is β D Dj )Di )U U ψ] = Tr [ψ(γ invariant under local gauge transformations, Aµ → iUDµ U −1 . It may be noted that we can now define a gauge field tensor for this theory as Fi0 = −i[D0 , κ TD (1 − β Dj Dj )Di ], Fij = −i[κ TD (1 − β Dk Dk )Di , κ TD (1 − β Dl Dl )Dj ].
(18)
It transforms as Fi0 → −i[UD0 U −1 , U κ TD (1 − β Dj Dj )Di U −1 ],
= UFi0 U −1 , Fij → −i[U κ TD U −1 (1 − β UDk U −1 UDk U − 1)UDi U −1 , U κ TD U −1 (1 − β UDl U −1 UDl )U −1 UDj U −1 ]
= UFij U −1 .
(19)
Now we can write the action for the gauge part of the action as follows, Sg = −
1 4
dd+1 x Tr [F µν Fµν ].
(20)
It may be noted that even though this action is non-local, it is invariant under local gauge transformations, Aµ → iUDµ U −1 , because, Tr [F µν Fµν ] → Tr [UF µν U −1 UFµν U −1 ] = Tr [F µν Fµν ]. Now we can write the gauge fixing term for this theory,
Sgh =
dd+1 x Tr [b∂ 0 A0 − bκ∂ i T∂ (1 − β∂ j ∂j )Ai ].
(21)
The ghost term for corresponding to this gauge fixing term, can be written as
Sgf =
dd+1 x Tr [¯c ∂ 0 D0 c − κ 2 c¯ ∂ i T∂ (1 − β∂ j ∂j )Di (1 − β Dk Dk )TD c ].
(22)
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4. Stochastic quantization In this section, the stochastic quantization of the Lifshitz fermionic theory based generalized uncertainty principle will be analysed. To perform this analysis an extra fictitious time variable τ ¯ x) → ψ( ¯ x, t ) [46–49]. We will also use an will be introduced, such that ψ(x) → ψ(x, t ), and ψ( appropriate Kernel K (x, y) to ensure the relaxation process is such that the systems will approach equilibrium as τ → 0. The anticommuting fermionic Gaussian noise, η(x, τ ) and η(x, τ ), will satisfy
⟨η(x, τ )⟩ = 0 ⟨η( ¯ x, τ )⟩ = 0, ⟨η(x′ , τ ′ )η( ¯ x, τ )⟩ = 2δ(τ − τ ′ )K (x, x′ )δ 4 (x − x′ ).
(23)
The Langevin equations for the Lifshitz fermionic theory based on the generalized uncertainty principle can be written as
¯ δ S [ψ, ψ] ∂ψ(x, τ ) = − d4 yK (x, y) + η(x, τ ), ∂τ δ ψ¯ ¯ x, τ ) 4 δ S [ψ, ψ] ¯ ∂ ψ( = d y K (x, y) + η( ¯ x, τ ). ∂τ δψ
(24)
Here the action for the Lifshitz fermionic theory based generalized uncertainty principle is given by S=
1
2
¯ 0 D0 + γ i κ TD (1 − β Dj Dj )Di )ψ]. dd+1 x Tr [ψ(γ
(25)
¯ are the dynamical variables, as we are analysing the system It may be noted that only ψ and ψ on a fixed background. The partition function for the Lifshitz fermionic theory based generalized uncertainty principle, can be expressed as
Dη¯ Dη exp −
Z =
1
2
d xd ydτ η( ¯ x, τ )K 4
4
−1
(x, y)η(y, τ ) .
(26)
This partition function can now be expressed as
Z =
¯ Dψ det Dψ
δ η¯ δ ψ¯
−1
δη det δψ
−1
1 × exp − d4 xd4 ydτ η( ¯ x, τ )K −1 (x, y)η(y, τ ) . 2
(27)
Finally, using Langevin equations, we obtain,
¯ ψ] −1 ¯ ψ] −1 ∂ δ 2 S [ψ, δ 2 S [ψ, ∂ − det K −1 + ¯ ∂τ ∂τ δ ψδψ δψδ ψ¯ ¯ ψ] ¯ ψ] 1 ∂ ψ¯ −1 δ S [ψ, ∂ψ δ S [ψ, × exp − . d4 xd4 ydτ K − +K 2 ∂τ δψ ∂τ δ ψ¯
Z =
¯ Dψ det K −1 Dψ
(28)
Here the determinant is defined as the regularized product of eigenvalues. This is done by using ghost fields (c1 , c¯1 , c2 , c¯2 ), and writing [56],
∂ δ2S − c1 = λ¯c1 c1 , ∂τ δψδ ψ¯ δ2S −1 ∂ c¯2 det K − c2 = λ¯c2 c2 . ∂τ δψδ ψ¯
c¯1 det K −1
(29)
M. Faizal, B. Majumder / Annals of Physics 357 (2015) 49–58
55
So, we get
¯ −1 ∂ψ ¯ ψ] δ S [ψ, ¯ ψ] ¯ ψ] ∂ 1 ∂ψ 1 δ S [ψ, δ 2 S [ψ, + c¯1 K −1 c1 K − K − ¯ 2 ∂τ ∂τ 2 δψ ∂τ δ ψ¯ δ ψδψ
Leff =
+ c¯2 K
−1
¯ ψ] ∂ δ 2 S [ψ, + c2 . ∂τ δψδ ψ¯
(30)
The auxiliary fields F¯ and F are introduced to write the partition function as [51,57]
Z =
4 ¯ ¯ Dψ Dψ Dc1 Dc¯1 Dc2 Dc¯2 DFDF exp − d xeLeff ,
(31)
where
¯ ψ] ¯ ψ] ∂ ψ¯ −1 δ S [ψ, ∂ψ δ S [ψ, K − F + iF¯ +K ∂τ δψ ∂τ δ ψ¯ 2 2 ¯ ψ] ¯ ψ] ∂ δ S [ψ, δ S [ψ, ∂ + c¯1 K −1 − + c1 + c¯2 K −1 c2 . ¯ ∂τ ∂τ δ ψδψ δψδ ψ¯
Leff = 2F¯ K −1 F + i
(32)
This action can be written using the superfield formalism. Thus, complex superfields superfields
Ω (x, τ , θ , θ¯ ) and Ω (x, τ , θ , θ¯ ), are defined as Ω (x, τ , θ , θ¯ ) = ψ(x, τ ) + θ¯ c1 (x, τ ) + c¯2 (x, τ )θ + iθ θ¯ F (x, τ ),
¯ x, τ ) + θ¯ c2 (x, τ ) + c¯1 (x, τ )θ + iθ¯ θ F¯ (x, τ ). ¯ (x, τ , θ , θ¯ ) = ψ( Ω
(33)
The following superderivatives and supercharges are also defined,
∂ ∂ −θ , ¯ ∂τ ∂θ ∂ ∂ Q¯ = + θ¯ , ∂θ ∂τ
∂ , ∂ θ¯ ¯ = ∂ . D ∂θ
D=
Q =
(34)
The commutators of these superderivatives and supercharges are given by
{D, D¯ } = −
∂ , ∂τ
{Q , Q¯ } =
∂ . ∂τ
(35)
Now we can define a superspace action S = S1 + S2 + S3 , where
S1 =
= S2 =
= S3 =
=
¯Ω ¯ DΩ d4 xedθ¯ dθ D
d4 xe c¯1
∂ c1 ¯ ∂ψ + F F + iF¯ , ∂τ ∂τ
(36)
¯ Ω DΩ ¯ d4 xedθ¯ dθ D
d4 xe c¯2
∂ c2 ∂ ψ¯ + F F¯ + iF¯ , ∂τ ∂τ
¯ , Ω] d4 xedθ¯ dθ S [Ω
d4 xe F¯
δ S3 δ S3 δ 2 S3 δ 2 S3 F + c¯1 + c1 + c¯2 c2 . δψ δ ψ¯ δψδ ψ¯ δψδ ψ¯
It may now be noted that this action S coincides with the component action, if we make the following ¯ K −1 → ψ, ¯ c¯1 K −1 → c¯1 , c¯2 K −1 → c¯2 , and F → F , ψ → ψ, c1 → changes of variables F¯ K −1 → F¯ , ψ c1 , c2 → c2 [37]. Thus, we have performed stochastic quantization of the Lifshitz fermionic theory based generalized uncertainty principle using superspace formalism. It may be noted that stochastic quantization of this system induces a supersymmetry corresponding to the extra fictitious time variable.
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5. Conclusion In this paper we deformed the Lifshitz field theories to make them consistent with the existence of a minimum measurable length. This was done by incorporating generalized uncertainty principle in them. We had to promote a parameter used in the theory to a background field with interesting scaling properties, to preserve the Lifshitz scaling of the deformed theory. We also analysed a deformed Lifshitz theory gauge theory based on the generalized uncertainty principle. We observed that even though this theory is non-local, it is invariant under local gauge transformations. We expect to obtain similar results, if we generalize this work by incorporating terms linear in the momentum, in the deformed Heisenberg algebra [58–62]. It would also be interesting to analyse the BRST symmetry for this theory. We also performed the stochastic quantization of the deformed Dirac equation semiclassically. It may be noted that the deformed version of the general relativity has been obtained based on generalized uncertainty principle [20]. It would be interesting to analyse the effect a combination of Lipschitz scaling and generalized uncertainty principle can have on general relativity. The holographic dual to the Lifshitz field theory has also been studied [63–67]. The dual of the field theory vacuum has a bulk metric, ds2 = −r 2z dt 2 + r 2 dx2 + L2 r −2 dr 2 ,
(37)
where L2 represents the overall curvature scale. It is obvious that for z = 1, this metric reduces to the usual AdS metric. In these Lifshitz theories, the renormalization group flow at finite temperature is used for evaluating the dependence of physical quantities such as the energy density on the momentum scale [68]. Furthermore, the holographic renormalization of gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling [69]. The holographic counter-terms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate value of the dynamical critical exponent. The holographic renormalization of Horava–Lifshitz gravity reproduces the full structure of the z = 2 anisotropic Weyl anomaly in dual field theories in three dimensions [70]. In fact, Lifshitz theories have also become important because of the development of Horava–Lifshitz gravity [71–75]. Horava–Lifshitz gravity is a renormalizable theory of gravity, in which unitarity is not spoiled. Even though gravity is not renormalizable, it can be made renormalizable by adding higher order curvature terms to it. However, the addition of higher order temporal derivatives spoils the unitarity of the theory. A way out of this problem is to add higher order spatial derivatives without adding any higher order temporal derivatives. Even though this breaks Lorentz symmetry, the Horava–Lifshitz theory of gravity reproduces General Relativity in the infrared limit. It may be noted that the string theory comes naturally equipped with a minimum measurable length scale, which is the string length scale. This is because the spacetime cannot be probed below this scale [1]. Furthermore, the existence of a minimum measurable length scale in a theory produces higher derivative correction terms, due to the existence of generalized uncertainty principle [19]. The CFT dual to a massive free scalar field theory with such higher derivative corrections has been analysed using the AdS /CFT correspondence [76]. So, it will be interesting to analyse the Lifshitz deformation of AdS /CFT correspondence, consistent with generalized uncertainty principle. As we have both the fermionic and bosonic actions, it will be interesting to analyse supersymmetric theories based on such deformations. This can be done by constructing various supersymmetric theories based on generalized uncertainty principle with Lifshitz scaling. It may be noted that Lifshitz supersymmetric theories have already been constructed [77]. In fact, according to AdS /CFT correspondence type IIB superstring on AdS 5 × S 5 is dual to the maximally supersymmetric N = 4 super-Yang–Mills theory in four dimensions [78–81], so, this result will can be used to study the gravity dual to such a theory. Furthermore, as AdS 5 × S 5 ∼ SO(2, 4)/SO(1, 4) × SO(6)/SO(5) ⊂ SU (2, 2|4)/SO(1, 4) × SO(5), so, the superisometries of this background are generated by the supergroup SU (2, 2|4), which also generates the superconformal invariance of N = 4 super-Yang–Mills theory in four dimensions. Here the four dimensional superconformal transformations are generated by SO(2, 4) and the R-symmetry is generated by SO(6) ∼ SU (4). Furthermore, N = 4 superYang–Mills theory, with U (N ) as the gauge group, is the low-energy limit for a stack of multiple coincident D3-branes on AdS 5 × S 5 . Here the transverse D3-brane coordinates give rise to six scalar
M. Faizal, B. Majumder / Annals of Physics 357 (2015) 49–58
57
fields in the N = 4 super-Yang–Mills theory. Apart from these six bosons, there are also sixteen fermions. Thus, a Lifshitz deformation of bulk theory, consistent with generalized uncertainty, may produce interesting deformation of the N = 4 super-Yang–Mills theory. It will be interesting to analyse such deformed super-Yang–Mills theories. Acknowledgements We would like to thank Ali Nassar for pointing out to us an interesting technique used in conformal field theories, i.e., the parameters in a conformal field theory can be promoted to background fields. These background fields can have interesting scaling properties. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
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