Rotationally Symmetric Yang-Mills Gauge Theory and Generalized Uncertainty Principle

Rotationally Symmetric Yang-Mills Gauge Theory and Generalized Uncertainty Principle

Atale Omprakash

Khandesh College Education Society’s Moolji Jaitha College, Jalgaon, Maharashtra, India (Corresponding author’s e-mail: atale.om@outlook.com)

Received: 7 February 2022, Revised: 20 June 2022, Accepted: 27 June 2022, Published: 15 November 2022

Abstract

Recently, it has been of considerably high interest to introduce the notion of minimal length in quantum mechanics and quantum field theories to get a proper description of quantum gravity. In this paper, we have attempted to unify the notion of minimal length with 𝑅 × 𝑆³ topological field theories which were introduced by Carmeli and Malin in 1985 [14]. We have collectively called this (𝑅 × 𝑆³)𝐻 topological field theories where the “H” in the sub-script stands for Heisenberg’s notion of minimal length. A proper description for equations like Schrodinger’s equation, Klein-Gordon equation and QED, QCD Lagrangian on (𝑅 × 𝑆³)𝐻 topology is derived.

Keywords: Quantum field theory, Minimal length, Schrodinger’s equation, Dirac’s equation, Klein- Gordon equation

Introduction

The holy grail of modern theoretical physics since mid-1900s is the problem of quantum gravity.

Numerous attempts since then were done to unify quantum mechanics and general relativity. One interesting consequence of this unification is that in quantum gravity there exists a minimal observable distance at the order of Plank’s length lp= √^Gℏ

c³= 10⁻³³. It is widely assumed by physicists from last few decades that an existence of a minimal length is necessary for the consistent formulation of quantum gravity theories and it has been observed from the studies of blackhole that for any type of quantum gravity theory to exist, there should first exist a minimum measurable length in the theory that is near the magnitude of Plank’s length. References for studying quantum theories at minimal length are [1-12]. Further exposure to the references regarding minimal length gravity theories alongside with quantum mechanics and quantum field theories can be found in [13]. The idea of the existence of a minimal length is not consistent with Heisenberg’s uncertainty principle because according to it, the minimum measurable length is 0. This inconsistency is removed by introducing a parameter in the uncertainty principle and the resulting uncertainty principle is known as generalized uncertainty principle [7]. Since the equation for uncertainty principal changes, the mathematical definition of underlying operators also changes. The generalized uncertainty principle can be written as:

Δx ≥ ^ℏ

Δp+ β^Δp

ℏ (1) where β is the deformation parameter which determines the strength of the modification of the uncertainty principle and is of order of Plank’s length. This leads to the following modification in the commutation relation between position and momentum operators:

[x̂^μp̂ν] = iℏδμν(1 + βp̂^ρp̂ρ) (2)

Operators in momentum space and position space now become [13] and,

x̂^μ= i(1 + βp̂^ρp̂_ρ) ^∂

∂p_μ, p̂_μ= p_μ, (3) and

x̂^μ= x^μ, p̂_μ= −iℏ(1 − β ∂^ρ∂_ρ) ∂_μ, (4)

respectively. The above representation of operators in fundamental in formulating theories at minimal length.

R × S³ topological field theory was established by Carmeli and Malka [20] a series of 7 papers [14- 20]. This attempt was motivated to describe those systems which are solely angular dependent rather than distance as in Minkowskian space-time. In R × S³ topology, x is replaced with θ = (θ¹, θ², θ³) which are 3 rotational angles such as Euler angles and L = (L₁, L₂, L₃) = (L_x, L_y, L_z) is the corresponding differential operator given by,

L1=^−sinθ3

sinθ²

∂

∂θ¹− cosθ^{3 ∂}

∂θ²+ cotθ³sinθ^{3 ∂}

∂θ³, (5)

L2=^−cosθ3

sinθ²

∂

∂θ¹− sinθ^{3 ∂}

∂θ²+ cotθ³cosθ^{3 ∂}

∂θ³, (6)

L₃= − ^∂

∂θ³. (7)

The operator L²= L²₁+ L²₂+ L²₃= L²_x+ L²_y+ L²_z is then give by,

L²= ¹

sinθ²

∂

∂θ²(sinθ^{2 ∂}

∂θ²) + ¹

sin²θ²( ^∂2

∂θ¹²− 2cosθ^{2 ∂2}

∂θ¹∂θ³+ ^∂2

∂θ³²). (8)

In relativistic notation θ^α= (θ⁰, θ¹, θ², θ³), θ⁰= ct and,

J^α= (J⁰, J^k) = (E, γJ), (9)

Jα= (J0, Jk) = (E, −γJ) (10)

is the angular momentum four vector with J = (Jx, Jy, Jz) and Jk= iℏγLk. Furthermore, using the definition of J^α, J_α and γ = c(m₀/I₀)^1/2 we can conclude the following relation:

EJ= J⁰= J0= ±(γ²J²+ I₀²γ⁴)^1/2, (11)

J^αJ_α= E_J²− γ²J²= I₀²γ⁴. (12)

here, m0 and I0 are rest mass and moment of inertia, respectively.

There are 2 ways in which we could construct (R × S³)H topological field theories. We get the same results however the approaches differ on priority basis. When R × S³ topological field theories were constructed by Carmeli and Malka [20], they did so by making the transition,

p = −iℏ∇→ J = −iℏL. (13)

Thus, for example, the standard form of Schrodinger’s equation in Cartesian coordinates was transformed into R × S³ topological form by simply making the transition from P to J and making the wave function depend on θ instead of x. Therefore, [14],

(^−ℏ2

2m₀∇²+ V) ψ = iℏ^∂ψ

∂t (14)

changes to

(^−ℏ2

2I₀L²+ V) ψ = iℏ^∂ψ

∂t. (15)

Now, the 2 methods that we mentioned are: First, writing down equations corresponding to minimal length and then making the transition to R × S³ topology. Second, making the transition to R × S³ topology and then formulating it in the framework of minimal length. We are going to follow the first route since most of the theories are already formulated in the framework of minimal length and now, we just have to make the transition to arrive on (R × S³)_H topological field theories.

Schrodinger’s equation and Klein-Gordon equation

The main aim of this section is to introduce Schrodinger’s equation and Klein-Gordon equation on (R × S³)H topology. Since now we are dealing with coordinated containing Euler’s angles, for the purpose of simplicity, we wouldn’t be writing the whole equivalent definition of differential operator every time, rather, we would use the short hand notation. Now, Eqs. (3) and (4) in R × S³ topological coordinates takes the following form:

θ̂^μ= i(1 + βĴ^ρĴ_ρ) ^∂

∂J_μ, Ĵ_μ= J_μ (16) and

θ̂^μ= θ^μ, Ĵ_μ= −iℏ(1 − βL^ρL_ρ)L_μ, (17)

respectively. There are 3 types of Schrodinger’s equation that we are going to study: First and second equation are in coordinate space whereas the third equation is in momentum space.

Consider the following minimal length Schrodinger’s equation from [21]:

(^pop2

2m₀+ V(x)) ψ_n(x) = E_nψ_n(x) (18)

where p_op= p(1 + βp²). If E_n⁽⁰⁾ is the eigenvalue of

H0= ^p2

2m₀+ V(x) (19)

then Eq. (18) can explicitly be written as

ℏ² 2m₀

d²ψ_n(x)

dx² + (En− 4m0β (E_n⁽⁰⁾− V(x))²− V(x)) ψn(x) = 0. (20)

This equation was originally introduced to calculate the scattering states of Woods-Saxon interaction. Now, using representation from Eq. (13), if E_n⁽⁰⁾ is now the eigenvalue of

H0= ^J2

2I₀+ V(θ) (21)

then Eq. (20) on (R × S³)_H can be written as

ℏ²

2I₀L²_xψ_n(θ) + (E_n− 4I₀β (E_n⁽⁰⁾− V(θ))²− V(θ)) ψ_n(θ) = 0. (22) Authors in [22] came up with a minimal length Schrodinger’s equation that can be solved by factorization method:

∂⁴ψ(x)

∂x⁴ − ³

2βℏ²

∂²ψ(x)

∂x² −^3m0E′

βℏ⁴ ψ(x) = 0 (23)

The corresponding momentum operator that they have used is slightly different form usually what it is,

pop= p (1 +^β

3p²). (24)

On (R × S³)H topology, Eq. (23) can be written as

L4xψ(x) − ³

2βℏ²L2xψ(x) −^3I0E′

βℏ⁴ ψ(x) = 0. (25)

Now let us focus our attention on stationary state minimal length Schrodinger’s equation for harmonic oscillator in momentum space [1]:

d²ψ(p) dp² + ^2βp

1+βp² dψ(p)

dp +_(1+βp¹₂₎₂(ε − η²p²)ψ(p) = 0 (26) where

ε = ^2E

m₀ℏ²ω² (27)

and

η =_(m ¹

0ℏω)². (28)

Corresponding Hamiltonian for above equation is

H = ^P2

2m₀+ m₀ω^{2 x2}

2. (29)

On (R × S³)H topology, Eq. (26) takes the form

d²ψ(J) dJ² + ^2βJ

1+βJ² dψ(J)

dJ +_(1+βJ¹₂₎₂(ε − η²J²)ψ(J) = 0 (30) where m0 in Eq. (27) and (28) is now replaced by I0 and the corresponding Hamiltonian is thus given by

H = ^J2

2I₀+ I0ω^{2 θ2}

2. (31)

We know that standard Klein-Gordon equation in terms of momentum can be written as (ℏ = c = 1)

(p^μp_μ+ m₀²)ϕ = 0 (32)

For a simpler case of 1 + 1 dimensions, consider the following form

(p_t²− px2+ m02)ϕ = 0. (33)

By the means of auxiliary variables [3], one can arrive at the following equation:

(∂^μ∂μ+ 2β(∂^μ∂μ)(∂^μ∂μ) + m02)ϕ = 0. (34)

Thus, on (R × S³)_H topology, one can write the Klein-Gordon equation as

(L^μLμ+ 2β(L^μLμ)(L^μLμ) + I02)ϕ = 0. (35) Dirac’s equation: Global, local and other complicated symmetries of the Lagrangian

Consider the standard Dirac’s Lagrangian

ℒ = ψ̅(iγ^μ∂_μ+ m)ψ. (36)

The corresponding minimal length Dirac’s Lagrangian thus can be given by [13]

ℒ = ψ̅(i(1 − β ∂^ρ∂ρ)γ^μ∂μ+ m)ψ (37) and thus, the corresponding (R × S³)H topological Dirac’s Lagrangian takes the following form [20,23]:

ℒ = ψ̅(i(1 − β ∂^ρ∂ρ)γ^μ∂μ+ m)ψ (38)

= ψ̅ ⁱ

2γ^μ L⃡ ψ − ψ_μ ̅ⁱ

2βL^ρL_ργ^μ L⃡ ψ +_μ ³

2aψ̅γ⁰γ⁵ψ − I₀ψ̅ψ. (39) here ψ̅ L⃡ ψ = ψμ ̅(Lμψ) − (L_μψ̅)ψ, I0 is the rest momentum of inertia of the particle and the additional term has been added because such a term does not violate parity nor parity + charge conjugation [23]. The constant a denotes radius of the universe. The additional term containing a gets smaller and smaller as time passes by but surely had played an important role in the beginning of the universe after the big bang.

The (R × S³)_H topological Dirac’s Lagrangian can be seen to be invariant under global U(1) phase transformation. Thus, if we apply the transformations

ψ(θ) → ψ′(θ) = e^−iξψ(θ) (40)

ψ̅(θ) → ψ̅̅̅(θ) = ψ^′ ̅(θ)e^iξ (41) or the infinitesimal form of it

δ_εψ(θ) = ψ′(θ) − ψ(θ) = −iεψ(θ) (42)

δ_εψ̅(θ) = ψ′̅̅̅(θ) − ψ̅(θ) = iεψ̅(θ) (43) then Lagrangian (39) transforms as

ℒ′ = ψ′̅̅̅ⁱ

2γ^μ L⃡ ψ′ − ψ′μ ̅̅̅ⁱ

2βL^ρLργ^μ L⃡ ψ′ +μ ³

2aψ′̅̅̅γ⁰γ⁵ψ′ − I0ψ′̅̅̅ψ′ (44)

= ψ̅e^{iξ i}

2γ^μ L⃡ eμ ^−iξψ − ψ̅e^{iξ i}

2βL^ρLργ^μ L⃡ eμ ^−iξψ + ³

2aψ̅e^iξγ⁰γ⁵e^−iξψ − I0ψ̅e^iξe^−iξψ (45)

= ψ̅ ⁱ

2γ^μ L⃡ ψ − ψμ ̅ⁱ

2βL^ρLργ^μ L⃡ ψ +μ ³

2aψ̅γ⁰γ⁵ψ − I0ψ̅ψ (46)

= ℒ (47)

This concludes that Lagrangian (39) is invariant under the global U(1) phase transformation. Now, we want to promote our global symmetry to local, namely

δεψ(θ) = −iε(θ)ψ(θ) (48)

δ_εψ̅(θ) = iε(θ)ψ̅(θ) (49) and

δ_ε(L_μψ(θ)) = L_μ(δ_εψ(θ)) = L_μ(−iε(θ)ψ(θ)) (50)

= −i ((Lμε(θ)) + ε(θ)Lμ) ψ(θ). (51)

Things get a little bit tricky here due to the presence of the term L⃡ μ in the Lagrangian. Therefore, it’s perhaps better to separate those terms in Lagrangian which will change under variation from those which will remain same. Thus, we can write Lagrangian (Eq. (39)) as,

ℒ = ψ̅ⁱ

2(1 − βL^ρL_ρ)γ^μ L⃡ ψ +_μ ³

2aψ̅γ⁰γ⁵ψ − I₀ψ̅ψ (52)

=ⁱ

2(1 − βL^ρL_ρ)γ^μ(ψ̅(L_μψ) − (L_μψ̅)ψ) + ³

2aψ̅γ⁰γ⁵ψ − I₀ψ̅ψ (53) and the Lagrangian now transforms as,

δℒ = ⁱ

2(1 − βL^ρLρ)γ^μ(δψ̅(Lμψ) + ψ̅(Lμδψ) − (Lμδψ̅)ψ − (Lμψ̅)δψ) + ³

2a(δψ̅γ⁰γ⁵ψ + ψ̅γ⁰γ⁵δψ) − I0δψ̅ψ − I0ψ̅δψ (54)

=ⁱ

2(1 − βL^ρLρ)γ^μ(iεψ̅(Lμψ) − iψ̅(Lμεψ + εLμψ) − i(Lμεψ̅ + εLμψ̅)ψ + (Lμψ̅)iεψ) +

3

2a(iεψ̅γ⁰γ⁵ψ − ψ̅γ⁰γ⁵iεψ) − I0iεψ̅ψ + I0ψ̅iεψ (55)

=ⁱ

2(1 − βL^ρL_ρ)γ^μ(iεψ̅(L_μψ) − iψ̅L_μεψ − iψ̅εL_μψ − iL_μεψ̅ψ − iεL_μψ̅ψ + (L_μψ̅)iεψ) (56)

= −(1 − βL^ρL_ρ)(γ^μψ̅L_μεψ) (57)

≠ 0 (58)

Thus, Lagrangian (Eq. (39)) is not invariant under the local phase transformation because there is no additional term whose variation will cancel out the product that we have got. To make the Lagrangian invariant under the local phase transformation, we introduce the following covariant derivative:

Dμψ(θ) = (Lμ+ ieAμ(θ)) ψ(θ) (59)

whose variation can be written as

δ (D_μψ(θ)) = −iε(θ)D_μψ(θ) (60)

and for the additional field A_μ

δ (A_μ(θ)) =¹

eL_με(θ). (61) Therefore, we have

ℒ =ⁱ

2(1 − βD^ρD_ρ)γ^μ(ψ̅(D_μψ) − (D_μψ̅)ψ) + ³

2aψ̅γ⁰γ⁵ψ − I₀ψ̅ψ (62)

= ψ̅ ⁱ

2(1 − βD^ρD_ρ)γ^μ D⃡ ψ +_{μ 2a}³ ψ̅γ⁰γ⁵ψ − I₀ψ̅ψ (63) After substituting the value of covariant derivative and opening the brackets, Eq. (63) can explicitly be written as

ℒ =ⁱ

2(γ^μψ̅L_μψ − γ^μψ̅ieA_μψ − γ^μL_μψ̅ψ + γ^μieA_μψ̅ψ − βL^ρL_ργ^μψ̅L_μψ + βL^ρL_ργ^μψ̅ieA_μψ + βL^ρL_ργ^μL_μψ̅ψ − βL^ρL_ργ^μieA_μψ̅ψ + βL^ρieA_ργ^μψ̅Lμψ − βL^ρieA_ργ^μψ̅ieAμψ − βL^ρieA_ργ^μL_μψ̅ψ + βL^ρieA_ργ^μieA_μψ̅ψ + βieA^ρL_ργ^μψ̅L_μψ − βieA^ρL_ργ^μψ̅ieA_μψ − βieA^ρL_ργ^μL_μψ̅ψ +

βieA^ρL_ργ^μieA_μψ̅ψ − βieA^ρieA_ργ^μψ̅L_μψ + βieA^ρieA_ργ^μψ̅ieA_μψ + βieA^ρieA_ργ^μL_μψ̅ψ − βieA^ρieA_ργ^μieA_μψ̅ψ) + ³

2aψ̅γ⁰γ⁵ψ − I₀ψ̅ψ. (64) One can see that the Lagrangian containing covariant derivatives and additional fields is now invariant under the local phase transformation.

Now, let us generalize our Lagrangian to more complicated symmetries. Consider the following Lagrangian containing several complex fields where ψ̅_k now belongs to a non trivial representation R of some internal symmetry group G:

ℒ = ψ̅_kⁱ

2(1 − βL^ρL_ρ)γ^μ L⃡ ψ_{μ k}+ ³

2aψ̅_kγ⁰γ⁵ψ_k− I₀ψ̅_kψ_k (65) where k = 1,2,3, . . . dimR is the internal symmetry index. If we apply the transformations

ψk→ ψ′k= (Uψ)k= (e^−iθaTaψ)_k (66)

ψ̅_k→ ψ′̅̅̅

k= (ψ̅U^†)_k= (ψ̅e^−iθaTa)

k (67)

or the infinitesimal form of it

δ_εψ_k= −iε^aT_kl^aψ_l (68)

δ_εψ̅_k= iε^aψ̅_l(T^a)_lk (69)

where a = 1,2,3. . . dimG, ε^a and θ^a are infinitesimal and T^a (Hermitian) is the generator of internal symmetry group G satisfying Lie algebra of Lie group G:

[T^a, T^b] = if^abcT^c, a, b, c = dimG. (70)

Thus

ℒ′ = ψ′̅̅̅

k i

2(1 − βL^ρL_ρ)γ^μ L⃡ ψ′_μ k+ ³

2aψ′̅̅̅

kγ⁰γ⁵ψ′_k− I₀ψ′̅̅̅

kψ′_k (71)

= (ψ̅U^†)_kⁱ

2(1 − βL^ρL_ρ)γ^μ L⃡ (Uψ)_μ k+ ³

2a(ψ̅U^†)_kγ⁰γ⁵(Uψ)_k− I₀(ψ̅U^†)_k(Uψ)_k (72)

= (ψ̅e^−iθaTa)_kⁱ

2(1 − βL^ρLρ)γ^μ L⃡ (eμ ^−iθaTaψ)_k+ ³

2a(ψ̅e^−iθaTa)_kγ⁰γ⁵(e^−iθaTaψ)_k− I₀(ψ̅e^−iθaTa)

k(e^−iθaTaψ)

k

(73)

= ψ̅_kⁱ

2(1 − βL^ρLρ)γ^μ L⃡ ψμ k+ ³

2aψ̅_kγ⁰γ⁵ψk− I0ψ̅_kψk (74)

= ℒ (75)

Or the other way around, using Eqs. (68) and (69), we have

δℒ = δψ̅_kⁱ

2(1 − βL^ρL_ρ)γ^μ L⃡ ψ_μ k+ ψ̅_kⁱ

2(1 − βL^ρL_ρ)γ^μ ↔ L_μδψ_k+ ³

2aδψ̅_kγ⁰γ⁵ψ_k+ ³

2aψ̅_kγ⁰γ⁵δψ_k−

I0δψ̅_kψk− I0ψ̅_kδψk (76)

= iε^aψ̅_l(T^a)_lkⁱ

2(1 − βL^ρLρ)γ^μ L⃡ ψμ k− ψ̅_kⁱ

2(1 − βL^ρLρ)γ^μ ↔ Lμiε^aT_kl^aψl+ ³

2aiε^aψ̅_l(T^a)_lkγ⁰γ⁵ψk−

3

2aψ̅_kγ⁰γ⁵iε^aT_kl^aψ_l− I₀iε^aψ̅_l(T^a)_lkψ_k+ I₀ψ̅_kiε^aT_kl^aψ_l (77)

= 0 (78)

where we have used the fact that (T^a)^†= T^a and that internal symmetry generators commute with gamma matrices. Let us now promote our global symmetry to local. For this, we apply the following transformations

δεψk= −iε^a(θ)T_kl^aψl (79)

δ_εψ̅_k= iε^a(θ)ψ̅_lT_lk^a (80)

δ(Lμψk) = Lμ(−iε^a(θ)T_kl^aψl) = −i (L_με^a(θ)) T_kl^aψl− iε^a(θ)T_kl^aLμψl (81)

δ(Lμψ̅_k) = Lμ(iε^a(θ)T_lk^aψ̅_l) = i (L_με^a(θ)) T_lk^aψ̅_l+ iε^a(θ)T_lk^aLμψ̅_l (82)

Thus, under the above local transformation, Lagrangian (Eq. (64)) transforms as δℒ =ⁱ 2(1 − βL^ρLρ)γ^μ(δψ̅_k(Lμψk) + ψ̅_k(Lμδψk) − (Lμδψ̅_k)ψk− (Lμψ̅_k)δψk) + ³ 2a(δψ̅_kγ⁰γ⁵ψk+ ψ̅_kγ⁰γ⁵δψ_k) − I₀δψ̅_kψ_k− I₀ψ̅_kδψ_k =ⁱ 2(1 − βL^ρL_ρ)γ^μ(iε^aψ̅_lT_lk^a(L_μψ_k) − ψ̅_kiL_με^aT_kl^aψ_l− ψ̅_kiε^aT_kl^aL_μψ_l− iL_με^aT_lk^aψ̅_lψ_k− iε^aT_lk^aL_μψ̅_lψ_k+ i(L_μψ̅_k)ε^aT_kl^aψ_l) +³ 2a(iε^a(θ)ψ̅_lT_lk^aγ⁰γ⁵ψ_k− ψ̅_kγ⁰γ⁵iε^a(θ)T_kl^aψ_l) − I₀iε^a(θ)ψ̅_lT_lk^aψ_k+ I₀ψ̅_kiε^a(θ)T_kl^aψ_l (83)

=¹ 2(1 − βL^ρL_ρ)γ^μ(ψ̅_kL_με^aT_kl^aψ_l+ L_με^aT_lk^aψ̅_lψ_k) (84)

≠ 0. (85)

Thus, as in the case of (53), there is no extra term here whose variation would cancel the extra term that we have obtained. Therefore, we now introduce the following covariant derivative with an appropriate coupling constant g: Dμψ = (Lμ+ igT^aAaμ)ψ (86)

whose variation is given by δ(D_μψ) k= −iε^a(θ)T_kl^a(D_μψ) l. (87)

We can further write (86) as D_μψ_k= L_μψ_k+ igT_kl^aA_μ^aψ_l= (L_μδ_kl+ igT_kl^aA^a_μ)ψ_l. (88)

Thus igT_kl^aδ(Aaμψl) = δ(Dμψk) − Lμδψk (89)

= −iε^aT_kl^a(D_μψ) l− L_μδψ_k (90)

= −iε^aT_kl^aδ(L_μψ_l+ igT_lm^b A^b_μψ_m) − L_μ(−iε^aT_kl^aψ_l) (91)

= −iε^aT_kl^aδ(L_μψ_l+ igT_lm^b A^b_μψ_m) + i(L_με^a)T_kl^aψ_l+ iε^aT_kl^aL_μψ_l (92)

= i(L_με^a)T_kl^aψ_l+ gε^aT_kl^aT_lm^b A^b_μψ_m. (93)

On the other hand, we can write igT_kl^aδ(A^a_μψ_l) as igT_kl^aδ(A^a_μψ_l) = igT_kl^aδA^a_μψ_l+ gT_kl^aA^a_μδψ_l (94)

= igT_kl^aδA^a_μψ_l+ gT_kl^aA^a_μ(−iε^b)T_lm^b ψ_m (95)

= igT_kl^aδAaμψl+ gε^aT_kl^bAbμT_lm^a ψm. (96)

Substituting the above result in (89), we get

igT_kl^aδ(Aaμψl) = i(Lμε^a)T_kl^aψl+ gε^a(T^aT^b)_kmAbμψm− gε^a(T^bT^a)_kmAbμψm (97)

= i(L_με^a)T_kl^aψ_l+ gε^a[T^a, T^b]

kmA^b_μψ_m (98)

= i(Lμε^a)T_kl^aψl+ gε^af^abc(T^c)_kmAbμψm. (99)

Thus, after further simplification one can obtain the following variation:

δA^a_μ=¹

gL_με^a− f^abcA^b_με^c. (100)

Therefore, using

δεψk= −iε^a(θ)T_kl^aψl (101)

δ_εψ̅_k= iε^a(θ)ψ̅_lT_lk^a (102)

and Eq. (97), one can easily arrive at the conclusion that the Lagrangian

ℒ = ψ̅_kⁱ

2(1 − βD^ρD_ρ)γ^μ D⃡ ψ_{μ k}+ ³

2aψ̅_kγ⁰γ⁵ψ_k− I₀ψ̅_kψ_k

(103)

is now invariant under the infinitesimal local phase transformation.

Electrodynamics and gauge fields

In this section, we are going to introduce the field strength tensor on (R × S³)_H topology and then we will expose it to different transformations and symmetries as we did in previous case for our modified Dirac’s Lagrangian in previous section. This will be a relatively short section as compare to the previous one because as we have now been already introduced to the concept of global symmetries, local symmetries and covariant derivative, we are going to omit obvious steps during calculations. Beginning from the field strength tensor on (R × S³) topology, we have [17],

F_μν= L_μA_ν− L_νA_μ. (104)

Now, the minimal length field strength tensor is given by [13],

ℱ_μν= −i[𝒟_μ, 𝒟_ν] = −i[(1 − βD^ρD_ρ)D_μ, (1 − βD^ρD_ρ)D_ν]. (105)

Thus, we get

ℱμν= Fμν− 2βD^ρDρFμν− β(D^ρFμρDν− D^ρFνρDμ) − β(FμρD^ρDν− FνρD^ρDμ) + O(β²) (106)

and

ℱ_μνℱ^μν= F_μνF^μν− 4βF_μνD^ρD_ρF^μν− 4βF_μνF^μρD_ρD^ν− 4βF_μνD_ρF^μρD^ν+ O(β²) (107)

where the field strength tensor in the above equation is now defined by (104). Inserting the expression for covariant derivative yields

FμνF^μν= FμνF^μν− β(4FμνLρL^ρF^μν+ 4iLρeA^ρFμνF^μν+ 8iFμνeAρL^ρF^μν+ 8iL^ρeAνFμρF^μν+ 4ieA^ρF_μρL_νF^μν− 8eA^ρF_μρeA_νF^μν− 4eA^ρeA_ρF_μνF^μν

(108)

and further inserting the value of field strength tensor yields

F_μνF^μν= L_μA_νL^μA^ν− L_μA_νL^νA^μ− L_νA_μL^μA^ν+ L_νA_μL^νA^μ− 4βL_μA_νL_ρL^ρL^μA^ν+ 4βL_νA_μL_ρL^ρL^μA^ν+ 4βLμAνLρL^ρL^νA^μ− 4βLνAμLρL^ρL^νA^μ− 4βiLρeA^ρLμAνL^μA^ν+ 4βiLρeA^ρLμAνL^νA^μ+

4βiL_ρeA^ρL_νA_μL^μA^ν− 4βiL_ρeA^ρL_νA_μL^νA^μ− 8βiL_μA_νeA_ρL^ρL^μA^ν+ 8βiL_νA_μeA_ρL^ρL^μA^ν+ 8βiLμAνeAρL^ρL^νA^μ− 8βiLνAμeAρL^ρL^νA^μ− 8βiL^ρeAνLμAρL^μA^ν+ 8βiL^ρeAνLρAμL^μA^ν+ 8βiL^ρeA_νL_μA_ρL^νA^μ− 8βiL^ρeA_νL_ρA_μL^νA^μ− 4βieA^ρL_μA_ρL_νL^μA^ν+ 4βieA^ρL_ρA_μL_νL^μA^ν+ 4βieA^ρLμAρLνL^νA^μ− 4βieA^ρLρAμLνL^νA^μ+ 8βeA^ρLμAρeAνL^μA^ν− 8βeA^ρLρAμeAνL^μA^ν− 8βeA^ρL_μA_ρeA_νL^νA^μ+ 8βeA^ρL_ρA_μeA_νL^νA^μ+ 4βeA^ρeA_ρL_μA_νL^μA^ν− 4βeA^ρeA_ρL_μA_νL^νA^μ− 4βeA^ρeAρLνAμL^μA^ν+ 4βeA^ρeAρLνAμL^νA^μ.

(109)

Note that the above representation of the field strength tensor on (R × S³)H topology is invariant under the infinitesimal local phase transformation. Using the above expression, we can construct the dynamical part of the gauge field which when added to Dirac’s Lagrangian would give a complete Lagrangian for quantum electrodynamics. For defining the finite symmetry transformation matrix, we can define a unitary representation of the group. Thus, let

U = e^{−iθa(θ)Ta} (110)

and define the following set of transformations

ψ → Uψ (111)

ψ̅ → ψ̅U (112) Aμ= UAμU⁻¹−¹

ig(LμU)U⁻¹. (113)

where A_μ= T^aA^a_μ. One can see that F_μν is not invariant under the above set of transformations:

F_μν= ¹

ig(L_μUL_νU⁻¹− L_νUL_μU⁻¹) + L_μUA_νU⁻¹+ UA_νL_μU⁻¹− L_νUA_μU⁻¹− UA_μL_νU⁻¹+

U(L_μA_ν− L_νA_μ)U⁻¹. (114)

Note that under the gauge transformation, ig[A_μ, A_ν] transforms as

ig[A_μ, A_ν] = igU[A_μ, A_ν]U⁻¹−¹

ig(L_μUL_νU⁻¹− L_νUL_μU⁻¹) − [L_μUA_νU⁻¹+ UA_νL_μU⁻¹− L_νUA_μU⁻¹−

UAμLνU⁻¹] (115)

Comparing (114) and (115) we can see that if we write field strength tensor as,

F_μν= L_μA_ν− L_νA_μ+ ig[A_μ, A_ν] (116)

then under that gauge transformations (111) - (113) F_μν transforms covariantly:

Fμν→ UFμνU⁻¹. (117)

In component form, writing Aμ as T^aAaμ, (116) takes the form

F_μν^a = L_μA^a_v− L_νA^a_μ− gf^abcA^b_μA^c_ν= −F_νμ^a . (118)

and thus

Fμνa F^μνa= (LμAav− LνAaμ− gf^abcAμbAcν)(L^μA^νa− L^vA^μa− gf^apqA^μpA^νq). (119)

Conclusions

In this paper, we have established (R × S³)_H topological field theories where we have unified the notion of minimal length in field theories on (R × S³)topology. In the beginning we derive a set of Schrodinger’s equations along with Klein-Gordon equation. From the above results, we can now construct quantum electrodynamics Lagrangian on (R × S³)H as

ℒQED= ψ̅ ⁱ

2(1 − βD^ρDρ)γ^μ D⃡ ψ +μ ³

2aψ̅γ⁰γ⁵ψ − I0ψ̅ψ −¹

4(FμνF^μν− β(4FμνLρL^ρF^μν+ 4iLρeA^ρFμνF^μν+ 8iFμνeAρL^ρF^μν+ 8iL^ρeAνFμρF^μν+ 4ieA^ρFμρLνF^μν− 8eA^ρFμρeAνF^μν−

4eA^ρeA_ρF_μνF^μν) (120)

The above Lagrangian is invariant under the infinitesimal local phase transformation:

δ_εψ(θ) = −iε(θ)ψ(θ) (121)

δεψ̅(θ) = iε(θ)ψ̅(θ) (122) and

δ (Aμ(θ)) =¹

eLμε(θ). (123) Similarly, we can construct quantum chromodynamics Lagrangian which will describe fermions integrating with a non-Abelian gauge field on (R × S³)_H topology. Thus

ℒ_QCD= ψ̅_kⁱ

2(1 − βD^ρD_ρ)γ^μ D⃡ ψ_μ k+ ³

2aψ̅_kγ⁰γ⁵ψ_k− I₀ψ̅_kψ_k−¹

4(F_μν^a F^μνa− β(4F_μν^a L_ρL^ρF^μνa+ 4iL_ρeA^ρF_μν^a F^μνa+ 8iF_μν^a eA_ρL^ρF^μνa+ 8iL^ρeA_νF_μρ^a F^μνa+ 4ieA^ρF_μρ^a L_νF^μνa− 8eA^ρF_μρ^a eA_νF^μνa−

4eA^ρeAρFμνa F^μνa). (124)

This Lagrangian is invariant under the local phase transformation:

ψ → Uψ (125)

ψ̅ → ψ̅U (126)

Aμ= UAμU⁻¹−¹

ig(LμU)U⁻¹. (127)

or the infinitesimal form of it. Here A_μ= T^aA^a_μ. The above presented model is based on a simple generalized uncertainty principle. There exist many other generalized formulations of the uncertainty principle constructed for different purposes.

The main reason of formulating field theories on R × S³ topology is due to the fact that the surface of the sphere S³ in a space has considerably more symmetries than other spaces. From the results that we have presented in this paper, it would be interesting to canonically quantize the above lightly presented version of Yang-Mills theory and introduce path integral formulation which would indeed help us to introduce Faddeev-Popov ghosts and BRST invariance in our theory.

References

[1] A Kempf, G Mangano and RB Mann. Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D 1995; 52, 1108.

[2] A Awad and A Ali. Planck-scale corrections to Friedmann equation. Cent. Eur. J. Phys. 2014; 12, 245-5.

[3] P Bosso, S Das and V Todorinov. Quantum field theory with the generalized uncertainty principle I:

Scalar electrodynamics. Ann. Phys. 2020; 422, 168319.

[4] P Bosso, S Das and V Todorinov. Quantum field theory with the generalized uncertainty principle II:

Quantum leectrodynamics. Ann. Phys. 2020; 424, 168350.

[5] LN Chang, Z Lewis, D Minic and T Takeuchi. On the minimal length uncertainty relation and the foundations of string theory. Adv. High Energ. Phys. 2011; 2011, 493514.

[6] H Hinrichsen and A Kempf. Maximal localization in the presence of minimal uncertainties in positions and in momenta. J. Math. Phys. 1996; 37, 2121-37.

[7] K Nozari and M Karami. Minimal length and generalized Dirac equation. Mod. Phys. Lett. A 2005;

20, 3095-03.

[8] A Kempf and G Mangano. Minimal length uncertainty relation and ultraviolet regularization. Phys.

Rev. D 1997; 55, 7909-20.

[9] M Lubo. Quantum minimal length and trans-Planckian photons. Phys. Rev. D 2000; 61, 124009.

[10] LN Chang, D Minic, N Okamura and T Takeuchi. Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations. Phys. Rev. D 2002; 65, 125027.

[11] S Hossenfelder, M Bleicher, S Hofmann, J Ruppert, S Scherer and H Stöcker. Signatures in the Planck regime. Phys. Lett. B 2003; 575, 85-99.

[12] U Harbach, S Hossenfelder, M Bleicher and H Stöcker. Probing the minimal length scale by precision tests of the muon g−2. Phys. Lett. B 2004; 584, 109-13.

[13] M Kober. Gauge theories under incorporation of a generalized uncertainty principle. Phys. Rev. D 2010; 82, 085017.

[14] M Carmeli. Field theory on R×S³ topology. I: The Klein-Gordon and Schrödinger equations. Found.

Phys. 1985; 15, 175-84.

[15] M Carmeli and S Malin. Field theory on R×S³ topology. II: The Weyl equation. Found. Phys. 1985;

15, 185-91.

[16] M Carmeli and S Malin. Field theory on R×S³ topology. III: The Dirac equation. Found. Phys. 1985;

15, 1019-29.

[17] M Carmeli and S Malin. Field theory on R×S³ topology. IV: Electrodynamics of magnetic moments.

Found. Phys. 1986; 16, 791-806.

[18] M Carmeli and S Malin. Field theory on R×S³ topology. V: SU 2-gauge theory. Found. Phys. 1987;

17, 193-200.

[19] M Carmeli and S Malin. Field theory on R×S³ topology. VI: Gravitation. Found. Phys. 1987; 17, 407- 17.

[20] M Carmeli and A Malka. Field theory on R×S³ topology: Lagrangian formulation. Found. Phys. 1990;

20, 71-110.

[21] H Hassanabadi, S Zarrinkamar and E Maghsoodi. Scattering states of Woods-Saxon interaction in minimal length quantum mechanics. Phys. Lett. B 2012; 718, 678-82.

[22] SA Khorram-Hosseini, S Zarrinkamar and H Panahi. Minimal length Schrödinger equation via factorisation approach. Pramana 2019; 92, 53.

[23] MA Dariescu, C Dariescu and I Gottlieb. Gauge theory of fermions on R×S³ spacetime. Found. Phys.

1995; 25, 959-63.