On the origin of generalized uncertainty principle from compactified   M5-brane

Alireza Sepehri; Anirudh Pradhan; A. Beesham

arXiv:1706.04639·gr-qc·July 18, 2017

On the origin of generalized uncertainty principle from compactified M5-brane

Alireza Sepehri, Anirudh Pradhan, A. Beesham

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TL;DR

This paper shows how compactification in M-theory naturally leads to a deformation of field theory consistent with the generalized uncertainty principle, linking high-dimensional theories to quantum gravity effects.

Contribution

It demonstrates that compactification in M-theory can produce a field theory deformation aligned with the GUP, providing a theoretical origin for the GUP from M-brane dynamics.

Findings

01

Higher derivative terms in M3-brane action relate to GUP deformations

02

Constructed a Heisenberg algebra consistent with GUP

03

Linked M-theory compactification to quantum gravity phenomenology

Abstract

In this paper, we demonstrate that compactification in M-theory can lead to a deformation of field theory consistent with the generalized uncertainty principle (GUP).We observe that the matter fields in the M3-brane action contain higher derivative terms. We demonstrate that such terms can also be constructed from a reformulation of the field theory by the GUP. In fact, we will construct the Heisenberg algebra consistent with this deformation, and explicitly demonstrate it to be the Heisenberg algebra obtained from the GUP. Thus, we use compactification in M-theory to motivate for the existence of the GUP.

Equations82

Δ x_{i} Δ p_{i}

Δ x_{i} Δ p_{i}

[x_{i}, p_{j}] = i ℏ (δ_{ij} + β δ_{ij} p^{2} + 2 β p_{i} p_{j}) .

[x_{i}, p_{j}] = i ℏ (δ_{ij} + β δ_{ij} p^{2} + 2 β p_{i} p_{j}) .

S = - T_{p} \int

S = - T_{p} \int

E_{mn} = G_{mn} + B_{mn}, Q_{j}^{i} = δ_{j}^{i} + iλ [X^{j}, X^{k}] E_{k j},

E_{mn} = G_{mn} + B_{mn}, Q_{j}^{i} = δ_{j}^{i} + iλ [X^{j}, X^{k}] E_{k j},

S_{D 0} = - \frac{1}{2 g _{s} l _{s}} \int d tT r (Σ_{i = 1}^{9} {\partial_{t} X^{i} \partial_{t} X^{i} - \frac{1}{4 π ^{2} l _{s}^{4}} [X^{i}, X^{j}]^{2}})

S_{D 0} = - \frac{1}{2 g _{s} l _{s}} \int d tT r (Σ_{i = 1}^{9} {\partial_{t} X^{i} \partial_{t} X^{i} - \frac{1}{4 π ^{2} l _{s}^{4}} [X^{i}, X^{j}]^{2}})

[X^{0}, X^{i}] = iλ \partial_{t} X^{i} [X^{m}, X^{m}] = 0

[X^{0}, X^{i}] = iλ \partial_{t} X^{i} [X^{m}, X^{m}] = 0

S_{D 0} = - \frac{1}{8 g _{s} π ^{2} l _{s}^{5}} \int d tT r (Σ_{m = 0}^{9} [X^{m}, X^{n}]^{2})

S_{D 0} = - \frac{1}{8 g _{s} π ^{2} l _{s}^{5}} \int d tT r (Σ_{m = 0}^{9} [X^{m}, X^{n}]^{2})

Σ_{m} \to \frac{1}{( 2 π ) ^{p}} \int d^{p + 1} σ Σ_{m - p - 1}

Σ_{m} \to \frac{1}{( 2 π ) ^{p}} \int d^{p + 1} σ Σ_{m - p - 1}

[X^{a}, X^{i}] = iλ \partial_{a} X^{i} [X^{a}, X^{b}] = i λ^{2} F^{ab}

i, j = p + 1, .., 9 a, b = 0, 1, ... p m, n = 0, 1, .., 9

S_{D p} = - T_{p} \int

S_{D p} = - T_{p} \int

S_{M 0} = T_{M 0} \int d tT r (Σ_{M, N, L = 0}^{10} [X^{M}, X^{N}, X^{L}]^{2})

S_{M 0} = T_{M 0} \int d tT r (Σ_{M, N, L = 0}^{10} [X^{M}, X^{N}, X^{L}]^{2})

S_{M 0}

S_{M 0}

[X^{a}, X^{b}, X^{i}]^{2} = \frac{1}{2} (\partial_{a} X^{i})^{2} [X^{a}, X^{b}, X^{c}]^{2} = (F^{ab c})^{2^{\overset{a}{^} \texteuro \textcent}}

[X^{a}, X^{b}, X^{i}]^{2} = \frac{1}{2} (\partial_{a} X^{i})^{2} [X^{a}, X^{b}, X^{c}]^{2} = (F^{ab c})^{2^{\overset{a}{^} \texteuro \textcent}}

Σ_{m} \to \frac{1}{( 2 π ) ^{p}} \int d^{p + 1} σ Σ_{m - p - 1}

i, j = p + 1, .., 10 a, b = 0, 1, ... p m, n = 0, .., 10

F_{ab c} = \partial_{a} A_{b c} - \partial_{b} A_{c a} + \partial_{c} A_{ab}

F_{ab c} = \partial_{a} A_{b c} - \partial_{b} A_{c a} + \partial_{c} A_{ab}

S_{M 5}

S_{M 5}

A_{ab} = \partial_{a} A_{b} - \partial_{b} A_{a}

A_{ab} = \partial_{a} A_{b} - \partial_{b} A_{a}

A_{a 4} = \partial_{a} X^{4} - \partial_{4} A_{a}

A_{a 4} = \partial_{a} X^{4} - \partial_{4} A_{a}

A_{a 5} = \partial_{a} X^{5} - \partial_{5} A_{a}

A_{a 5} = \partial_{a} X^{5} - \partial_{5} A_{a}

F_{aa s} = \partial_{a} A_{a s} - \partial_{a} A_{s a} + \partial_{s} A_{aa}

F_{aa s} = \partial_{a} A_{a s} - \partial_{a} A_{s a} + \partial_{s} A_{aa}

F_{aa 4} = 2 \partial_{a}^{2} X^{4} - 2 \partial_{4} \partial_{a} A_{a}

F_{aa 4} = 2 \partial_{a}^{2} X^{4} - 2 \partial_{4} \partial_{a} A_{a}

F_{a 44} = 2 \partial_{4}^{2} A_{a} - 2 \partial_{4} \partial_{a} X^{4}

F_{ab 4} = 2 \partial_{a} \partial_{b} X^{4} - 2 \partial_{4} \partial_{a} A_{b} + 2 \partial_{4} \partial_{b} A_{a}

F_{aa 5} = 2 \partial_{a}^{2} X^{5} - 2 \partial_{5} \partial_{a} A_{a}

F_{aa 5} = 2 \partial_{a}^{2} X^{5} - 2 \partial_{5} \partial_{a} A_{a}

F_{a 55} = 2 \partial_{5}^{2} A_{a} - 2 \partial_{5} \partial_{a} X^{5}

F_{ab 5} = 2 \partial_{a} \partial_{b} X^{5} - 2 \partial_{5} \partial_{a} A_{b} + 2 \partial_{5} \partial_{b} A_{a}

\partial_{4} A_{a} = 0 \partial_{5} A_{a} = 0

\partial_{4} A_{a} = 0 \partial_{5} A_{a} = 0

\partial_{4} \partial_{a} A_{a} = 0 \partial_{5} \partial_{a} A_{a} = 0 \to

F_{aa 4} = 2 \partial_{a}^{2} X^{4} F_{aa 5} = 2 \partial_{a}^{2} X^{5}

F_{a 44} = - 2 \partial_{4} \partial_{a} X^{4} F_{a 55} = - 2 \partial_{5} \partial_{a} X^{5}

F_{ab 4} = 2 \partial_{a} \partial_{b} X^{4} F_{aa 5} = 2 \partial_{a} \partial_{b} X^{5}

x^{4} = x^{5} = 2 π R \to

x^{4} = x^{5} = 2 π R \to

\partial_{4} = \partial_{5} = \frac{1}{2 π} \partial_{R}, \int d x^{4} = \int d x^{5} = 2 π \int d r \to

\int d x^{4} F_{a 44} = - 2 \int d x^{4} \partial_{4} \partial_{a} X^{4} =

- 2 \int d x^{5} \partial_{4} \partial_{a} X^{4} = 2 \partial_{a} X^{4}

\int d x^{5} F_{a 55} = - 2 \int d x^{5} \partial_{5} \partial_{a} X^{5} =

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On the origin of generalized uncertainty principle from compactified $M5$ -brane

Alireza Sepehri 1[email protected]

1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran,P.O.Box:55134-441.

Anirudh Pradhan 2[email protected]

2Department of Mathematics, Institute of Applied Sciences and Humanities, G L A University, Mathura-281 406, Uttar Pradesh, India.

A. Beesham 3[email protected]

3 Department of Mathematical Sciences, University of Zululand, Kwa-Dlangezwa 3886, South Africa.

Abstract

In this paper, we demonstrate that compactification in M-theory can lead to a deformation of field theory consistent with the generalized uncertainty principle (GUP). We observe that the matter fields in the $M3$ -brane action contain higher derivative terms. We demonstrate that such terms can also be constructed from a reformulation of the field theory by the GUP. In fact, we will construct the Heisenberg algebra consistent with this deformation, and explicitly demonstrate it to be the Heisenberg algebra obtained from the GUP. Thus, we use compactification in $M$ -theory to motivate for the existence of the GUP.

PACS numbers: 98.80.-k, 04.50.Gh, 11.25.Yb, 11.25.-w

Keywords: GUP, M3 and M5-branes, Lie 3-algebra

I Introduction

Despite the fact that we as yet do not have a full theory of quantum gravity (QG), various different approaches to quantum gravity have been proposed. These approaches to quantum gravity can be used to gain phenomenological understanding of the nature of the actual quantum theory of gravity. It is interesting to note that various different approaches to quantum gravity have predicted the existence of a minimum measurable length scale. These approaches include string theory and the semi-classical physics of black holes. The existence of a minimum length deforms the usual uncertainty principle to a (GUP) q1 ; q2

[TABLE]

where $p^{2}=\sum\limits_{j}p_{j}p_{j}$ , $\beta=\beta_{0}/(M_{p}c)^{2}=\beta_{0}\frac{\ell_{p}^{2}}{\hbar^{2}}$ , $M_{p}=$ Planck mass, and $M_{p}c^{2}=$ Planck energy. However, as the uncertainty principle bears close resemblance to the Heisenberg algebra, the deformation of the uncertainty principle also deforms the Heisenberg algebra to q3 ; q4

[TABLE]

This form ensures, via the Jacobi identity, that $[x_{i},x_{j}]=0=[p_{i},p_{j}]$ q3 ; q4 .

The GUP has opened up an interesting window for quantum gravity phenomenology because it implies a modification of the fundamental commutation relation between position and momentum and this in turn also produces correction terms for all quantum mechanical systems q12 ; q14 . It may be noted that the GUP deformation has been generalized to include the full spacetime deformation, and the quantum field theory corresponding to such a deformation has also been constructed w1 ; w2 .

Even though the GUP has been motivated from various approaches to quantum gravity including string theory, it has not been studied within the framework of compactification. On the other hand, compactification seems to deform spacetime geometry by a fundamental length scale, so it is expected to produce a deformation similar to the GUP deformed field theory. In this paper, we demonstrate that compactification can be used to motivate the deformation of matter fields consistent with the GUP. To investigate this, we have to obtain a relevant action for the $M3$ -brane.

The gauge field in the action for two M2-branes is known to be valued in a Lie $3$ -algebra rather than a Lie algebra q17 ; q20 . The action for two $M2$ -branes can also be constructed without using a Lie $3$ -algebra as long as the matter fields transform in the bi-fundamental representation. However, the Lie $3$ -algebra valued gauged fields are important in the construction of a $M5$ -brane action. This proposal for the $M5$ -brane action is based on an infinite dimensional Lie $3$ -algebra and a Nambu-Poisson structure on three dimensional manifolds. This model is expected to contain self-dual $2$ -form gauge fields in six dimensions, and so the resulting action may be interpreted as the action for $M5$ -brane world-volume q22 .

Motivated by these results, we propose a new model for constructing M3-branes from $M0$ -branes. This is done by replacing the Nambu-Poisson structure in two dimensions by the structure in three dimensions. Then, we demonstrate that $N$ $M0$ -branes join each other and form a $M5$ -brane. This $M5$ -brane action is then compactified with a brane on two circles. Thus, we obtain a DBI action for $M3$ -branes. This action contains extra higher order derivative terms. We observe that these higher order derivative terms are similar to the terms formed from a GUP deformation of the field theory. In fact, we will demonstrate that the explicit form of the modified commutation relations between position and momenta in this model is the same as that produced by the GUP. It is worth mentioning that the foundations of quantum commutation relations can be motivated from string theory Bars:2014jca .

The outline of our paper is as follows. In section II, we construct the M0-brane from the $D0$ -brane and consider the relation between their algebras. We also show that $M0$ -branes can unite and form a $M5$ -brane . In section III, we will compactify the $M5$ -brane on two circles and obtain the relevant action for the $M3$ -brane. We also demonstrate the relation between the GUP and compactification in this section. The last section is devoted to the summary and conclusion.

II $M0$ -branes growing into $M5$ -branes

In this section, inspired by the Lagrangian for $M2$ -branes and $M5$ -branes, we will replace the two dimensional Nambu-Poisson bracket by three one in $D0$ -branes and build the DBI action for $M0$ -branes with a Lie $3$ -algebra as the internal symmetry. $D0$ -branes are $0+1$ dimensional branes in a ten dimensional space-time of string theory which has nine transverse scalars corresponding to the nine transverse directions in their action. However, $M0$ -branes are $0+1$ dimensional branes in the eleven dimensional space-time of M-theory which has ten transverse scalars corresponding to the ten transverse directions. We will show that these M0-branes can join each other, grow and then make a transition to a $M5$ -brane. Let us begin with the Born-Infeld action for a $Dp$ -brane p1 ; pv1 ; p2 ; q23 ; q24 ; qv24 ; q25 :

[TABLE]

where

[TABLE]

$\lambda=2\pi l_{s}^{2}$ and $X^{i}$ are scalar fields of dimension mass. Here $a,b=0,1,...,p$ are the world-volume indices of the Dp-branes, $i,j,k=p+1,...,9$ are indices of the transverse space, and $m$ , $n$ are the ten-dimensional space-time indices. Also, $T_{p}=\frac{1}{g_{s}(2\pi)^{p}l_{s}^{p+1}}$ is the tension of $Dp$ -brane, $l_{s}$ is the string length and $g_{s}$ is the string coupling. Using this equation and assuming $G_{ab}=\eta_{ab}+\partial_{a}X^{i}\partial_{b}X^{i}$ , we can approximate Infeld action for $D0$ -branes p1 ; pv1 ; q26

[TABLE]

where $X^{i}(i=1,3,...9)$ are transverse scalars. Following rules for commutation relations, we can write p1 ; pv1 ; p2 ; q27 ; q28

[TABLE]

Replacing $\partial_{t}X^{i}$ by $[X^{0},X^{i}]$ in equation (6), we get p1 ; pv1 ; p2

[TABLE]

Now, from the $D0$ -branes, we can construct other $Dp$ -branes by substituting the following rules in the action (7) p1 ; p2 ; q27 ; q28 :

[TABLE]

Doing some calculations, we obtain

[TABLE]

which is in agreement with studies done for $D1$ , $D3$ and $D5$ -branes p1 ; p2 ; q24 ; q26 ; q27 ; q28 . Now, we can extend this mechanism to $M$ -Theory and obtain the relevant action for $Mp$ -branes by replacing the two dimensional Nambu-Poisson bracket by three one-dimensional ones in the action (7) p1 ; pv1 ; p2 .

[TABLE]

By the compactification of $M$ -theory on a circle of radius $R$ , this action will make a transition to the ten dimensional action (7). To show this, we make use of the method used in q29 , and define $<X^{8}>=\frac{R}{l_{p}^{3/2}}$ , where $l_{p}$ is the Planck length. Thus, we obtain p1 ; pv1 ; p2

[TABLE]

where we define $T_{D0}=6T_{M0}(\frac{R^{2}}{l_{p}^{3}})$ . The terms of order $O(X^{6})$ have not been written explicitly because they will be ignorable at large radius ( $R\rightarrow\infty$ ). Clearly, in this limit, the action (11) for compactified $M$ -theory is in agreement with the action (7).

Different $Mp$ -branes can be built from $M0$ -branes (similar to $Dp$ -branes) by using the following rules q22

[TABLE]

where

[TABLE]

and $A_{ab}$ is a $2$ -form gauge field. We can obtain the relevant action for the $M5$ -brane by substituting (12) in the action (10):

[TABLE]

Earlier studies done on the $M5$ -branes q22 agree with the above. The derivation of the action for other $Mp$ -branes can be done by this mechanism. So, we can also use it to obtain a suitable action for $Dp$ -branes by compacting $M$ -branes on a circle.

III Emergence of GUP in compactified $M5$ -branes

In this section, we compactify the $M5$ -brane on two circles to obtain $M3$ -branes. We do this because our universe has $3+1$ dimensions, and can be thought to be located on a $3+1$ dimensional brane like a $M3$ -brane. In order to do this, we will first write the $2$ -form gauge fields in $M$ -branes in terms of $1$ -form gauge fields:

[TABLE]

For compacting the $M5$ -brane to a $M3$ -brane, we need to replace two gauge fields by two scalar fields p1 ; p2 ; q17 ; q20 ; q22 . We assume that $A^{4}=X^{4}$ and $A^{5}=X^{5}$ . Thus, we obtain

[TABLE]

where $a=0,1,2,3.$ Substituting (16) and (17) in (13), we obtain

[TABLE]

where $s=4,5.$ We use the fact that $A_{as}=-A_{sa}$ and $A_{aa}=0$ to obtain

[TABLE]

and

[TABLE]

It may be noted that by compacting to a $M3$ -brane, the gauge fields do not depend on the fourth and fifth dimensions. Thus, we can write

[TABLE]

Using the above equations and assuming that the radius of the compactified dimensions are the same, we can calculate the integrals

[TABLE]

Now we can also obtain the rules for commutation as follows

[TABLE]

Substituting equations (21), (22) and (23) in equation (14), we can obtain the relevant action for the compactified $M3$ -brane as

[TABLE]

From this action, we can obtain the equations of motion for $X^{4}$ and $X^{5}$ :

[TABLE]

Replacing $\partial_{a}=P^{a}$ and $\frac{\partial^{2}V}{\partial(x^{a})^{2}}=m^{2}$ , we obtain

[TABLE]

We can compare these equations with the usual equations for scalar field theory

[TABLE]

As a result, we can redefine the momentum $\tilde{P}$ as

[TABLE]

Near the Planck scale, our model modifies the GUP as follows:

[TABLE]

This equation is the same as Eq. (2). Thus, the compactification of a $M5$ -brane action to a $M3$ -brane action produces the deformed GUP of Heisenberg. We can argue that compactification in $M$ -theory can also be used as a motivation for GUP. Also, our calculations show that the GUP occurs only for momentum of scalar fields which are produced from compactification.

IV Summary and Discussion

In this paper, we have demonstrated that all $Dp$ -branes can be formed from joining $D0$ -branes. We generalized our model to $M$ -theory and showed that $M5$ -branes can also be obtained from $M0$ -branes. Then, we compactified an $M5$ -brane on two circles by replacing two gauge fields with two scalars. Thus, we were able to calculate the relevant action for $M3$ -branes. We observed that extra derivatives terms are produced due to compactification. These higher derivative terms can be expressed in terms of higher order momentum terms in the equation of motion for new scalar fields. These terms can be obtained by modifying the usual uncertainty principle to the GUP. So, we have demonstrated that compactification in $M$ -theory can be used as an motivation for the GUP.

Acknowledgments

The work of Alireza Sepehri has been supported financially by Research Institute for Astronomy-Astrophysics of Maragha (RIAAM) under research project NO.1/4717-98. A. Pradhan would also like to thank the University of Zululand, Kwa-Dlangezwa 3886, South Africa for providing facilities and support where part of this work has been done. The authors are heartily grateful the anonymous referee for his valuable comments which have helped to improve and correct the paper in present form.

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

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On the origin of generalized uncertainty principle from compactified M5M5M5-brane

Abstract

I Introduction

II M0M0M0-branes growing into M5M5M5-branes

III Emergence of GUP in compactified M5M5M5-branes

IV Summary and Discussion

Acknowledgments

On the origin of generalized uncertainty principle from compactified $M5$ -brane

II $M0$ -branes growing into $M5$ -branes

III Emergence of GUP in compactified $M5$ -branes