# The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem

###### Abstract

We investigate the effect of the minimal length uncertainty relation, motivated by perturbative string theory, on the density of states in momentum space. The relation is implemented through the modified commutation relation . We point out that this relation, which is an example of an UV/IR relation, implies the finiteness of the cosmological constant. While our result does not solve the cosmological constant problem, it does shed new light on the relation between this outstanding problem and UV/IR correspondence. We also point out that the blackbody radiation spectrum will be modified at higher frequencies, but the effect is too small to be observed in the cosmic microwave background spectrum.

###### pacs:

02.40.Gh,03.65.Sq,98.70.Vc,98.80.Es^{†}

^{†}preprint: VPI–IPPAP–01–06

## I Introduction

In this paper, we continue our investigation CMOT1 of the consequences of the commutation relation

(1) |

which leads to the minimal length uncertainty relation

(2) |

As reviewed in Ref. CMOT1 , Eq. (2) has appeared in the context of perturbative string theory gross where it is implicit in the fact that strings cannot probe distances below the string scale . It should be noted that the precise theoretical framework for such a minimal length uncertainty relation is not understood in string theory. In particular, it is not clear whether Eq. (1) represents the correct quantum mechanical implementation of Eq. (2). Indeed, Kempf has shown that the commutation relation which implies the existence of a minimal length is not unique Kempf:2001 .

Furthermore, Eq. (2) does not seem to be universally valid. For example, both in the realms of perturbative and non-perturbative string theory (where distances shorter than string scale can be probed by -branes joep ), another type of uncertainty relation involving both spatial and time coordinates has been found to hold stu . The distinction (and relation) between the minimal length uncertainty relation and the space-time uncertainty relation has been clearly emphasized by Yoneya yoneya .

Notwithstanding these caveats, the minimum length uncertainty formula does exhibit the basic features of UV/IR correspondence: when is large, is proportional to , a fact which seems counterintuitive from the point of view of local quantum field theory. As is well known, this kind of UV/IR correspondence has been previously encountered in various contexts: the correspondence adscft , non-commutative field theory ncft , and more recently in the attempts to understand quantum gravity in asymptotically de Sitter spaces dscft ; dsrg .

Moreover, it has been argued by various authors banks that the UV/IR correspondence, described by Eq. (2), is relevant for the understanding of the cosmological constant problem cosmoc . Likewise, it has been suggested in the literature that some kind of UV/IR relation is necessary to understand observable implications of short distance physics on inflationary cosmology Kempf:2001 ; greene .

In this paper we ask the question whether the cosmological constant problem could be understood by utilizing a concrete UV/IR relation, such as Eq. (1). In particular, we study the implication of the commutation relation on the effective density of states in the vacuum and consequently on the cosmological constant problem. We point out that the commutation relation implies the finiteness of the cosmological constant and the modification of the blackbody radiation spectrum. While we do not present a solution to the cosmological constant problem, our results offer a new perspective from which this outstanding problem may be addressed.

## Ii The Classical Limit and the Liouville Theorem

The observation we would like to make is that the right hand side of Eq. (1) can be considered to define an ‘effective’ value of which is -dependent. This means that the size of the unit cell that each quantum state occupies in phase space can be thought of as being also -dependent. This will change the -dependence of the density of states and affect the calculation of the cosmological constant, the blackbody radiation spectrum, etc. Lubo:2000yj . For this interpretation to make sense, we must first check that any volume of phase space evolves in such a way that the number of states inside does not change with time. What we are looking for here is the analog of the Liouville theorem. To place the discussion in a general context, we begin by extending Eq. (1) to higher dimensions.

In -dimensions, Eq. (1) is extended to the tensorial form Kempf:1995su :

(3) |

If the components of the momentum are assumed to commute with each other,

(4) |

then the commutation relations among the coordinates are almost uniquely determined by the Jacobi Identity (up to possible extensions) as

(5) |

Let us take a look at what happens in the classical limit. Recall that the quantum mechanical commutator corresponds to the Poisson bracket in classical mechanics via

(6) |

So the classical limits of Eqs. (3)–(5) read

(7) | |||||

(8) | |||||

(9) |

The time evolutions of the coordinates and momenta are governed by

(10) | |||||

(11) |

The analog of the Liouville theorem in this case states that the weighted phase space volume

(12) |

is invariant under time evolution. To see this, consider an infinitesimal time interval . The evolution of the coordinates and momenta during are

(13) | |||||

(14) |

with

(15) | |||||

(16) |

An infinitesimal phase space volume after this infinitesimal evolution is

(17) |

Since

(18) |

the Jacobian to first order in is

(19) |

We find

(20) | |||||

(21) | |||||

(23) | |||||

(24) | |||||

(25) | |||||

(26) |

Therefore, to first order in

(27) |

On the other hand,

(28) | |||||

(29) | |||||

(30) | |||||

(31) | |||||

(32) | |||||

(33) |

and

(34) | |||||

(35) | |||||

(36) | |||||

(37) | |||||

(38) | |||||

(39) |

Therefore, to first order in

(40) | |||||

(42) | |||||

From Eqs. (27) and (42), we deduce that the weighted phase space volume Eq. (12) is invariant. Note that in addition to the non-canonical Poisson brackets between the coordinates and momenta, those among the coordinates themselves (and thus the non–commutative geometry of the problem) is crucial in arriving at this result. When , Eq. (12) simplifies to

(43) |

As a concrete example, consider the harmonic oscillator with the Hamiltonian

(44) |

The equations of motion are

(45) | |||||

(46) |

These equations can be solved to yield

(47) | |||||

(48) |

where

(49) |

Note that the period of oscillation, , is now energy (and thus amplitude) dependent:

(50) |

Now, consider the infinitesimal phase space volume sandwiched between the equal–energy contours and , and the equal–time contours and . It is straightforward to show that

(51) |

The left–hand side of this equation is time–independent by definition, so the right–hand side must be also.

Finally, note that the semi–classical quantization ( limit) of the harmonic oscillator is consistent with the full quantum mechanical result derived in our previous paper, Ref. CMOT1 .

## Iii Density of States

From this point on, we will only consider the case for the sake of simplicity.

Integrating over the coordinates, the invariant phase space volume Eq. (43) becomes

(52) |

where is the coordinate space volume. This implies that upon quantization, the number of quantum states per momentum space volume should be assumed to be

(53) |

Eq. (53) indicates that the density of states in momentum space must be modified by the extra factor of . This factor effectively cuts off the integral beyond . Indeed, in the weight factor is

(54) |

the plot of which is shown in Fig. 1. We look at the consequence of this modification in the calculation of the cosmological constant and the blackbody radiation spectrum in the following.