On the Minimal Length Uncertainty Relation and the Foundations of String Theory

Abstract

We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.

1. Introduction

One of the unequivocal characteristics of string theory [1–3] is its possession of a fundamental length scale which determines the typical spacetime extension of a fundamental string. This is ℓ_𝑠=𝛼^′, where ℏ𝑐/𝛼^′ is the string tension. Such a feature is to be expected of any candidate theory of quantum gravity, since gravity itself is characterized by the Planck length ℓ_𝑃=ℏ𝐺_𝑁/𝑐³. Moreover, ℓ_𝑃∼ℓ_𝑠 is understood to be the minimal length below which spacetime distances cannot be resolved [4–7]𝛿𝑠≳ℓ_𝑃∼ℓ_𝑠. (1.1) Quantum theory, on the other hand, is completely oblivious to the presence of such a scale, despite its being the putative infrared limit of string theory. A natural question to ask is, therefore, whether the formalism of quantum theory can be deformed or extended in such a way as to consistently incorporate the minimal length. If it is at all possible, the precise manner in which quantum theory must be modified may point to solutions of yet unresolved mysteries such as the cosmological constant problem [8–12], which is quantum gravitational in its origin. It should also illuminate the nature of string theory [13], whence quantum theory must emerge [14].

The idea of introducing a minimal length into quantum theory has a fascinating and long history. It was used by Heisenberg in 1930 [15, 16] to address the infinities of the newly formulated theory of quantum electrodynamics [17]. Over the years, the idea has been picked up by many authors in a plethora of contexts, for example, [18–42] to list just a few. Various ways to deform or extend quantum theory have also been suggested [43–47]. In this paper, we focus our attention on how a minimal length can be introduced into quantum mechanics by modifying its algebraic structure [48–50].

The starting point of our analysis is the minimal length uncertainty relation (MLUR) [51, 52],ℏ𝛿𝑥∼𝛿𝑝+𝛼^′𝛿𝑝ℏ, (1.2) which is suggested by a resummed perturbation expansion of the string-string scattering amplitude in a flat spacetime background [53–56]. This is essentially a Heisenberg microscope argument [57] in the 𝑆-matrix language [58–61] with fundamental strings used to probe fundamental strings. The first term inside the parentheses on the right-hand side is the usual Heisenberg term coming from the shortening of the probe wavelength as momentum is increased, while the second term can be understood as due to the lengthening of the probe string as more energy is pumped into it 𝛿𝑝=𝛿𝐸𝑐∼ℏ𝛼^′𝛿𝑥. (1.3) Equation (1.2) implies that the uncertainty in position, 𝛿𝑥, is bounded from below by the string length scale,𝛿𝑥≳𝛼^′=ℓ_𝑠, (1.4) where the minimum occurs atℏ𝛿𝑝∼𝛼^′=ℏℓ_𝑠≡𝜇_𝑠. (1.5) Thus, ℓ_𝑠 is the minimal length below which spatial distances cannot be resolved, consistent with (1.1). In fact, the MLUR can be motivated by fairly elementary general relativistic considerations independent of string theory, which suggests that it is a universal feature of quantum gravity [4–7].

Note that in the trans-Planckian momentum region 𝛿𝑝≫𝜇_𝑠, the MLUR is dominated by the behavior of (1.3), which implies that large 𝛿𝑝 (UV) corresponds to large 𝛿𝑥 (IR), and that there exists a correspondence between UV and IR physics. Such UV/IR relations have been observed in various string dualities [1–3], and in the context of AdS/CFT correspondence [62, 63] (albeit between the bulk and boundary theories). Thus, the MLUR captures another distinguishing feature of string theory.

In addition to the MLUR, another uncertainty relation has been pointed out by Yoneya as characteristic of string theory. This is the so-called spacetime uncertainty relation (STUR)ℓ𝛿𝑥𝛿𝑡∼^2𝑠𝑐, (1.6) which can be motivated in a somewhat hand-waving manner by combining the usual energy-time uncertainty relation 𝛿𝐸𝛿𝑡∼ℏ [64–66] with (1.3). However, it can also be supported via an analysis of D0-brane scattering in certain backgrounds in which 𝛿𝑥 can be made arbitrary small at the expense of making the duration of the interaction 𝛿𝑡 arbitrary large [67–73]. While the MLUR pertains to dynamics of a particle in a nondynamic spacetime, the STUR can be interpreted to pertain to the dynamics of spacetime itself in which the size of a quantized spacetime cell is preserved.

In the following, we discuss how the MLUR and STUR may be incorporated into quantum mechanics via a deformation and/or extension of its algebraic structure. In Section 2, we introduce a deformation of the canonical commutation relation between 𝑥 and 𝑝 which leads to the MLUR and discuss its phenomenological consequences. In Section 3, we take the classical limit by replacing commutation relations with Poisson brackets and derive the analogue of Liouville's theorem in the deformed mechanics. We then discuss the effect this has on the density of states in phase space. In Section 4, we discuss the implications of the MLUR on the cosmological constant problem. We conclude in Section 5 with some speculations on how the STUR may be incorporated via a Nambu triple bracket and comment on the lessons for the foundations of string theory and on the question “What is string theory?”

2. Quantum Mechanical Model of the Minimal Length

2.1. Deformed Commutation Relations

To place the MLUR, (1.2), on firmer ground, we begin by rewriting it asℏ𝛿𝑥𝛿𝑝≥21+𝛽𝛿𝑝², (2.1) where we have introduced the parameter 𝛽=𝛼′/ℏ². The minimum value of 𝛿𝑥 as a function of 𝛿𝑝 is plotted in Figure 1. This uncertainty relation can be reproduced by deforming the canonical commutation relation between 𝑥 and 𝑝 to1𝑝1𝑖ℏ𝑥,=1⟶𝑝𝑝𝑖ℏ𝑥,=𝐴², (2.2) with 𝐴(𝑝²)=1+𝛽𝑝². Indeed, we find1𝛿𝑥𝛿𝑝≥2𝑝=ℏ𝑥,21+𝛽𝑝²≥ℏ21+𝛽𝛿𝑝², (2.3) since 𝛿𝑝²=⟨𝑝²⟩−⟨𝑝⟩². The function 𝐴(𝑝²) can actually be more generic, with 𝛽𝑝² being the linear term in its expansion in 𝑝².

Figure 1 

The 𝛿𝑝-dependence of the lower bound of 𝛿𝑥 under the minimal length uncertainty relation (2.1) (red curve). The bound for the usual Heisenberg relation 𝛿𝑥≥ℏ/(2𝛿𝑝) is shown in blue, and the linear bound 𝛿𝑥≥(ℏ𝛽/2)𝛿𝑝 is shown in green.

When we have more than one spatial dimension, the above commutation relation can be generalized to1𝑥𝑖ℏ^𝑖,𝑝_𝑗=𝐴𝐩²𝛿_𝑖𝑗+𝐵𝐩²𝑝_𝑖𝑝_𝑗, (2.4) where 𝐩²=∑_𝑖𝑝²_𝑖. The right-hand side is the most general form that depends only on the momentum and respects rotational symmetry. Assuming that the components of the momentum commute among themselves,𝑝_𝑖,𝑝_𝑗=0, (2.5) the Jacobi identity demands that1𝑥𝑖ℏ^𝑖,𝑥_𝑗=−2𝐴+𝐵𝐩²𝐴^′−𝐴𝐵𝐿_𝑖𝑗, (2.6) where we have used the shorthand 𝐴=𝐴(𝐩²), 𝐴^′=(𝑑𝐴/𝑑𝐩²)(𝐩²), 𝐵=𝐵(𝐩²), and 𝐿_𝑖𝑗=(𝑥_𝑖𝑝_𝑗−𝑥_𝑗𝑝_𝑖)/𝐴. That 𝐿_𝑖𝑗 generates rotations can be seen from the following:1𝐿𝑖ℏ^𝑖𝑗𝑥_𝑘=𝛿_𝑖𝑘𝑥_𝑗−𝛿_𝑗𝑘𝑥_𝑖,1𝐿𝑖ℏ^𝑖𝑗𝑝_𝑘=𝛿_𝑖𝑘𝑝_𝑗−𝛿_𝑗𝑘𝑝_𝑖,1𝐿𝑖ℏ^𝑖𝑗𝐿_𝑘ℓ=𝛿_𝑖𝑘𝐿_𝑗ℓ−𝛿_𝑖ℓ𝐿_𝑗𝑘+𝛿_𝑗ℓ𝐿_𝑖𝑘−𝛿_𝑗𝑘𝐿_𝑖ℓ. (2.7) Note that the noncommutativity of the components of position can be interpreted as a reflection of the dynamic nature of space itself, as would be expected in quantum gravity.

Various choices for the functions 𝐴(𝐩²) and 𝐵(𝐩²) have been considered in the literature. Maggiore [48, 49] proposed that 𝐴𝐩²=1+2𝛽𝐩²,𝐵𝐩²1=0,𝑥𝑖ℏ^𝑖,𝑥_𝑗=−2𝛽𝐿_𝑖𝑗, (2.8) while Kempf et al. [50] assumed that 𝐴𝐩²=1+𝛽𝐩²,𝐵𝐩²=𝛽^′=constant, (2.9) in which case1𝑥𝑖ℏ^𝑖,𝑥_𝑗=−2𝛽−𝛽^′+𝛽2𝛽+𝛽^′𝐩²𝐿_𝑖𝑗. (2.10) Kempf's choice encompasses the algebra of Snyder [19, 20]𝐴𝐩²=1,𝐵𝐩²=𝛽^′,1𝑥𝑖ℏ^𝑖,𝑥_𝑗=𝛽^′𝐿_𝑖𝑗, (2.11) and that of Brau and Buisseret [74, 75]𝐴𝐩²=1+𝛽𝐩²,𝐵𝐩²1=2𝛽,𝑥𝑖ℏ^𝑖,𝑥_𝑗=𝑂𝛽², (2.12) for which the components of the position approximately commute. In our treatment, we follow Kempf and use (2.9).

2.2. Shifts in the Energy Levels

Let us see whether the above deformed commutation relations led to a reasonable quantum mechanics, with well-defined energy eigenvalues and eigenstates. Given a Hamiltonian in terms of the deformed position and momentum operators, 𝐻(𝐱,𝐩), we would like to solve the time-independent Schrödinger equation𝐻(𝐱,𝐩)|𝐸⟩=𝐸|𝐸⟩. (2.13) The operators which satisfy (2.4), (2.5), and (2.6), subject to the choice (2.9), can be represented using operators which obey the canonical commutation relation [𝑞_𝑖,𝑝_𝑗]=𝑖ℏ𝛿_𝑖𝑗 as [50, 76]𝑥_𝑖=𝑞_𝑖𝑝+𝛽²𝑞_𝑖+𝑞_𝑖𝑝²2+𝛽^′𝑝_𝑖𝑝⋅𝑞+𝑞⋅𝑝𝑝_𝑖2,𝑝_𝑖=𝑝_𝑖. (2.14) The 𝛽 and 𝛽^′ terms are symmetrized to ensure the hermiticity of 𝑥_𝑖. Note that this representation allows us to write the Hamiltonian in terms of canonical 𝑞_𝑖’s and 𝑝_𝑖’s𝐻^′(𝐪,𝐩)≡𝐻(𝐱(𝐪,𝐩),𝐩). (2.15) Thus, our deformation of the canonical commutation relations is mathematically equivalent to a deformation of the Hamiltonian. (In this work, we do not address the question of whether the dependence of the Hamiltonian on the position and momentum operators also need be modified in the presence of a minimal length. Lacking in any guideline to do so, we simply keep them fixed to their standard forms.)

By the standard replacements𝑞_𝑖=𝑞_𝑖,𝑝_𝑖=ℏ𝑖𝜕𝜕𝑞_𝑖,or𝑞_𝑖𝜕=𝑖ℏ𝜕𝑝_𝑖,𝑝_𝑖=𝑝_𝑖, (2.16)𝑥_𝑖 and 𝑝_𝑗 can be represented as differential operators acting on a Hilbert space of 𝐿² functions in either the 𝑞_𝑖's or the 𝑝_𝑖's, and one can write down a Schrödinger equation for a given Hamiltonian in either 𝑞-space or 𝑝-space to solve for the energy eigenvalues. Note, however, that while the 𝑝_𝑖's are the eigenvalues of the momentum operators 𝑝_𝑖, the 𝑞_𝑖's are not the eigenvalues of the position operator 𝑥_𝑖. In fact, the existence of the minimal length implies that 𝑥_𝑖 cannot have any eigenfunctions within either Hilbert spaces. Therefore, the meaning of the wave function in 𝑞-space is somewhat ambiguous. Nevertheless, the 𝑞-space representation is particularly useful when the Schrödinger equation cannot be solved exactly, since one can treatΔ𝐻(𝐪,𝐩)=𝐻^′(𝐪,𝐩)−𝐻(𝐪,𝐩) (2.17) as a perturbation and calculate the shifts in the energies via perturbation theory in 𝑞-space.

In the following, we look at the energy shifts induced by nonzero 𝛽 and 𝛽^′ in the harmonic oscillator [77, 78], the Hydrogen atom [74, 79], and a particle in a uniform gravitational well [75, 76]. Since detailed derivations can be found in the respective references, we only provide an outline of the results in each case.

2.2.1. Harmonic Oscillator

Consider a 𝐷-dimensional isotropic harmonic oscillator. The Hamiltonian is of course𝐩𝐻=²+12𝑚2𝑚𝜔²𝐱². (2.18) The 𝑝-space representation of the operators is𝑥_𝑖=𝑖ℏ1+𝛽𝑝²𝜕𝜕𝑝_𝑖+𝛽^′𝑝_𝑖𝑝_𝑗𝜕𝜕𝑝_𝑗+𝛽+𝛽^′𝐷+12−𝛿𝛽+𝛽^′𝑝_𝑖,𝑝_𝑖=𝑝_𝑖. (2.19) Here, 𝛿 is an arbitrary real parameter which can be used to simplify the representation of the operator 𝑥_𝑖 at the expense of modifying the definition of the inner product in 𝑝-space to𝑓∣𝑔_𝛿=𝑑^𝐷𝐩1+𝛽+𝛽^′𝐩^2𝛿𝑓^∗(𝐩)𝑔(𝐩). (2.20) The introduction of 𝛿 is a canonical transformation which does not affect the energy eigenvalues [76]. The choice𝛿=𝛽+𝛽^′((𝐷+1)/2)𝛽+𝛽^′ (2.21) eliminates the third term in the expression for 𝑥_𝑖.

The rotational symmetry of the Hamiltonian, (2.18), allows us to write the wave function in 𝑝-space as a product of a radial wave-function and a 𝐷-dimensional spherical harmonic:Ψ_𝐷(𝐩)=𝑅𝑝𝑌_{ℓ𝑚𝐷−2𝑚𝐷−3⋯𝑚2𝑚1}(Ω),𝑝≡|𝐩|. (2.22) The radial Schrödinger equation is then−𝑚ℏ𝜔[1+(𝛽+𝛽′)𝑝²]𝜕𝜕𝑝²+(𝐷−1)1+𝛽𝑝²1+𝛽+𝛽^′𝑝²𝑝𝜕−𝐿𝜕𝑝₂1+𝛽𝑝²²𝑝²𝑅𝑝+1𝑝𝑚ℏ𝜔²𝑅𝑝=2𝐸𝑅𝑝,ℏ𝜔 (2.23) where𝐿²=ℓ(ℓ+𝐷−2),ℓ=0,1,2,… (2.24) is the eigenvalue of the angular momentum operator in 𝐷 dimensions. The solution to (2.23) has been worked out in detail in [78], and the energy eigenvalues are𝐸_𝑛ℓ𝐷=ℏ𝜔𝑛+21+𝛽²𝐿²+𝐷𝛽+𝛽^′24𝑚²ℏ²𝜔²+𝛽+𝛽^′𝐷𝑛+2²+𝛽−𝛽^′𝐿²+𝐷²4+𝛽^′𝐷2𝑚ℏ𝜔2, (2.25) with eigenfunctions given by𝑅_𝑛ℓ𝑝=𝛽+𝛽^′𝐷/42(2𝑘+𝑎+𝑏+1)𝑘!Γ(𝑘+𝑎+𝑏+1)Γ(𝑘+𝑎+1)Γ(𝑘+𝑏+1)1−𝑧2^𝜆/21+𝑧2^ℓ/2𝑃_𝑘^(𝑎,𝑏)(𝑧). (2.26) Here, 𝑃_𝑘^(𝑎,𝑏)(𝑧) is the Jacobi polynomial of order 𝑘=(𝑛−ℓ)/2 with argument𝑧=𝛽+𝛽^′𝑝²−1𝛽+𝛽^′𝑝²,1+1𝑎=𝑚ℏ𝜔𝛽+𝛽′1+𝛽²𝐿²+𝐷𝛽+𝛽^′24𝑚²ℏ²𝜔²𝐷,𝑏=2+ℓ−1,𝜆=𝐷𝛽+𝛽^′2𝛽+𝛽^′+𝑎. (2.27) Note that due to the (𝑛+𝐷/2)²-dependent term in (2.25), the energy levels are no longer uniformly spaced. Note also that, due to the explicit 𝐿² dependence, the original(𝐷+𝑛−1)!(𝐷−1)!𝑛! (2.28) fold degeneracy of the 𝑛th energy level, which was due to states with different 𝑘 and ℓ sharing the same 𝑛=2𝑘+ℓ, is resolved, leaving only the(𝐷+ℓ−1)!()−()!𝐷−1!ℓ!𝐷+ℓ−3(𝐷−1)!(ℓ−2)! (2.29) fold degeneracy for each value of ℓ due to rotational symmetry alone [80–82]. For example, in 𝐷=2 dimensions, the (𝑛+1)-fold degeneracy of the 𝑛th level breaks down to the 2-fold degeneracies between the pairs of 𝑚=±ℓ states. This is illustrated in Figure 2.

(a)

(b)

Figure 2 

The energy levels of the 2D isotropic harmonic oscillator for the cases 𝛽′=0 (left) and 𝛽=0 (right). The purple solid lines indicate 𝑠-wave states which are singlets. The dashed lines are doublets with the color indicating that ℓ=1 (blue), ℓ=2 (green), ℓ=3 (yellow), ℓ=4 (orange), and ℓ=5 (red). 𝛽𝑚ℏ𝜔 is the ratio of the minimal length ℏ𝛽to the characteristic length scale ℏ/𝑚𝜔 of the system.

2.2.2. Hydrogen Atom

The introduction of a minimal length to the coulomb potential problem was first discussed by Born in 1933 [18]. There, it was argued that the singularity at 𝑟=0 will be blurred out. Here, we find a similar effect. We consider the usual Hydrogen atom Hamiltonian in 𝐷 dimensions:𝐩𝐻=²−𝑒2𝑚𝑟²̂, (2.30) where the operator 1/𝑟̂ is defined as the inverse of the square root of the operator𝑟²̂=^𝐷_𝑖=1𝑥²_𝑖. (2.31)1/𝑟̂ will be best represented in the basis in which 𝑟²̂ is diagonal. The eigenvalues of 𝑟²̂ can be obtained from those of the harmonic oscillator, (2.25), by taking the limit 𝑚→∞:𝑟²_𝑘ℓ=lim_𝑚→∞2𝐸_𝑛ℓ𝑚𝜔²=ℏ²𝛽+𝛽^′𝐷2𝑘+ℓ+2+1𝛽+𝛽^′𝛽²𝐿²+𝐷𝛽+𝛽^′24²−𝛽^′𝛽+𝛽^′𝐿²+(𝐷−1)²4. (2.32) The corresponding eigenfunctions are given by the same expression as (2.26) except with 𝑎 replaced with1𝑎=𝛽+𝛽^′𝛽²𝐿²+𝐷𝛽+𝛽^′24. (2.33) Denoting these eigenfunctions as 𝑅_𝑘ℓ(𝑝), we can define1𝑟̂𝑅_𝑘ℓ𝑝=1𝑟_𝑘ℓ𝑅_𝑘ℓ𝑝. (2.34) As in the harmonic oscillator case, the rotational symmetry of the Hamiltonian allows us to write an energy eigenstate wave function as a product of a radial wave function and a spherical harmonic. The radial wave function can then be expressed as a superposition of the 𝑟²̂ eigenfunctions with fixed ℓ:𝑅_ℓ𝑝=^∞_𝑘=0𝑓_𝑘𝑅_𝑘ℓ𝑝. (2.35) The radial Schrödinger equation will impose a recursion relation on the coefficients 𝑓_𝑛, which can be solved numerically on a computer. The condition that the resulting function be square integrable determines the eigenvalues 𝐸. The detailed procedure can be found in [76, 79]. Here, we only display the results for the 𝐷=3 case in Figure 3. As can be seen, the degeneracy between difference angular momentum states is lifted, just as in the harmonic oscillator case.

(a)

(b)

Figure 3 

Energy shifts of the 𝑛=1,2, and 3 states of the Hydrogen atom for the 𝛽′=0 (left) and 𝛽=0 (right) cases. 𝑎0=ℏ2/𝑚𝑒2 is the Bohr radius, and the energy is in units of the Rydberg constant 𝑒2/2𝑎0. The color of the lines indicates the orbital angular momentum: 𝑠 (red), 𝑝 (green), and 𝑑 (blue). The 𝑠-wave states are affected nonperturbatively even for very small 𝛽 or 𝛽′, indicating their sensitivity to the singularity of the Coulomb potential at the origin.

It is also possible to calculate the energy shifts perturbatively using the 𝑞-space representation for the cases 𝐷≥4 or ℓ≠0. The unperturbed energy eigenfunctions in 𝐷 dimensions are𝑅_𝑛ℓ𝑞=2^2𝐷𝑎^𝐷₀(2𝑛+𝐷−3)^𝐷+1(𝑛−ℓ−1)!()!𝑒𝑛+ℓ+𝐷−3^−𝜌/2𝜌^ℓ𝐿^{(2ℓ+𝐷−2)}_{𝑛−ℓ−1}𝜌, (2.36) where 𝑎₀=ℏ²/𝑚𝑒² is the Bohr radius, 𝐿_𝑘^(𝜆)(𝜌) the order 𝑘 Laguerre polynomial, and𝜌=2𝑞𝑎₀(𝑛+((𝐷−3)/2)). (2.37) The eigenvalues are𝐸_𝑛𝑒=−²2𝑎₀(𝑛+((𝐷−3)/2))²,𝑛=1,2,3,…. (2.38) The operator 1/𝑟̂ can be expanded in powers of 𝛽 and 𝛽′ as [76]1𝑟̂=1𝑞+ℏ²𝛽1𝑞𝜕²𝜕𝑞²+𝐷−2𝑞²𝜕−𝐿𝜕𝑞²+𝐷−2𝑞³+ℏ²𝛽^′1𝑞𝜕²𝜕𝑞²+𝐷−2𝑞²𝜕+𝐷𝜕𝑞²−5𝐷+84𝑞³+⋯, (2.39) and the expectation value of the extra terms converges for ℓ≠0 or 𝐷≥4, yieldingΔ𝐸_𝑛ℓ=𝑒²𝑎₀(𝑛+((𝐷−3)/2))³ℏ²𝑎²₀(𝐷−1)2𝛽−𝛽^′4()))+ℓ+((𝐷−3/2))(ℓ+((𝐷−2/2))(ℓ+((𝐷−1/2))2𝛽+𝛽^′()−ℓ+((𝐷−2/2))𝛽+𝛽^′(),𝑛+((𝐷−3/2)) (2.40) which agrees very well with the numerical results for all cases to which it is applicable. For 𝐷=3, this formula reduces toΔ𝐸_𝑛ℓ=𝑒²𝑎₀𝑛³ℏ²𝑎²₀2𝛽−𝛽^′(()+2ℓℓ+1/2))(ℓ+12𝛽+𝛽^′((−ℓ+1/2))𝛽+𝛽^′𝑛, (2.41) which is clearly problematic for ℓ=0. This is due to the breakdown of the expansion equation (2.39) near 𝑞=0 for 𝐷≤3. Physically, this can be interpreted to mean that the 𝑠-wave in 3D and lower dimensions is sensitive to the nonperturbative resolution of the singularity at the origin due to the minimal length. Interestingly, in 4D and higher, there are enough spatial dimensions for the wavefunction to spread out around the origin so that even the 𝑠-wave is insensitive to the singularity, and the effect of the minimal length becomes perturbative.

2.2.3. Uniform Gravitational Potential

This subsection is based on unpublished material by Benczik in [76]. Consider the 1D motion of a particle in a linear potential𝑉(𝑥)=𝑚𝑔𝑥𝑥>0,∞𝑥≤0. (2.42) The Hamiltonian is𝑝𝐻=²2𝑚+𝑚𝑔𝑥. (2.43) Since 𝑥 does not have any eigenstates within the Hilbert space, the condition 𝑥>0 is replaced with ⟨𝑥⟩>0. In the 𝑞-space representation, the operators are given by𝑥=𝑞1−ℏ²𝛽𝑑²𝑑𝑞²,ℏ𝑝=𝑖𝑑,𝑑𝑞 (2.44) and the Schrödinger equation becomesℏ𝐻𝜓=−²𝑑2𝑚²𝜓𝑑𝑞²+𝑚𝑔𝑞1−ℏ²𝛽𝑑²𝑑𝑞²𝜓=𝐸𝜓. (2.45) The condition ⟨𝑥⟩>0 can be imposed by restricting the domain of 𝑞 to 𝑞>0 and demanding that the wave function vanish at 𝑞=0. The solution to the 𝛽=0 case is given by the Airy function𝜓_𝑛𝑞=1Ai^′𝛼_𝑛𝑞Ai𝑎+𝛼_𝑛ℏ,𝑎=²2𝑚²𝑔^1/3, (2.46) with eigenvalues𝐸_𝑛𝑚𝑔𝑎=−𝛼_𝑛, (2.47) where⋯<𝛼₃<𝛼₂<𝛼₁<0 (2.48) are the zeroes of Ai(𝑧). The solution to the 𝛽≠0 case is given in terms of the confluent hypergeometric function of the second kind [83–85]𝜓𝑞∝𝑒^−𝑞/𝑏𝑈−12𝐸+𝑎𝑚𝑔𝑏𝑏³³𝑎;0;2³𝑏³+𝑞𝑏ℏ,𝑎=²2𝑚²𝑔^1/3,𝑏=ℏ𝛽. (2.49) The energy eigenvalues are determined by the condition𝑈−12𝐸+𝑎𝑚𝑔𝑏𝑏³³;0;2𝑎³𝑏³=0, (2.50) which can be solved numerically using Mathematica. In Figure 4, we plot the 𝑏 dependence of the energies of the lowest lying states. The energies of higher-dimensional cases, in which there are one or more spatial dimensions orthogonal to the potential direction, are discussed in [75, 76].

Figure 4 

The 𝑏 dependence of the lowest energy levels of a particle of mass 𝑚 in a linear gravitaional potential 𝑉(𝑥)=𝑚𝑔𝑥 with 𝑥>0. 𝑎=[ℏ2/2𝑚2𝑔]1/3 is the characteristic length scale of the system, and 𝑏=ℏ𝛽 is the minimal length.

2.3. Experimental Constraints

As these three examples show, the main effect of the introduction of the minimal length into quantum mechanical systems is the shifts in energy levels which also leads to the breaking of well-known degeneracies. The natural question arises whether these shifts can be used to constrain the minimal length experimentally. Of course, if the minimal length is at the Planck scale, detecting its actual effect would be impossible. However, the exercise is of interest to models of large extra dimensions which possess a much lower effective Planck scale than the 4D value [86–89].

In the case of the harmonic oscillator, actual physical systems are never completely harmonic, so it is difficult to distinguish the shift in energy due to an harmonicity with that due to a possible minimal length. Reference [78] considers using the energy levels of an electron in a Penning trap to constrain 𝛽 and finds that even under highly optimistic and unrealistic assumptions, the best bound that can be hoped for is1𝛽≳1GeV/𝑐. (2.51)

References [76, 79] consider placing a bound on 𝛽 using the 1𝑆 Lamb shift of the hydrogen atom. The current best experimental value is that given by Schwob et al. in [90]𝐿^exp_1𝑠=8172.837(22)MHz. (2.52) This is to be compared with the theoretical value, for which we use that given in [91, 92]𝐿^th_1𝑠=8172.731(40)MHz. (2.53) The calculation requires the experimentally determined proton rms charge radius 𝑟_𝑝 as an input, and the error on 𝐿^th_1𝑠 is dominated by the experimental error on 𝑟_𝑝. Here, the value of 𝑟_𝑝=0.862(12)fm [93] was used. Attributing the entire discrepancy to 𝛽 (𝛽^′=0), [76, 79] cite1𝛽≳7GeV/𝑐, (2.54) which is only slightly better than (2.51). There is no bound on 𝛽^′(𝛽=0) since the shift is in the wrong direction as can be seen in Figure 3.

The energy levels of neutrons in a linear gravitational potential have been measured by Nesvizhevsky et al. [94–96]. However, as analyzed by Brau and Buisseret [75], the experimental precision is still very many orders of magnitude above what is necessary to place a meaningful bound on 𝛽. The current lower bound on 1/𝛽 is on the order of 100eV/𝑐.

3. Classical Limit: The Liouville Theorem and the Density of States

Note that rewriting our 1D-deformed commutator as𝑥,𝑝=𝑖ℏ𝐴𝑝² (3.1) suggests that ℏ𝐴(𝑝²) takes on the role of a momentum-dependent Planck constant. Given that-ℏ determines the size of a quantum mechanical state in phase space, a momentum-dependent ℏ would imply that the size of this state must scale according to 𝐴(𝑝²) as it evolves. To see whether this interpretation makes sense, we formally take the naive classical limit by replacing commutators with Poisson brackets,1𝑝𝑝𝑖ℏ𝑥,=𝐴²⟶𝑥,𝑝=𝐴𝑝², (3.2) and proceed to derive the analogue of Liouville's theorem [97]. The Poisson brackets among the 𝑥_𝑖's and 𝑝_𝑖's for the multidimensional case are𝑥_𝑖,𝑝_𝑗=𝐴𝛿_𝑖𝑗+𝐵𝑝_𝑖𝑝_𝑗,𝑝_𝑖,𝑝_𝑗𝑥=0,_𝑖,𝑥_𝑗2=−𝐴+𝐵𝐩²𝐴𝑑𝐴𝑑𝐩²−𝐵𝑥_𝑖𝑝_𝑗−𝑥_𝑗𝑝_𝑖. (3.3) The generic Poisson bracket of arbitrary functions of the coordinates and momenta can then be defined as{𝐹,𝐺}=𝜕𝐹𝜕𝑥_𝑖𝜕𝐺𝜕𝑝_𝑗−𝜕𝐹𝜕𝑝_𝑖𝜕𝐺𝜕𝑥_𝑗𝑥_𝑖,𝑝_𝑗+𝜕𝐹𝜕𝑥_𝑖𝜕𝐺𝜕𝑥_𝑗𝑥_𝑖,𝑥_𝑗. (3.4) Here, we use the convention that repeated indices are summed. Assuming that the equations of motion of 𝑥_𝑖 and 𝑝_𝑖 are given formally by𝑥̇_𝑖=𝑥_𝑖,𝐻=𝑥_𝑖,𝑝_𝑗𝜕𝐻𝜕𝑝_𝑗+𝑥_𝑖,𝑥_𝑗𝜕𝐻𝜕𝑥_𝑗,𝑝̇_𝑖=𝑝_𝑖,𝐻=−𝑥_𝑗,𝑝_𝑖𝜕𝐻𝜕𝑥_𝑗, (3.5) the evolution of 𝑥_𝑖 and 𝑝_𝑖 during an infinitesimal time interval 𝛿𝑡 is found to be𝑥^′_𝑖=𝑥_𝑖+𝑥̇_𝑖𝛿𝑡=𝑥_𝑖+𝑥_𝑖,𝑝_𝑗𝜕𝐻𝜕𝑝_𝑗+𝑥_𝑖,𝑥_𝑗𝜕𝐻𝜕𝑥_𝑗𝑝𝛿𝑡,_′𝑖=𝑝_𝑖+𝑝̇_𝑖𝛿𝑡=𝑝_𝑖−𝑥_𝑗,𝑝_𝑖𝜕𝐻𝜕𝑥_𝑗𝛿𝑡. (3.6) To find the change in phase-space volume associated with this evolution, we calculate the Jacobian of the transformation from (𝑥₁,𝑥₂,…,𝑥_𝐷;𝑝₁,𝑝₂,…,𝑝_𝐷) to (𝑥^′₁,𝑥^′₂,…,𝑥^′_𝐷;𝑝^′₁,𝑝^′₂,…,𝑝^′_𝐷)𝑑^𝐷𝐱^′𝑑^𝐷𝐩^′=𝜕𝑥^′₁,𝑥^′₂,…,𝑥^′_𝐷;𝑝^′₁,𝑝^′₂,…,𝑝^′_𝐷𝜕𝑥₁,𝑥₂,…,𝑥_𝐷;𝑝₁,𝑝₂,…,𝑝_𝐷𝑑^𝐷𝐱𝑑^𝐷𝐩. (3.7) Since𝜕𝑥^′_𝑖𝜕𝑥_𝑗=𝛿_𝑖𝑗+𝜕𝑥̇_𝑖𝜕𝑥_𝑗𝛿𝑡,𝜕𝑥^′_𝑖𝜕𝑝_𝑗=𝜕𝑥̇_𝑖𝜕𝑝_𝑗𝛿𝑡,𝜕𝑝^′_𝑖𝜕𝑥_𝑗=𝜕𝑝̇_𝑖𝜕𝑥_𝑗𝛿𝑡,𝜕𝑝^′_𝑖𝜕𝑝_𝑗=𝛿_𝑖𝑗+𝜕𝑝̇_𝑖𝜕𝑝_𝑗𝛿𝑡, (3.8) we find that𝜕𝑥^′₁,𝑥^′₂,…,𝑥^′_𝐷;𝑝^′₁,𝑝^′₂,…,𝑝^′_𝐷𝜕𝑥₁,𝑥₂,…,𝑥_𝐷;𝑝₁,𝑝₂,…,𝑝_𝐷𝜕𝑥̇=1+^𝑖𝜕𝑥_𝑖+𝜕𝑝̇_𝑖𝜕𝑝_𝑖𝛿𝑡+𝑂𝛿𝑡², (3.9) where𝜕𝑥̇_𝑖𝜕𝑥_𝑖+𝜕𝑝̇_𝑖𝜕𝑝_𝑖=𝜕𝜕𝑥_𝑖𝑥_𝑖,𝑝_𝑗𝜕𝐻𝜕𝑝_𝑗+𝑥_𝑖,𝑥_𝑗𝜕𝐻𝜕𝑥_𝑗+𝜕𝜕𝑝_𝑖−𝑥_𝑗,𝑝_𝑖𝜕𝐻𝜕𝑥_𝑗=𝜕𝜕𝑥_𝑖𝑥_𝑖,𝑥_𝑗𝜕𝐻𝜕𝑥_𝑗−𝜕𝜕𝑝_𝑖𝑥_𝑗,𝑝_𝑖𝜕𝐻𝜕𝑥_𝑗(2=−𝐷−1)𝐴+𝐵𝐩²𝐴𝑑𝐴𝑑𝐩²−𝐵𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗−2𝑑𝐴𝑑𝐩²+2𝑑𝐵𝑑𝐩²𝐩²+(𝐷+1)𝐵𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗)2=−(𝐷−1𝐴+𝐵𝐩²𝐴𝑑𝐴𝑑𝐩²+2𝑑𝐴𝑑𝐩²+𝑑𝐵𝑑𝐩²𝐩²+𝐵𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗. (3.10) On the other hand, using 𝛿𝐩²=2𝑝_𝑖𝛿𝑝_𝑖=2𝑝_𝑖𝑝_𝑖̇𝛿𝑡=−2𝐴+𝐵𝐩²𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗𝛿𝑡, (3.11) we have𝐴^′=𝐴+𝑑𝐴𝑑𝐩²𝛿𝐩²2=𝐴1−𝐴+𝐵𝐩²𝐴𝑑𝐴𝑑𝐩²𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗,𝐴𝛿𝑡_′+𝐵^′𝐩^′𝟐=𝐴+𝐵𝐩²+𝑑𝐴𝑑𝐩²+𝑑𝐵𝑑𝐩²𝐩²+𝐵𝛿𝐩²=𝐴+𝐵𝐩²1−2𝑑𝐴d𝐩²+𝑑𝐵𝑑𝐩²𝐩²+𝐵𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗𝛿𝑡, (3.12) where we have used the shorthand 𝐴^′=𝐴(𝐩^′𝟐) and 𝐵^′=𝐵(𝐩^′𝟐). Thus,𝐴^′𝐷−1𝐴^′+𝐵^′𝐩^′𝟐𝐴^𝐷−1𝐴+𝐵𝐩²=21−(𝐷−1)𝐴+𝐵𝐩²𝐴𝑑𝐴𝑑𝐩²+2𝑑𝐴𝑑𝐩²+𝑑𝐵𝑑𝐩²𝐩²+𝐵𝑝_𝑗𝜕𝐻𝜕𝑥_𝑗𝛿𝑡. (3.13) Comparing (3.10) and (3.13), it is clear that the ratio𝑑^𝐷𝐱𝑑^𝐷𝐩𝐴^𝐷−1𝐴+𝐵𝐩² (3.14) is invariant under time evolution.

This behavior of the phase space volume can be demonstrated using simple Hamiltonians. In [98], we solve the harmonic oscillator and coulomb potential problems for the case 𝐴=1+𝛽𝐩² and 𝐵=𝛽^′. There, in addition to the behavior of the phase space, it is found that the orbits of particles in these potentials no longer close on themselves. This is consistent with the breaking of degeneracies observed in the quantum cases which are associated with the conservation of the Runge-Lenz vector.

For the case 𝐵=0, (3.14) reduces to 𝑑^𝐷𝐱𝑑^𝐷𝐩/𝐴^𝐷, and our interpretation of ℏ𝐴(𝑝²) as the momentum dependent Planck constant which determines the size of a unit quantum cell becomes apparent. Integrating (3.14) over space,1𝑉𝑑^𝐷𝐱𝑑^𝐷𝐩𝐴^𝐷−1𝐴+𝐵𝐩²=𝑑^𝐷𝐩𝐴^𝐷−1𝐴+𝐵𝐩², (3.15) we can identify𝜌𝐩²=1𝐴^𝐷−1𝐴+𝐵𝐩² (3.16) as the density of states in momentum space. At high momentum where 𝐴 and 𝐵𝐩² become large, 𝜌(𝐩²) will be suppressed. We look at the impact of this suppression on the cosmological constant problem next.

4. Vacuum Energy and the Minimal Length

4.1. The Cosmological Constant and the Density of States

The origin of the cosmological constant Λ=3𝐻²₀Ω_Λ remains a mystery, and its understanding presents a major challenge to theoretical physics [8–12]. It is a contentious issue for string theory, notwithstanding its being the leading candidate for quantum gravity, though various hints exist that may point towards its resolution [99, 100]. Furthermore, the problem has recently assumed added urgency due to observations that the cosmological constant is small, positive, and clearly nonzero [101, 102]. In terms of the parameter Ω_Λ, the most-up-to date value is Ω_Λ∼0.73. With the Hubble parameter ℎ∼0.7 (the parameter ℎ is defined as ℎ=𝐻₀/(100km/s/Mpc)), we obtain as the vacuum energy density𝑐²Λ8𝜋𝐺_𝑁=𝑐²𝜌_critΩ_Λ=3𝐻²₀𝑐²8𝜋𝐺_𝑁Ω_Λ=8.096×10⁻⁴⁷GeV⁴/ℏ³𝑐³Ω_Λℎ²∼10⁻⁴⁷GeV⁴/ℏ³𝑐³. (4.1) The order of magnitude of this result is set by the dimensionful prefactor in the parentheses which can be expressed in terms of the Planck length ℓ_P=ℏ/𝜇_𝑃=ℏ𝐺_𝑁/𝑐³∼10⁻³⁵ m, and the scale of the visible universe ℓ₀=ℏ/𝜇₀≡𝑐/𝐻₀∼10²⁶ m as𝐻²₀𝑐²𝐺_𝑁=𝑐ℏ³𝜇²_𝑃𝜇²₀=ℏ𝑐ℓ²_𝑃ℓ²₀. (4.2)

In quantum field theory (QFT), the cosmological constant is calculated as the sum of the vacuum fluctuation energies of all momentum states. This is clearly infinite, so the integral is usually cut off at the Planck scale 𝜇_P=ℏ/ℓ_𝑃 beyond which spacetime itself is expected to become foamy [4], and the calculation untrustworthy. For a massless particle, we find that 1(2𝜋ℏ)^3𝜇P𝑑³𝐩12ℏ𝜔_𝑝=𝑐4𝜋²ℏ^3𝜇P0𝑑𝑝𝑝³=𝑐16𝜋²ℏ³𝜇⁴_𝑃=ℏ𝑐16𝜋²1ℓ⁴_𝑃∼10⁷⁴GeV⁴/ℏ³𝑐³, (4.3) which is about 120 orders of magnitude above the measured value. Note that this difference is essentially a factor of (ℓ₀/ℓ_𝑃)², the scale of the visible universe in Planck units squared. The change in the density of states suggested by the MLUR would change this calculation to1(2𝜋ℏ)^3∞𝑑³𝐩𝜌𝐩²12ℏ𝜔_𝑝=𝑐4𝜋²ℏ^3∞0𝑝𝑑𝑝³𝐴𝑝²²𝐴𝑝²+𝑝²𝐵𝑝². (4.4) For the case 𝐴(𝑝²)=1+𝛽𝑝², 𝐵(𝑝²)=0, we find [97] that 𝑐4𝜋²ℏ^3∞0𝑝𝑑𝑝³1+𝛽𝑝²³=𝑐16𝜋²ℏ³𝛽²=𝑐16𝜋²ℏ³𝜇⁴_𝑠=ℏ𝑐16𝜋²1ℓ⁴_𝑠,ℓ_𝑠=ℏ𝜇_𝑠=ℏ𝛽. (4.5) The integral is finite, without a UV cutoff, due to the suppression of the contribution of high momentum states. (There is an intriguing similarity here with Planck's resolution of the UV catastrophe of the black body radiation.) However, if we make the identification ℓ_𝑠=ℓ_𝑃, then this result is identical to (4.3), and nothing is gained. Of course, this is not surprising given that ℓ_𝑠 is the only scale in the calculation and effectively plays the role of the UV cutoff. To obtain the correct value of the cosmological constant from the above expression, we must choose ℓ_𝑠∼ℓ_𝑃ℓ₀∼10⁻⁵ m, which is too large to be the minimal length, or equivalently, 𝜇_𝑠=ℏ/ℓ_𝑠∼√𝜇_𝑃𝜇₀∼10⁻³eV/𝑐, which is too small to be the UV cutoff. However, we mention in passing that ℓ_𝑃ℓ₀ can be considered the uncertainty in measuring ℓ₀ due to the foaminess of spacetime [4, 103, 104] and has been argued as the possible size of a spacetime quantum cell when quantum gravity is properly taken into account [105–114]. At the moment, this point of view seems difficult to reconcile with phenomenological considerations.

We could introduce a second scale into the problem by letting 𝐵(𝑝²)=𝛽^′≠0. This leads to𝑐4𝜋²ℏ^3∞0𝑝𝑑𝑝³1+𝛽𝑝²²1+𝛽+𝛽^′𝑝²=𝑐8𝜋²ℏ³1𝛽𝛽^′𝛽1−𝛽^′𝛽ln1+^′_{𝛽′≫𝛽}𝛽𝑐−−−−→8𝜋²ℏ³1=𝑐𝛽𝛽′8𝜋²ℏ³𝜇²_𝑠𝜇_𝑠^′𝟐=ℏ𝑐8𝜋²1ℓ²_𝑠ℓ_𝑠^′2, (4.6) where ℓ^′_𝑠=ℏ/𝜇^′_𝑠=ℏ𝛽^′. If we identify ℓ_𝑠=ℓ_𝑃, then we must have ℓ^′_𝑠∼ℓ₀, which is even more problematic than ℓ_𝑃ℓ₀.

As these considerations show, our simple choice for 𝐴(𝑝²) and 𝐵(𝑝²) succeeds in rendering the cosmological constant finite but does not provide an adequate suppression. Would some other choice of 𝐴(𝑝²) and 𝐵(𝑝²) do better? To this end, let us try to see whether we can reverse engineer these functions so that the correct order of magnitude is obtained. Let us write𝜖⁴=^∞₀𝑑𝑝𝜌𝑝²𝑝³. (4.7) To generate the correct value for the cosmological constant, we must have 𝜖∼√𝜇_𝑃𝜇₀=10⁻³eV/𝑐, as we have seen. At this point, we invoke some numerology and note that if the SUSY breaking scale 𝜇_SUSY is on the order of a few TeV/𝑐, then the seesaw formula,𝜇𝜖∼^2SUSY𝜇_𝑃∼10⁻³eV/𝑐, (4.8) would give the correct size for 𝜖 as observed by Banks [115]. This expression is reminiscent of the well-known seesaw mechanism used to explain the smallness of neutrino masses [116–119]. One way to obtain this result is to have the density of states scale as 𝜌(𝑝²)∼𝑝⁴/𝜇⁴_𝑃 and place the UV cutoff at 𝜇_SUSY, beyond which the bosonic and fermionic contributions cancel. This would yield 𝜖⁴∼𝜇⁸_SUSY/𝜇⁴_𝑃. Unfortunately, this density of states is problematic since 𝑝⁴/𝜇⁴_𝑃≪1 for the entire integration region, so we are effectively suppressing everything. Furthermore, to obtain this suppression, we must have 𝐴(𝑝²)∼(𝜇_𝑃/𝑝)^4/3≫1, making the effective value of ℏ, and thus the size of the quantum cell, huge at low energies in clear contradiction to reality.

In retrospect, this result is not surprising since raising the UV cutoff from √𝜇_𝑃𝜇₀∼10⁻³eV/𝑐 to much higher values naturally requires the drastic suppression of contributions from below the cutoff. Thus, it is clear that the modification to the density of states, as suggested by the MLUR, by itself cannot solve the cosmological constant problem.

4.2. Need for a UV/IR Relation and a Dynamical Energy-Momentum Space

In the above discussion of summing over momentum states, the unstated assumption was that states at different momentum scales were independent, and that their total effect on the vacuum energy was the simple sum of their individual contributions. Of course, this assumption is the basis of the decoupling between small (IR) and large (UV) momentum scales which underlies our use of effective field theories. However, there are hints that this assumption is what needs to be reevaluated in order to solve the cosmological constant problem.

First and foremost, the expression for the vacuum energy density itself, 𝐻²₀𝑐²/𝐺_𝑁=ℏ𝑐/ℓ²_𝑃ℓ²₀, is dependent upon an IR scale ℓ₀ and a UV scale ℓ_𝑃, suggesting that whatever theory that explains its value must be aware of both scales and have some type of dynamical connection between them. Note that effective QFT's are not of this type but string theory is, given the UV/IR mixing relations discovered in several contexts as mentioned in the introduction.

Second, the contributions of the sub-Planckian modes (𝑝<𝜇_𝑃) independently by themselves are clearly too large, and there is a limit to the tweaking that can be done to the density of states in the IR since those modes undeniably exist. The only way out of the dilemma would be to cancel the contribution of the IR sub-Planckian modes against those of something else, say that coming from the UV trans-Planckian modes (𝑝>𝜇_𝑃) by introducing a dynamical connection between the two regimes [115].

That the sub-Planckian (𝑝<𝜇_𝑃) and trans-Planckian (𝑝>𝜇_𝑃) modes should cancel against each other is also suggested by the following argument: consider how the MLUR, (2.1), would be realized in field theory. The usual Heisenberg relation 𝛿𝑥𝛿𝑝=ℏ/2 is a simple consequence of the fact that coordinate and momentum spaces are Fourier transforms of each other. The more one wishes to localize a wave packet in coordinate space (smaller 𝛿𝑥), the more momentum states one must superimpose (larger 𝛿𝑝). In the usual case, there is no lower bound to 𝛿𝑥 one may localize the wave packet as much as one likes by simply superimposing states with ever larger momentum, and thus ever shorter wavelength, to cancel out the tails of the coordinate space distributions. On the other hand, the MLUR implies that if one keeps on superimposing states with momenta beyond 𝜇_𝑃=1/𝛽, then 𝛿𝑥 ceases to decrease and starts increasing instead. (See Figure 1.) The natural interpretation of such a phenomenon would be that the trans-Planckian modes (𝑝>𝜇_𝑃) when superimposed with the sub-Planckian ones (𝑝<𝜇_𝑃) would “jam” the sub-Planckian modes and prevent them from canceling out the tails of the wave-packets effectively. The mechanism we envision here is analogous to the “jamming” behavior seen in nonequilibrium statistical physics, in which systems are found to freeze with increasing temperature [120–123]. In fact, it has been argued that such “freezing by heating” could be characteristic of a background-independent quantum theory of gravity [124–128].

We should also note, that in our calculation presented above, the phase space over which the integration was performed was fixed and flat. Quantum gravity will naturally change the situation, leading to a fluctuating dynamical spacetime background. Furthermore, the MLUR implies that energy-momentum space will be a fluctuating dynamical entity as well [129–134]. First, the necessity of “jamming” between the sub-Planckian and trans-Planckian modes to implement the MLUR in field theory clearly illustrates that momentum space cannot be the simple Fourier transform of coordinate space but must rather be an independent entity. (Introducing a momentum space independent from coordinate space in field theory would make the wave-particle duality more complete in a sense, since for particles, momenta and coordinates are independent until the equation of motion, is imposed [135].) Second, the quantum properties of spacetime geometry may be understood in terms of effective expressions that involve the spacetime uncertainties:g_𝑎𝑏(𝑥)𝑑𝑥^𝑎𝑑𝑥^𝑏⟶𝑔_𝑎𝑏(𝑥)𝛿𝑥^𝑎𝛿𝑥^𝑏. (4.9) The UV/IR relation 𝛿𝑥∼ℏ𝛽𝛿𝑝 in the trans-Planckian region implies that this geometry of spacetime uncertainties can be transferred directly to the space of energy-momentum uncertainties, endowing it with a geometry as well:𝑔_𝑎𝑏(𝑥)𝛿𝑥^𝑎𝛿𝑥^𝑏⟶𝐺_𝑎𝑏𝑝𝛿𝑝^𝑎𝛿𝑝^𝑏. (4.10) The usual intuition that local properties in spacetime correspond to non-local features of energy-momentum space (as implied by the canonical uncertainty relations) is obviated by the linear relation between the uncertainties in coordinate space and momentum space.

What would a dynamical energy-momentum space entail? Let us speculate. It has been argued that a dynamical spacetime, with its foamy UV structure [4], would manifest itself in the IR via the uncertainties in the measurements of global spacetime distances as [103–114]𝛿ℓ∼ℓℓ_𝑃, (4.11) a relation which is reminiscent of the famous result for Brownian motion derived by Einstein [136] and is also covariant in 3+1 dimensions. Let us assume that a similar “Brownian” relation holds in energy-momentum space due to its “foaminess” [134]𝛿𝜇∼√𝜇𝜇_𝑃. (4.12) If the energy-momentum space has a finite size, a natural UV cutoff, at 𝜇₊≫𝜇_P, (a maximum energy/momentum is introduced in deformed special relativity [137–140]), then its fluctuation 𝛿𝜇₊ will be given by 𝛿𝜇₊=√𝜇₊𝜇_P≫𝜇_P. The MLUR implies that the mode at this scale must cancel, or “jam,” against another which shares the same 𝛿𝑥, namely, the mode with an uncertainty given by 𝛿𝜇₋=𝜇²_𝑃/𝛿𝜇₊=𝜇_𝑃𝜇_𝑃/𝜇₊=√𝜇₋𝜇_𝑃≪𝜇_𝑃, that is,𝜇₋=𝜇²_𝑃𝜇₊=𝛿𝜇²₋𝜇_𝑃≪𝜇_𝑃. (4.13) All modes between 𝜇₋ and 𝜇₊ will “jam.” Therefore, 𝜇₋ will be the effective UV cutoff of the momentum integral and not 𝜇₊, which would yield𝜖⁴∼𝜇⁴₋∼𝛿𝜇⁸₋𝜇⁴_𝑃∼𝜇⁸_𝑃𝜇⁴₊. (4.14) This reproduces the seesaw formula, (4.8), and if 𝛿𝜇₋∼fewTeV/𝑐, we obtain the correct cosmological constant.

5. Outlook: What Is String Theory?

In the concluding section, we wish to discuss a few implications of our work for nonperturbative string theory and the question: what is string theory [13]? Our discussion of this difficult question, being limited by the scope of our work on the minimal length, will neccesarily be a bit speculative.

Our toy model for the MLUR was essentially algebraic. As such, it raises the possibility that more general algebraic structures may play a key role in nonperturbative string theory. In the introductory section, we mentioned that the MLUR is motivated by the scattering of string-like excitations in first quantized string theory. If one takes into account other objects in nonperturbative string theory, such as 𝐷-branes, one is led to the STUR, (1.6), proposed by Yoneya. The STUR generalizes the MLUR, and can be further generalized to a cubic form (motivated by 𝑀-theory) [67–73]𝛿𝑥𝛿𝑦𝛿𝑡∼ℓ³_𝑃/𝑐. (5.1) Given the usual interpretation of the canonical Heisenberg uncertainty relations in terms of fundamental commutators, one might look for the associated cubic algebraic structures in string theory.

Another hint of cubic algebraic structure appears in the nonperturbative formulation of open string field theory by Witten et al. [141, 142]. The Witten action for the classical open string field, Φ, is of an abstract Chern-Simons type𝑆_o(Φ)=Φ⋆Φ⋆Φ. (5.2) Here, the star product is defined by the world-sheet path integral,(𝑋)𝐺(𝑋)𝑖𝐹⋆𝐺=𝐷𝑋𝐹exp𝛼^′𝑆_𝑃(𝑋), (5.3) which is in turn determined by the world-sheet Polyakov action𝑆_𝑃(𝑋)=12𝑑²𝜎√−𝑔𝑔^𝑎𝑏𝜕_𝑎𝑋^𝑖𝜕_𝑏𝑋^𝑗𝐺_𝑖𝑗+⋯. (5.4) The fully quantum open string field theory is then, in principle, defined by yet another path integral in the infinite dimensional space of Φ, that is,𝑖𝐷Φexp𝑔_𝑐𝑆_o(Φ). (5.5) A more general, and in principle nonassociative structure, appears in Strominger's formulation of closed string field theory, which is also cubic [143]. Strominger's paper mentions the relevance of the 3-cocycle structure for this formulation of closed string field theory. Very schematically𝑆_c(Ψ)=Ψ×(Ψ×Ψ), (5.6) where × is a nonassociative product defined in [143]. (For the role of nonassociativity in the theories of gravity, and a relation between Einstein's gravity and nonassociative Chern-Simons theory, see [144]).

Is there an underlying algebraic structure that could give rise to these cubic structures? In our toy model, the 2-bracket appears quite naturally. Such structures can be naturally generalized into 3-bracket. For example, the usual Lie algebra structure known from gauge theories, [𝑇_𝐴,𝑇_𝐵]=𝑓_𝐴𝐵𝐶𝑇_𝐶, where the structure constants 𝑓_𝐴𝐵𝐶 satisfy the usual Jacobi identity, seems to be naturally generalized to a triple algebraic structure𝑇_𝐴,𝑇_𝐵,𝑇_𝐶=𝑓_{𝐴𝐵𝐶𝐷}𝑇_𝐷, (5.7) where𝐴_𝑖,𝐴_𝑗,𝐴_𝑘≡𝜖^𝑎𝑏𝑐𝐴_𝑎𝐴_𝑏𝐴_𝑐, (5.8) with the structure constants 𝑓_{𝐴𝐵𝐶𝐷} satisfying a quartic fundamental identity [145–149]. These structures occur in the context of the theory of 𝑁-membranes [150]. They are also present in more elementary examples. Consider a charged particle 𝑒 of mass 𝑚 in the external magnetic field 𝐁. As is well known, the velocities 𝑣_𝑎 satisfy the commutation relation𝑣_𝑖,𝑣_𝑗=𝑖𝑒ℏ𝑚²𝜖_𝑖𝑗𝑘𝐵_𝑘, (5.9) as well as the triple commutation relation, the associator, given by [151]𝑣₁,𝑣₂,𝑣₃+𝑣₂,𝑣₃,𝑣₁+𝑣₃,𝑣₁,𝑣₂=𝑒ℏ²𝑚³𝜕_𝑖𝐵_𝑖. (5.10) This associator is zero, and thus trivial, in the absence of magnetic monopoles: 𝜕_𝑖𝐵_𝑖=0. Note that the triple bracket defined in (5.8) is “one-half” of the associator since𝐴,𝐵,𝐶≡𝜖^𝑎𝑏𝑐𝐴_𝑎𝐴_𝑏𝐴_𝑐=𝐴[𝐵,𝐶]+𝐵[𝐶,𝐴]+𝐶[𝐴,𝐵],𝐴,𝐵,𝐶≡𝜖^𝑎𝑏𝑐𝐴_𝑎𝐴_𝑏𝐴_𝑐=[𝐵,𝐶]𝐴+[𝐶,𝐴]𝐵+[𝐴,𝐵]𝐶. (5.11) The presence of monopoles is an indicator of a 3-cocycle [151]. The triple commutator has also been encountered in the study of closed string dynamics [156].

What would be the role of such a general algebraic structure for the foundations of string theory? Given the general open-closed string relation (the closed strings being in some sense the bounds states of open strings) the noncommutative and nonassociative algebraic structures might be related as in some very general and abstract form of the celebrated AdS/CFT duality [152–155]. We recall that in the AdS/CFT correspondence, one computes the on-shell bulk action 𝑆_bulk and relates it to the appropriate boundary correlators. The conjecture is that the generating functional of the vacuum correlators of the operator 𝑂 for a 𝑑-dimensional conformal field theory (CFT) is given by the partition function 𝑍(𝜙) in (Anti-de-Sitter) AdS_𝑑+1 spaceexp𝐽𝑂=𝑍𝜙⟶exp−𝑆_bulk𝑔,𝜙,…, (5.12) where, in the semiclassical limit, the partition function becomes 𝑍=exp(−𝑆_bulk). Here 𝑔 denotes the metric of the AdS_𝑑+1 space, and the boundary values of the bulk field, 𝜙, are given by the sources, 𝐽, of the boundary CFT. Essentially, one reinterprets the RG flow of the boundary nongravitational theory in terms of bulk gravitational equations of motion and then rewrites the generating functional of vacuum correlators of the boundary theory in terms of a semiclassical wave function of the bulk “universe” with specific boundary conditions.

In view of our comments on the general algebraic structures in string theory, it is tempting to propose an extension of this duality in a more abstract sense of open and closed string field theory, and the relationship between the non-commutative and nonassociative structuresexp𝐽𝑂(Φ)_o=𝑍(Ψ)⟶exp−𝑆_𝑐(Ψ). (5.13) The “boundary” in this abstract case has to be defined algebraically, as a region of the closed string Hilbert space on which the 3-cocycle anomaly vanishes. Inside the region, the 3-cocycle would be nonzero. In this way, we would have more abstract definitions of the “boundary” and “bulk.” In some sense, this relation would look like a generalized Laplace transform of an exponential of a cubic expression giving another exponential of a cubic expression, as with the asymptotics of the Airy function ∫𝑑𝑥exp(𝑡𝑥−𝑥³)∼exp(−𝑡³/2).

Finally, following our discussion of the vacuum energy problem in the previous section, it seems natural that any more fundamental formulation of string theory would have to work on a curved momentum space. This would mesh nicely with the ideas presented in [124–133]. If curved energy-momentum space is crucial in quantum gravity (and thus string theory) for the solution of the vacuum energy problem, then we are naturally led to question the usual formulation of string theory as a canonical quantum theory. Also, if the vacuum energy can be made small, what physical principle selects such a vacuum? This leads to the question of background independence and vacuum selection. The issue of background independence in string theory is that the fundamental equations should not select a quantum theory the same way Einstein's gravitational equations do not select any geometry; only asymptotic or symmetry conditions select a geometry. Again, we are back to the questions regarding the role of general quantum theory in the most fundamental formulation of string theory. Note that such discussion of general quantum theory also sheds light on the question of time evolution and the problem of time in string theory [124–128].

Acknowledgments

The authors wish to thank Yang-Hiu He for inviting us to write this review. DM, TT and ZL are supported in part by the U.S. Department of Energy, grant DE-FG05-92ER40677, task A.