Space-time Uncertainty Relation Research Articles

Stringy spacetime uncertainty principle and a modified trans-Planckian censorship criterion

We study the implications of the stringy spacetime uncertainty relation (STUR) for inflationary cosmology. By demanding that no fluctuation modes that exit the Hubble radius are affected by the nonlocality resulting from the STUR, we find an upper bound on the number of e-foldings of inflation. The bound is a factor of 2 weaker than what results from the trans-Planckian censorship criterion (TCC). By demanding that the inflationary phase is simultaneously consistent with STUR and sufficiently long for inflation to provide a causal explanation of structure on the scale of the current Hubble radius, we find an upper bound on the energy scale of inflation. The bound is less restrictive than what follows from the TCC, but it remains in conflict with canonical single-field inflation models. Published by the American Physical Society 2024

Energy-Time Uncertainty Relations in Quantum Measurements

Quantum measurement is a physical process. A system and an apparatus interact for a certain time period (measurement time), and during this interaction, information about an observable is transferred from the system to the apparatus. In this study, we quantify the energy fluctuation of the quantum apparatus required for this physical process to occur autonomously. We first examine the so-called standard model of measurement, which is free from any non-trivial energy-time uncertainty relation, to find that it needs an external system that switches on the interaction between the system and the apparatus. In such a sense this model is not closed. Therefore to treat a measurement process in a fully quantum manner we need to consider a "larger" quantum apparatus which works also as a timing device switching on the interaction. In this setting we prove that a trade-off relation (energy-time uncertainty relation), $\tau\cdot \Delta H_A \geq \frac{\pi \hbar}{4}$, holds between the energy fluctuation $\Delta H_A$ of the quantum apparatus and the measurement time $\tau$. We use this trade-off relation to discuss the spacetime uncertainty relation concerning the operational meaning of the microscopic structure of spacetime. In addition, we derive another trade-off inequality between the measurement time and the strength of interaction between the system and the apparatus.

K-Inflation in noncommutative space–time

The power spectra of the scalar and tensor perturbations in the noncommutative k-inflation model are calculated in this paper. In this model, all the modes created when the stringy space-time uncertainty relation is satisfied are generated inside the sound/Hubble horizon during inflation for the scalar/tensor perturbations. It turns out that a linear term describing the noncommutative space-time effect contributes to the power spectra of the scalar and tensor perturbations. Confronting the general noncommutative k-inflation model with latest results from \textit{Planck} and BICEP2, and taking $c_S$ and $\lambda$ as free parameters, we find that it is well consistent with observations. However, for the two specific models, i.e. the tachyon and DBI inflation models, it is found that the DBI model is not favored, while the tachyon model lies inside the $1\sigma$ contour, if the e-folds number is assumed to be around $50\sim60$.

Field propagation in a stochastic background space: The rate of light incoherence in stellar interferometry

We present Hilbert space representation for a relatively broad class of minimum-length deformed quantum mechanical models obtained by incorporating a space-time uncertainty relation into quantum mechanics. The correspondingly modified field theory is used for estimating the deviation of the light incoherence rate from distant astrophysical sources from the standard case.

Space–time uncertainty relation from quantum and gravitational principles

By collecting both quantum and gravitational principles, a space–time uncertainty relation ( δ t ) ( δ r ) 3 ⩾ π r 2 l p 2 is derived. It can be used to facilitate the discussion of several profound questions, such as computational capacity and thermodynamic properties of the universe and the origin of holographic dark energy. The universality and validity of the proposed relation are illustrated via these examples.

Dark energy due to effective quantum field theory

In the cosmological context an effective quantum field theory describing the behavior of visible matter in the Universe is characterized with its inherent UV cutoff and also with an IR scale that is set by the cosmological (particle) horizon. This UV–IR relation naturally defines a space–time grid over a horizon scale. Using the approach for determining of dark energy through the space–time uncertainty relation versus such a space–time grid, we estimate the energy density and pressure of a dark energy defined by this UV–IR relation. Such a dark energy shows up to decay linearly with time and exhibits a negative pressure only recently.

Extended fermion representation of multi-charge 1/2-BPS operators in AdS/CFT — towards field theory of D-branes

We extend the fermion representation of single-charge 1/2-BPS operators in the four-dimensional N=4 super Yang-Mills theory to general (multi-charge) 1/2-BPS operators such that all six directions of scalar fields play roles on an equal footing. This enables us to construct a field-theorectic representation for a second-quantized system of spherical D3-branes in the 1/2-BPS sector. The Fock space of D3-branes is characterized by a novel exclusion principle (called `Dexclusion' principle), and also by a nonlocality which is consistent with the spacetime uncertainty relation. The Dexclusion principle is realized by composites of two operators, obeying the usual canonical anticommutation relation and the Cuntz algebra, respectively. The nonlocality appears as a consequence of a superselction rule associated with a symmetry which is related to the scale invariance of the super Yang-Mills theory. The entropy of the so-called superstars, with multiple charges, which have been proposed to be geometries corresponding to the condensation of giant gravitons is discussed from our viewpoint and is argued to be consistent with the Dexclusion principle. Our construction may be regarded as a first step towards a possible new framework of general D-brane field theory.

Power spectra in spacetime noncommutative inflation

The spacetime uncertainty relation, which deviates from general relativity, emerges in string/M-theory. It is possible to observe this deviation through cosmological experiments, in particular through the measurements of CMB power spectrum. This paper extends some previous observations to more general inflation schemes. We find that the noncommutative spacetime effects always suppress the power spectrum of both the scalar and tensor perturbations, and may provide a large enough running of the spectral index to fit the results of WMAP in the inflation model. (c) 2005 Elsevier B.V. All rights reserved.

Non-commutative power-law inflation: mode equation and spectra index

Following an elegant approach that merges the effects of the stringy spacetime uncertainty relation into primordial perturbations suggested by Brandenberger and Ho, we show the mode equation up to the first order of non-commutative parameter. A new approximation is provided to calculate the mode functions analytically in the non-commutative power-law inflation models. It turns out that non-commutativity of spacetime can provide small corrections to the power spectrum of primordial fluctuations as the first-year results of WMAP indicate. Moreover, using the WMAP data, we obtain the value of expansion parameter, non-commutative parameter and find the approximation is viable. In addition, we determined the string scale l s ≃ 2.0 × 10 − 29 cm .

A note on curvature fluctuation of noncommutative inflation

An elegant approach, which incorporates the effect of the stringy spacetime uncertainty relation, to calculate power spectra of fluctuations during inflation has been suggested by Brandenberger and Ho. In this approach, one of important features is the appearance of an upper bound on the comoving momentum $k$, at which the stringy spacetime uncertainty relation is saturated. As a result, the time-dependent upper bound leads us to choose naturally a set of initial vacua for each mode, in which the stringy uncertainty relation is saturated. In this note, with that set of vacua we calculate power spectrum of curvature fluctuation for a power law inflation, up to the leading order of a parameter describing the spacetime noncommutativity. It turns out that this choice of initial vacuum has a significant effect on the power spectrum of fluctuations.

Noncommutative spacetime, stringy spacetime uncertainty principle, and density fluctuations

We propose a variation of spacetime noncommutative field theory to realize the stringy spacetime uncertainty relation without breaking any of the global symmetries of the homogeneous isotropic universe. We study the spectrum of metric perturbations in this model for a wide class of accelerating background cosmologies. Spacetime noncommutativity leads to a coupling between the fluctuation modes and the background cosmology which is nonlocal in time. For each mode, there is a critical time at which the spacetime uncertainty relation is saturated. This is the time when the mode is generated. These effects lead to a spectrum of fluctuations whose spectral index is different from what is obtained for commutative spacetime in the infrared region, but is unchanged in the ultraviolet region. In the special case of an exponentially expanding background, we find a scale-invariant spectrum. but with a different magnitude than in the context of commutative spacetime if the Hubble constant is above the string scale.

S-duality, noncritical open string and noncommutative gauge theory

We examine several aspects of S-duality of four-dimensional noncommutative gauge theory. By making duality transformation explicit, we find that S-dual of noncommutative gauge theory is defined in terms of dual noncommutative deformation. In `magnetic' noncommutative U(1) gauge theory, we show that, in addition to gauge bosons, open D-strings constitute important low-energy excitations: noncritical open D-strings. Upon S-duality, they are mapped to noncritical open F-strings. We propose that, in dual `electric' noncommutative U(1) gauge theory, the latters are identified with gauge-invariant, open Wilson lines. We argue that the open Wilson lines are chiral due to underlying parity noninvariance and obey spacetime uncertainty relation. We finally argue that, at high energy-momentum limit, the `electric' noncommutative gauge theory describes correctly dynamics of delocalized multiple F-strings.

String Theory and the Space-Time Uncertainty Principle

The notion of a space-time uncertainty principle in string theory is clarified and further developed. The motivation and the derivation of the principle are first reviewed in a reasonably self-contained way. It is then shown that the nonperturbative (Borel summed) high-energy and high-momentum transfer behavior of string scattering is consistent with the space-time uncertainty principle. It is also shown that, in consequence of the principle, string theories in 10 dimensions generically exhibit a characteristic length scale which is equal to the well-known 11 dimensional Planck length $g_s^{1/3}\ell_s$ of M-theory as the scale at which stringy effects take over the effects of classical supergravity, even without involving D-branes directly. The meanings of the space-time uncertainty relation in connection with D-branes and black holes are discussed and reinterpreted. Finally, we present a novel interpretation of the Schild-gauge action for strings from a viewpoint of noncommutative geometry, which conforms to the space-time uncertainty relation by manifestly exhibiting a noncommutativity of quantized string coordinates dominantly between space and time. We also discuss the consistency of the space-time uncertainty relation with S and T dualities.

Space-time uncertainty relation and Lorentz invariance

We discuss a Lorentz covariant space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period delta t is larger than the Planck time lp. At delta t < lp, this uncertainty relation takes roughly the form delta t delta x > lp^2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the representation space. In four-dimensional space-time, the Jacobi identity should be violated in the algebraic representation of the uncertainty relation.

Short-distance space-time structure and black holes in string theory: a short review of the present status

We briefly review the present status of string theory from the viewpoint of its implications on the short-distance space-time structure and black hole physics. Special emphases are given on two closely related issues in recent developments towards nonperturbative string theory, namely, the role of the space-time uncertainty relation as a qualitative but universal characterization of the short-distance structure of string theory and the microscopic formulation of black-hole entropies. We will also suggest that the space-time uncertainty relation can be an underlying principle for the holographic property of M theory, by showing that the space-time uncertainty relation naturally explains the UV/IR relation used in a recent derivation of the holographic bound for D3 brane by Susskind and Witten.

Supersymmetric IIB matrix models from space-time uncertainty principle and topological symmetry

Starting with topological field theory, we derive space-time uncertainty relation proposed by Yoneya through breakdown of topological symmetry in the large $N$ matrix model. Next, on the basis of only two basic principles, those are, generalized space-time uncertainty principle containing spinor field and topological symmetry, we construct a new matrix model. If we furthermore impose a requirement of N=2 supersymmetry, this new matrix model exactly reduces to the IKKT model or the Yoneya model for IIB superstring depending on an appropriate choice for a scalar function. A key feature of these formulations is an appearance of the nontrivial "dynamical" theory through breakdown of topological symmetry in the matrix model. It is closely examined why the nontrivial "dynamical" theory appears from the trivial topological field theory.

SPACE–TIME UNCERTAINTY PRINCIPLE FROM BREAKDOWN OF TOPOLOGICAL SYMMETRY

Starting from topological quantum field theory, we derive space–time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large-N matrix model. This work suggests that the topological symmetry might be an underlying higher symmetry behind the space–time uncertainty principle of string theory.

On the Space–Time Uncertainty Relations of Liouville Strings and D-Branes

Within a Liouville approach to noncritical string theory, we argue for a nontrivial commutation relation between space and time observables, leading to a nonzero space–time uncertainty relation δx δt&gt;0, which vanishes in the limit of weak string coupling.

Strings from IIB matrices

D-string action is constructed from IIB matrices, a spacetime commutator is essential in this construction. This hints at the central role of the spacetime uncertainty relation in a unified formulation of strings. Vertex operators of fundamental strings are also discussed.

Pointlike D-brane Dynamics and Space-Time Uncertainty Relation

We argue that the space-time uncertainty relation of the form $\ensuremath{\Delta}X\ensuremath{\Delta}T\ensuremath{\gtrsim}{\ensuremath{\alpha}}^{\ensuremath{'}}$ for the observability of the distances with respect to time, $\ensuremath{\Delta}T$, and space, $\ensuremath{\Delta}X$, is universally valid in string theory including D-branes. This relation has been previously proposed by one (T. Y.) of the present authors as a simple qualitative representation of the perturbative short-distance structure of fundamental string theory. We show that the relation, combined with the usual quantum mechanical uncertainty principle, explains the key qualitative features of D-particle dynamics.