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String Theory and the Space-Time Uncertainty Principle

Tamiaki Yoneya

TL;DR

The paper posits a universal space-time uncertainty principle for string/M-theory, $ ext{Δ}X\text{Δ}T \gtrsim \,\ell_s^2$, developed from both energy-time reinterpretations and conformal-invariance arguments. It analyzes high-energy string scattering, showing that nonperturbative Borel-summed amplitudes are compatible with STUR and identify a characteristic M-theory scale $\text{ℓ}_M = g_s^{1/3}\text{ℓ}_s$, emerging from black-hole considerations and D-particle dynamics. The STUR is further connected to microscopic black hole physics, holography, and UV-IR relations, suggesting a deep nonlocal structure underlying gravity in string theory. A noncommutative-geometric formulation based on the Schild action and a world-sheet bivector is proposed as a route to a nonperturbative framework consistent with STUR. Overall, the work argues that space-time nonlocality, encoded in STUR, captures essential qualitative features of string/M-theory, including dualities, black-hole physics, and holography, and points toward a nonperturbative, geometrically grounded formulation.

Abstract

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.

String Theory and the Space-Time Uncertainty Principle

TL;DR

The paper posits a universal space-time uncertainty principle for string/M-theory, , developed from both energy-time reinterpretations and conformal-invariance arguments. It analyzes high-energy string scattering, showing that nonperturbative Borel-summed amplitudes are compatible with STUR and identify a characteristic M-theory scale , emerging from black-hole considerations and D-particle dynamics. The STUR is further connected to microscopic black hole physics, holography, and UV-IR relations, suggesting a deep nonlocal structure underlying gravity in string theory. A noncommutative-geometric formulation based on the Schild action and a world-sheet bivector is proposed as a route to a nonperturbative framework consistent with STUR. Overall, the work argues that space-time nonlocality, encoded in STUR, captures essential qualitative features of string/M-theory, including dualities, black-hole physics, and holography, and points toward a nonperturbative, geometrically grounded formulation.

Abstract

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 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.

Paper Structure

This paper contains 16 sections, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Derivation of the space-time uncertainty relation
  3. A reinterpretation of energy-time uncertainty relation in terms of strings
  4. The nature of the space-time uncertainty relation
  5. Conformal symmetry and the space-time uncertainty relation
  6. High-energy scattering of strings and the space-time uncertainties
  7. How do we detect the interaction region from S-matix?
  8. High-energy and high-momentum transfer behavior of string scattering
  9. The Regge limit and space-time uncertainties
  10. A remark: minimum nonlocality ?
  11. The meaning of space-time uncertainty relation
  12. The characteristic scale for microscopic black holes in string theory
  13. The characteristic scale of D-particle dynamics
  14. Interpretation of black-hole complementarity and UV-IR correspondence
  15. Toward a noncommutative geometric formulation
  16. ...and 1 more sections

Figures (1)

  • Figure 1: This diagram schematically shows the structure of the space-time uncertainty relation and the black hole uncertainty relation. The critical point is where the two relations meet.