CAN SPACE TIME BE PROBED BELOW THE STRING SIZE?

CAN SPACE TIME BE PROBED BELOW THE STRING SIZE?

October 1988 | D. Amati, M. Ciafaloni, G. Veneziano
Can space-time be probed below the string size? D. Amati, M. Ciafaloni, G. Veneziano Abstract: Strings may be explored through scattering at Planckian energies down to distances of the order of the string size. This is possible if the energy is not extreme and the scattering angle approaches a critical value. Above this angle, the distance increases, departing from the usual position-momentum uncertainty relation. The resolution cannot be smaller than the string length, suggesting that below the Planck scale, the concept of space-time changes. In field theories, short-distance behavior of local operators provides information on fundamental degrees of freedom. String theory deals with physical amplitudes, so local operators and Green's functions are unnatural. Insights on short distances must come from hard processes like fixed-angle high-energy scattering and high-temperature behavior. Using explicit calculations, the theory's reaction to large angles is studied. It is not possible to test distances shorter than the string length λs = √(κα'). This is due to the softness of the string, which generates a different realization of hard processes. The gravitational radius R(E) ~ (GN E)^{1/(D-3)} is dynamically generated. If R(E) > λs, new contributions at distances of order R(E) indicate a classical gravitational instability. If R(E) < λs, these contributions are irrelevant, and no obstacle arises in analyzing short distances. Larger momentum transfers do not always correspond to shorter distances. The analysis of the angle-distance relationship suggests a modification of the uncertainty relation at the Planck scale, leading to a minimal observable length of the order of the string size. The high-energy regime studied has large effective coupling g²α' s and small loop expansion parameter. The smallness of g² allows a hierarchy of high-energy behaviors in the loop expansion. The leading term gives powers of g² s, while subleading terms are suppressed by extra powers of g². The eikonal function δ(b, s) is derived, showing characteristic power behavior for large b and infrared singularity for D = 4. The finite string size smooths Re δ for small b and contributes to the absorptive part (Im δ). String excitations provide dominant absorptive contributions even when Im δ dies out. The Fourier transform of the amplitude is discussed to uncover the distance-angle relationship. For small angles, the larger saddle point b_> dominates, leading to classical Einstein deflection in an Aichelburg-Sexl metric. For large angles, the saddle points become complex, and the behavior simplifies. The results are valid under consistency conditions, ensuring the angular region explored is reliable. The physics of large momentum transfers in string theory is characterized by two phenomena: increasing loop number <N> leading to sizable q/<N> but not large, and a change in regime at θ = θ_M, modifying the distance-momentum transfer relation and suggestingCan space-time be probed below the string size? D. Amati, M. Ciafaloni, G. Veneziano Abstract: Strings may be explored through scattering at Planckian energies down to distances of the order of the string size. This is possible if the energy is not extreme and the scattering angle approaches a critical value. Above this angle, the distance increases, departing from the usual position-momentum uncertainty relation. The resolution cannot be smaller than the string length, suggesting that below the Planck scale, the concept of space-time changes. In field theories, short-distance behavior of local operators provides information on fundamental degrees of freedom. String theory deals with physical amplitudes, so local operators and Green's functions are unnatural. Insights on short distances must come from hard processes like fixed-angle high-energy scattering and high-temperature behavior. Using explicit calculations, the theory's reaction to large angles is studied. It is not possible to test distances shorter than the string length λs = √(κα'). This is due to the softness of the string, which generates a different realization of hard processes. The gravitational radius R(E) ~ (GN E)^{1/(D-3)} is dynamically generated. If R(E) > λs, new contributions at distances of order R(E) indicate a classical gravitational instability. If R(E) < λs, these contributions are irrelevant, and no obstacle arises in analyzing short distances. Larger momentum transfers do not always correspond to shorter distances. The analysis of the angle-distance relationship suggests a modification of the uncertainty relation at the Planck scale, leading to a minimal observable length of the order of the string size. The high-energy regime studied has large effective coupling g²α' s and small loop expansion parameter. The smallness of g² allows a hierarchy of high-energy behaviors in the loop expansion. The leading term gives powers of g² s, while subleading terms are suppressed by extra powers of g². The eikonal function δ(b, s) is derived, showing characteristic power behavior for large b and infrared singularity for D = 4. The finite string size smooths Re δ for small b and contributes to the absorptive part (Im δ). String excitations provide dominant absorptive contributions even when Im δ dies out. The Fourier transform of the amplitude is discussed to uncover the distance-angle relationship. For small angles, the larger saddle point b_> dominates, leading to classical Einstein deflection in an Aichelburg-Sexl metric. For large angles, the saddle points become complex, and the behavior simplifies. The results are valid under consistency conditions, ensuring the angular region explored is reliable. The physics of large momentum transfers in string theory is characterized by two phenomena: increasing loop number <N> leading to sizable q/<N> but not large, and a change in regime at θ = θ_M, modifying the distance-momentum transfer relation and suggesting
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