Discretized Finsler Structure: An Approach to Quantizing the First Fundamental Form

Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundamental form could be proposed. For precise accurate measurement of the first fundamental form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><msup><mi>s</mi><mn>2</mn></msup><mo>=</mo><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>d</mi><msup><mi>x</mi><mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi><mi>ν</mi></msup></mrow></semantics></math></inline-formula>, the author derived a quantum-induced revision of the fundamental tensor. To this end, the four-dimensional Riemann manifold is extended to the eight-dimensional Finsler manifold, in which the quadratic restriction on the length measure is relaxed, especially in the relativistic regime; the minimum measurable length could be imposed ad hoc on the Finsler structure. The present script introduces an approach to quantize the fundamental tensor and first fundamental form. Based on gravitized quantum mechanics, the resulting relativistic generalized uncertainty principle (RGUP) is directly imposed on the Finsler structure, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><msubsup><mover accent="true"><mi>x</mi><mo>^</mo></mover><mn>0</mn><mi>μ</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>p</mi><mo>^</mo></mover><mn>0</mn><mi>ν</mi></msubsup><mo>)</mo></mrow></semantics></math></inline-formula>, which is obviously homogeneous to one degree in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mover accent="true"><mi>p</mi><mo>^</mo></mover><mn>0</mn><mi>μ</mi></msubsup></semantics></math></inline-formula>. The momentum of a test particle with mass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>m</mi><mo>¯</mo></mover><mo>=</mo><mi>m</mi><mo>/</mo><msub><mi>m</mi><mi mathvariant="monospace">p</mi></msub></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi mathvariant="monospace">p</mi></msub></semantics></math></inline-formula> is the Planck mass. This unambiguously results in the quantized first fundamental form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><msup><mover accent="true"><mi>s</mi><mo>˜</mo></mover><mn>2</mn></msup><mrow><mo>=</mo><mo>[</mo><mn>1</mn><mo>+</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>β</mi><msubsup><mover accent="true"><mi>p</mi><mo>^</mo></mover><mn>0</mn><mi>ρ</mi></msubsup><msub><mover accent="true"><mi>p</mi><mo>^</mo></mover><mrow><mn>0</mn><mi>ρ</mi></mrow></msub><mo>)</mo></mrow><msup><mover accent="true"><mi>m</mi><mo>¯</mo></mover><mn>2</mn></msup><mrow><mo>(</mo><mo>|</mo></mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><msup><mrow><mo>|</mo><mo>/</mo><mi mathvariant="script">A</mi><mo>)</mo></mrow><mn>2</mn></msup><mrow><mo>]</mo></mrow><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>d</mi><msup><mover accent="true"><mi>x</mi><mo>^</mo></mover><mi>μ</mi></msup><mi>d</mi><msup><mover accent="true"><mi>x</mi><mo>^</mo></mover><mi>ν</mi></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>x</mi><mo>¨</mo></mover></semantics></math></inline-formula> is the proper spacelike four-acceleration, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is the maximal proper acceleration, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is the RGUP parameter. We conclude that an additional source of curvature associated with the mass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>m</mi><mo>¯</mo></mover></semantics></math></inline-formula>, whose test particle is accelerated at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo></mrow><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>, apparently emerges. Thereby, quantizations of the fundamental tensor and first fundamental form are feasible.