Published equation contexts
Imagine each mass taking a left or right path. Pick branch i for mass A and branch j for mass B. Their separation sets a gravitational interaction energy; over time that energy rotates the complex amplitude of that joint path by the angle shown here. Different paths can acquire different angles.
None of this makes the experiment ambiguous, and this is the paper’s second and sharper result. The branch separations that enter the gravitational phase each mass’s superposition picks up are relational quantities — distances between mass A’s branches and mass B’s branches — and relational quantities do not change when you change which system you have designated the reference frame. Write = G t / ( ) for the phase a pair of branches i,j \{L,R\} accumulates over interaction time t at separation , with G Newton’s constant and the reduced Planck constant. Three of the four possible combinations of these branch-pair phases can always be removed…
The dimensionless phase angle, in radians, acquired by the joint path in which mass A uses branch i and mass B uses branch j. Its individual value depends on phase convention; differences between branch-pair phases can affect interference and entanglement.
Read this term in its guide →Each label is L or R. The first chooses mass A’s path and the second chooses mass B’s path, giving LL, LR, RL, and RR.
Read this term in its guide →The constant that sets the strength of Newtonian gravity. Multiplying G by both masses and dividing by their separation gives the magnitude of their interaction energy.
Read this term in its guide →The mass of the first object, measured in kilograms. The phase grows in proportion to this mass while the other quantities are fixed.
Read this term in its guide →The mass of the second object, measured in kilograms. The phase grows in proportion to the product .
Read this term in its guide →How long the two branches maintain the gravitational interaction. Doubling this duration doubles the angle in this idealized constant-separation model.
Read this term in its guide →The separation between mass A on branch i and mass B on branch j. A smaller separation gives a larger gravitational interaction energy and therefore a larger accumulated phase.
Read this term in its guide →The quantum constant that converts energy multiplied by time (action) into a phase angle. Dividing the gravitational energy-time product by ħ leaves a dimensionless number of radians.
Read this term in its guide →This is an idealized geometry: mass A and mass B each have a left and right branch. Matching branches are separated by a gap d; crossed branches are farther apart by a horizontal offset Δx. Change the values to see which phase difference survives.
Illustrative symmetric geometry. The phase and concurrence calculations assume coherent, stationary branches and no other interactions. The article’s physical experiment has additional constraints.
The two masses each have L and R spatial branches. Their joint state has four branch pairs: LL, LR, RL, and RR. Each pair has a potentially different distance ; consequently, each acquires a potentially different phase φ_{ij}. The phase is attached to the amplitude of a possible joint path, not to a classical trajectory that has already been observed.
The interaction-energy magnitude is G /. The masses multiply, the distance appears in the denominator, and time accumulates the effect. If all four branch distances were equal, all four phases would be equal and this gravitational interaction would supply no relative phase between the paths.
This is the weak-field, Newtonian, fixed-separation approximation for coherent, localized branches. Motion during the interaction, other forces, environmental decoherence, and a more complete gravitational treatment require additional analysis. It is an idealized phase model, not by itself a prediction of a completed experiment.
The Newtonian potential energy for a branch pair is −G /. Quantum time evolution turns energy multiplied by time into phase: −t/ħ gives the positive angle written here under this sign convention. The energy-time product has units of joule-seconds, matching ħ, so the result is an angle rather than a force or a distance. Only relative phases enter observable interference; one branch phase alone is not an entanglement measure.
A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.
Equation 1 · Quantum Relativity
The gravitational interaction gives each pair of spatial branches a phase angle. Mass, interaction time, and inverse separation set its size.
None of this makes the experiment ambiguous, and this is the paper’s second and sharper result. The branch separations that enter the gravitational phase each mass’s superposition picks up are relational quantities — distances between mass A’s branches and mass B’s branches — and relational quantities do not change when you change which system you have designated the reference frame. Write = G t / ( ) for the phase a pair of branches i,j \{L,R\} accumulates over interaction time t at separation , with G Newton’s constant and the reduced Planck constant. Three of the four possible combinations of these branch-pair phases can always be removed…