Born reciprocity on The Broken-Born program

The canonical commutator and the uncertainty principle

Mon, 01 Jan 0001 00:00:00 +0000

The canonical commutator [x, p] = iℏ is the rule of quantum mechanics that position and momentum do not commute: measuring one and then the other is not the same as the reverse. Its best-known consequence is the uncertainty principle: no state has sharp values of both, and the product of the two spreads cannot fall below a bound set by Planck’s constant ℏ. The rule is unchanged when position and momentum are exchanged by the quarter-turn Born reciprocity, so on its own it treats the two variables alike. The program leans on it twice. It argues that because nothing can be sharp in both variables, any world with local physics has to favor one of them; and it reads the fixed value of ℏ as the one rigid structure of phase space.

The symplectic form and Sp(2d, ℝ)

Mon, 01 Jan 0001 00:00:00 +0000

The symplectic form ω is the antisymmetric pairing of positions with momenta that underlies Hamiltonian mechanics and the quantum commutator; in the plane of one position and one momentum it measures signed area. The symplectic group Sp(2d, ℝ) is the set of linear maps of phase space (d positions, d momenta) that preserve it. The group contains rotations of phase space — the quarter-turn Born reciprocity is one — together with stretches that lengthen one direction and shorten another, called boosts and squeezes. In the program ω is the one structure held fixed: its corrected postulate reads the rigidity of ω as the constancy of Planck’s constant, and every other piece of phase-space geometry is allowed to vary.

The Fourier transform as a phase-space rotation

Mon, 01 Jan 0001 00:00:00 +0000

The Fourier transform is the exact, reversible map that turns a quantum state’s description in position into its description in momentum — the same operation that turns a sound’s waveform into its spectrum. Neither description approximates the other; each contains the whole state. Applied once, the transform carries out Born reciprocity, the quarter-turn ρ: (x, p) → (p, −x) of phase space; applied twice, it reflects x → −x; applied four times, it returns every state unchanged. The program takes the analogy literally: position and momentum are two descriptions of one state, and its toy models use the transform to relate a vacuum in which a matter coupling is local in position to its mirror image, in which the coupling is local in momentum.

Born reciprocity ρ: (x, p) → (p, −x)

Mon, 01 Jan 0001 00:00:00 +0000

Born reciprocity is the map ρ: (x, p) → (p, −x), which exchanges position and momentum, with one sign change, as a quarter-turn of phase space. The rules of quantum kinematics are unchanged by it, and the Fourier transform carries it out on wavefunctions. Max Born proposed in 1938 and 1949 that a fundamental theory should respect it, and hoped for a physics with a fundamental mass and a fundamental length on equal footing. Observed dynamics does not respect it — interactions are local in position — and the program’s starting assumption, its hard core, is that ρ is an exact symmetry of the laws that the vacuum breaks.

Unitary equivalence versus Noether symmetry

Mon, 01 Jan 0001 00:00:00 +0000

Unitary equivalence and Noether symmetry are the two different ways a map like Born reciprocity can relate to a physical theory. It can be a change of description, a unitary equivalence — the same physics written in other variables, as a Fourier transform rewrites a waveform as a spectrum — or it can be a symmetry of the dynamics: an operation that leaves a specific action unchanged, and therefore constrains what the theory can do (a Noether symmetry). In ordinary quantum mechanics ρ is the first kind, because the Stone–von Neumann theorem makes all the standard ways of representing the position–momentum commutation rules equivalent. The program needs the second kind, since only a symmetry of an action can be spontaneously broken. Its referees made exhibiting such an action the first acceptance criterion for anything it proposes, and #11 records the criterion as met, in the referees’ wording, by the toy models.

Lagrangian subspaces and polarizations

Mon, 01 Jan 0001 00:00:00 +0000

A Lagrangian subspace of phase space is a set of directions, half the dimension of phase space, on which the symplectic form vanishes: no two of its directions form a position–momentum pair. Choosing one, a polarization, is choosing which coordinates count as “position”. The ordinary position axis is one Lagrangian subspace, the momentum axis is another, and a continuous family of tilted ones lies in between. A quantum state can be written as a wavefunction over any of them, and the different descriptions are related by unitary transforms such as the Fourier transform. The program’s postulates say that the vacuum carries one such choice, and that ours is tilted toward position.

The Wigner function and Weyl symbols

Mon, 01 Jan 0001 00:00:00 +0000

The Wigner function is a way of drawing a quantum state as a distribution over phase space, with position on one axis and momentum on the other. Adding it up over all momenta gives the probability of each position, and adding it up over all positions gives the probability of each momentum, so one picture holds both descriptions of the state; unlike a true probability, it can be negative. Weyl symbols extend the same idea from states to operators, turning each operator into a function on phase space. The program’s toy-model note uses the Wigner function to build a matter coupling that reads one axis of phase space as “position”, and to check that a coupling blind to that choice treats both axes alike.

Momentum fibers and the cotangent bundle

Mon, 01 Jan 0001 00:00:00 +0000

A momentum fiber is the set of all momenta a particle can have at one point of space-time. Phase space, the space of positions and momenta taken together, is made of these fibers, one attached to each point, much as a spectrogram attaches a whole frequency axis to each instant of a recording; geometers call the structure the cotangent bundle of space-time. In general relativity every fiber is flat. Momenta at a point are measured with the inverse of the space-time metric, which is the same everywhere along the fiber, so the curvature of space-time changes how neighbouring fibers are joined but not the geometry inside any one of them. The program’s checkpoint lists this flatness among the places where the laws treat position and momentum differently and calls it “an architectural assumption, not a measurement” (Born’s question). In the program’s own geometry the compatibility theorem fixes the momentum metric as the inverse metric on the vacuum, any independent momentum-space geometry is assigned to the momentum graviton X, and the dark-matter gate acts through the width of conjugate-matter quanta along the fibers. How X propagates along the fibers, which would fix the shape of that gate, has not been computed.

Curved momentum space

Mon, 01 Jan 0001 00:00:00 +0000

Curved momentum space is the idea, developed in a line of quantum-gravity research, that the space of momenta can carry a geometry of its own instead of being flat. Snyder used it in 1947 to build a quantized space-time; in quantum gravity with two space dimensions, which can be solved exactly, the momenta of a particle live on a curved space; and “relative locality” made the geometry of momentum space a principle. General relativity curves space-time but leaves momentum space flat, which the program counts as an assumption rather than a measurement; its own version of momentum-space geometry is the momentum graviton X. One known difficulty of the idea, the soccer-ball problem, was listed as open in the lab notebook and never resolved.

Broken Born duality: the Freidel–Leigh–Minic precedent

Mon, 01 Jan 0001 00:00:00 +0000

Broken Born duality is Freidel, Leigh and Minic’s phrase for what happens when all physical probes agree on which phase-space directions are space-time: the agreement is a preferred polarization, a preferred choice of which directions count as position, and it breaks the position–momentum symmetry they call Born duality. In their work on Born geometry and metastring theory they made the neutral metric η dynamical and described space-time as a choice of Lagrangian subspace, but they wrote no potential for η and treated the breaking as kinematical, not as spontaneous symmetry breaking. The program’s central idea therefore has a literature precedent. Since version 2.2 the program’s checkpoint credits them, and claims as its own only the reading of the breaking as spontaneous, with η as the order parameter.

The Born length λ

Mon, 01 Jan 0001 00:00:00 +0000

The Born length λ is a reference length that, together with Planck’s constant ℏ, puts position and momentum on a common scale: positions are counted in units of λ and momenta in units of ε = ℏ/λ. A length and a momentum have different units, so a geometry that combines them, or a map that exchanges them such as Born reciprocity, has a meaning only once some such pair of scales is chosen. The name comes from #13, the program’s paper on conjugate special relativity, in which λ fixes the slope of the momentum-space light cone and with it the maximum force, c̃ = ℏc/λ². #13 sets λ equal to the Planck length, in one sentence that carries no tag, and the maximum force becomes c⁴/G. The program’s lab notebook writes its phase-space structures in units in which the same pair of scales is 1, without naming λ or giving it a value. The identification with the Planck length is alive but unreviewed: no review round has examined it, no script checks it, and the checkpoint #11 does not mention the Born length.