Spontaneous breaking of reciprocity on The Broken-Born program

Spontaneous symmetry breaking

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

Spontaneous symmetry breaking is the situation in which the equations of a system are unchanged by some transformation but its lowest-energy state is not. A cooled magnet is the standard example: the laws governing its atoms treat every direction alike, yet the magnet ends up magnetized along one direction, and any other direction would have been an equally good ground state. The symmetry is not lost. It now relates the possible ground states to one another, and the system occupies one of them. The Broken-Born program uses this pattern for its hard core: it proposes that the exchange of position and momentum, an exact symmetry of quantum kinematics, is broken by the vacuum rather than by the laws. This page is the textbook background and the program’s operational version of it; the program’s own claims follow on the next pages of this section.

Topological defects and the vacuum manifold

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

A topological defect is a wall, line or point where a field that has broken a symmetry is held out of its lowest-energy states, because the vacuum values it takes around that place cannot be joined smoothly across it: a domain wall between two regions that settled into different vacua, a string around which the field’s vacuum value runs once round a loop, or a point-like monopole around which it covers a sphere. The set of lowest-energy states is the vacuum manifold. When a symmetry group G is spontaneously broken, the vacuum manifold is G/G₀, the group taken modulo the subgroup G₀ that leaves one vacuum in place. Its dimension counts the massless Goldstone modes of a relativistic theory, and its shape, measured by its homotopy groups, decides which defects a cooling universe can leave and why, once formed, they cannot decay on their own. In the Broken-Born program the vacuum manifold of the polarization, the Lagrangian Grassmannian Λ(d) of all possible choices of which phase-space directions count as position, contains a loop that cannot be shrunk, and the program takes the periodicity of its dark-energy field τ from that loop. The retired dark-matter candidate was a defect of the monopole kind in a hidden gauge sector; it died on the count of how many of them a transition makes, not on their stability. One open question, without a candidate, asks whether topology could keep a single object localized in momentum.

Goldstone modes and the Higgs mechanism

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

A Goldstone mode is a massless excitation that appears when a continuous symmetry of the laws is broken by the ground state. Moving along the family of equally good ground states costs no energy, so a slow, long-wavelength wave of that motion costs arbitrarily little, and the particle it describes has no mass. If a small explicit term tilts the family, the mode acquires a small mass and is called a pseudo-Goldstone mode. When the broken symmetry is a gauge symmetry, the Higgs mechanism applies instead: the gauge field absorbs the would-be Goldstone modes as extra polarizations and becomes massive. The program uses all three ideas. In its coset toy model the polarization angle is a Goldstone direction, which an added potential turns into a pseudo-Goldstone mode, and its momentum graviton X gets its mass from a gravitational version of the Higgs mechanism.

Stückelberg fields

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

Stückelberg fields are extra fields written into a theory of a massive gauge field so that the gauge symmetry its mass term breaks holds again. For a vector field A_μ the mass term m²A_μA^μ changes when A_μ is shifted by the gradient of a function; written in terms of the combination A_μ + ∂_μπ, where π is a new scalar field that shifts back by the same function, it does not. Setting π to zero, the choice called the unitary gauge, returns the original theory, so the construction changes the description and not the physics. The new fields are the would-be Goldstone modes of the Higgs mechanism written out as fields: the extra polarizations that a gauge field absorbs, or “eats”, when it becomes massive. A massive spin-2 field needs a vector of them, and the vector’s gradient part carries the polarization of helicity zero, the one with no spin along the direction of motion. In the Broken-Born program the momentum graviton X becomes massive by absorbing the would-be Goldstone modes of its broken dual diffeomorphisms (the local shifts of X), and checkpoint #11 calls these modes “the Stückelberg sector”. The dark-energy field τ lies outside that sector at leading order; how strongly the two mix has not been computed and is open problem 6 (OP6).

Inverse-Higgs constraints

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

An inverse-Higgs constraint is a relation that fixes one would-be Goldstone field as a derivative of another, so that the first drops out of the theory as an independent field. Such relations arise when the symmetries broken by the ground state act on space and time themselves, as translations, rotations and boosts do. A flat membrane is the standard example. It breaks the translation perpendicular to itself and the rotations that would tilt it, but its local tilt is set by the slope of its local displacement, so the displacement is its only massless field. In a relativistic theory each broken generator of an internal symmetry brings its own massless Goldstone mode; when spacetime symmetries break, there can be fewer modes than broken generators. The program’s record mentions the idea in a single document: calc note #5 expects such constraints to remove most of the nine directions, besides that of the dark-energy field τ, in which the vacuum’s polarization can vary. No document works this out, and the question is open.

The Mermin–Wagner–Coleman theorem

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

The Mermin–Wagner–Coleman theorem says that in two dimensions a continuous symmetry of the compact kind, such as rotation through an angle, cannot be spontaneously broken. The two dimensions can be two of space at nonzero temperature, or one of space and one of time in a quantum field theory. Slow, long-wavelength wobbles of the would-be ordered direction cost so little energy that they grow without limit as the system gets larger, and they wash the order out. Discrete symmetries escape the theorem, as the two-dimensional Ising magnet shows, and so do some non-compact symmetries. With only a few degrees of freedom, as in ordinary quantum mechanics (0+1 dimensions), not even a discrete symmetry breaks, because tunnelling mixes the would-be ground states. These limits determined how the program’s toy models were built: they start in 1+1 dimensions, and their compact polarization angle can order only after a potential reduces its symmetry to a discrete one.

Mandatory breaking: the uncertainty argument

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

Mandatory breaking is the program’s argument that Born reciprocity, the exchange of position and momentum, cannot remain unbroken in any world that contains local physics. The uncertainty principle rules out anything being sharp in both position and momentum at once, much as a waveform and its spectrum cannot both be narrow. A world whose interactions are local must therefore be local in one variable at the cost of the other; in #11’s phrase, “the magnet must point”. The argument is why the program treats the breaking as forced rather than optional, while leaving open which of the two variables ends up as position. Since v2.2 it carries qualifications from the toy models: the polarization is a continuous angle, a definite polarization needs an added potential, and the breaking has physical effect only once matter couples to the sign of the order parameter.

The order parameter: J, η and K

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

An order parameter is the quantity that tells a broken-symmetry state from a symmetric one, as a magnet’s magnetization does: it is zero while the symmetry is intact, and once the symmetry is broken it takes a definite value that the symmetry would change. In the Broken-Born program that role belongs to the neutral metric η, the structure on phase space that marks which directions count as position and which as momentum. Born reciprocity reverses the sign of η, so a vacuum that carries a definite η is not reciprocity-symmetric, and its mirror image carries −η. The program has written this order parameter in three notations over its history, using J, η and K, and the documents do not agree on which of J and K to call “the real structure”. An early description of the breaking as a discrete pattern has been replaced by the Goldstone-angle picture.

The polarization as a Goldstone angle

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

The polarization as a Goldstone angle is the toy models’ finding that the vacuum’s polarization, which fixes the direction in the position–momentum plane that counts as position, is an angle that can turn smoothly rather than a switch with two settings. Born reciprocity is a quarter turn of that plane, the operation the Fourier transform performs when it turns a waveform into its spectrum, and in the toy models it is one member of a continuous family of symmetries. Turning the angle costs no energy, which makes it a Goldstone direction: the position-sharp and momentum-sharp vacua lie on one continuous family. This finding, recorded in #11 v2.2, is why a definite polarization needs an added potential on the angle. In the coset model the angle is the same circle that the program identifies with its dark-energy field τ, which shows that the structures fit together but does not test the cosmology. It is evidence from two toy models, not a theorem.

Open: the origin of the ρ-preserving potential

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

This open problem asks where the potential that fixes the polarization angle comes from. In the program’s coset toy model the vacuum’s polarization, the choice of which direction in the position–momentum plane counts as position, is an angle that can take any value at no cost in energy. In wave terms the angle runs continuously from describing a state by its waveform to describing it by its spectrum, through the intermediate descriptions that fractional Fourier transforms give. A definite polarization appears only after a potential is added that keeps Born reciprocity as a symmetry but pins the angle to one of two opposite settings. The program gives its dark-energy field τ a potential of the same kind. Such a potential brings in a fixed, unordered pair of axes, which is more fixed structure than the program’s postulate P2′ (“only ω is rigid”) provides, and nothing in the program derives it. Its first proposed origin, explicit breaking by the rigid background of the original postulate P2, did not survive the replacement of P2.

The twisted element ρ∘(K → −K)

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

The twisted element is a combined operation in the coset toy model of broken reciprocity: apply Born reciprocity, the exchange of position and momentum, and at the same time reverse the sign of the real structure K, the field whose value marks which of two lines in the position–momentum plane counts as position. Reciprocity alone carries every vacuum of the model to a different one, which is the sense in which the vacuum breaks it. The combination carries the reference vacuum back to itself, and it acts on the model’s metric H exactly as reciprocity does. So as long as nothing in the theory depends on the sign of K, the vacuum keeps a symmetry that cannot be told apart from reciprocity, and calling reciprocity broken is a choice of labels. The statement gains physical content once matter couples to the sign of K, which in the program is the role of postulate P4. The round-4 referee found the point, the adjudication upheld it, and checkpoint #11 now states it in its postulate P2′.

The section condition: what η fixes

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

A section condition is a rule, taken from double field theory, that makes every field depend on only half of a doubled set of coordinates, so that physics lives on one “section” of the doubled space. In the program’s coset toy model the neutral metric η, the order parameter of broken reciprocity, singles out exactly two lines in the position–momentum plane: its null lines, which are the two eigenlines of the real structure K. A section condition built from η forces each matter field to depend on the coordinate along one of those lines only. It cannot say which of the two is position, because the same fields pass whether the condition is built from η or from −η. So the condition fixes the pair of candidate directions, and matter’s choice within the pair has to come from somewhere else. The clean two-way choice holds only when phase space is a plane (one position and one momentum coordinate); with more dimensions the literature finds a continuous family of sections.

Locality from an (H, K)-built coupling

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

The coupling I_K is a particular interaction for matter in the coset toy model of broken reciprocity. It is built from the model’s two fields, the metric H and the real structure K, and it is local in position in one vacuum and local in momentum in the mirror vacuum. Evaluated on a single quantum state, it reduces in the first vacuum to an integral over the state’s position profile, its waveform, and in the second to the same integral over its momentum profile, the spectrum of that waveform, so one state can give different values in the two vacua. Because the interaction depends on the sign of K, it also gives the breaking of reciprocity a physical effect. The arrangement parallels the Standard Model’s fermion masses, which follow from the Higgs field’s vacuum value once the fermions’ Yukawa couplings to it are given. Nothing in the toy requires matter to couple this way, which is why this is only the weak form of the locality criterion S5.

Open: forced locality and the P4 lock

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

This open problem asks whether anything forces matter to be local in position, rather than leaving that as an input. In the coset toy model the vacuum’s order parameter fixes a pair of candidate directions in the position–momentum plane and marks one of them as position, and a suitable coupling makes matter local in whichever direction is marked; but nothing in the model requires matter to use that coupling. Without such a link, the line a matter field lives on and the sign of the order parameter are two independent choices. In the toy the link is supplied by postulate P4, which places ordinary matter in the position-polarized representation and its mirror image in the momentum-polarized one. The toy therefore explains locality in position only as part of a package that includes a momentum-local mirror sector, the program’s dark-matter candidate, and does not derive it. The toy-model worklog notes that the question is also open in the literature.