Survey of two-time physics

The 11-dimensional extended superalgebra provides a model-independent signal for 12 dimensions in M-theory with (10,2) signature.

“It was noted that this structure provides a model-independent signal for 12 dimensions in Mtheory with (10, 2) signature [13], since the 32 supercharges may be viewed as a Weyl spinor in 12 dimensions and the 528 bosonic charges may be viewed as a 2-form plus a self-dual 6-form in 12 dimensions,”

• cites [13] sourcePartly supported

  The Discussion (pp. 15–16) describes an OSp(1/32) algebra relevant to M-theory in an SO(10,2) basis, with a 32-component spinor and two-form and six-form generators. It does not describe the 11-dimensional extended superalgebra as a model-independent signal for 12 dimensions, or state the count of 528 bosonic charges. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The simplest action with an Sp(2,R) gauge symmetry is L = Ẋ1 • X2 - 1/2 A_ij X_i • X_j.

“The simplest action with an Sp(2, R) gauge symmetry is [1] L = Ẋ1 • X 2 -1 2 A ij X i • X j (3 )”

• cites [1] sourceSupported

Using the procedure, the Schrödinger equation and other equations corresponding to first quantized field equations for the systems in figure 1 follow from (13).

“Using this procedure, it was shown in [11] that the Schrödinger equation and other equations that correspond to the first quantized field equations for the systems in figure 1 follow from (13).”

• cites [11] sourcePartly supported

  The Introduction (p. 2) describes non-relativistic particles among the systems obtained by gauge fixing, and Basic Formalism (p. 5) says their equations of motion reduce to the expected equations. Our copy does not specifically say that this procedure yields the Schrödinger equation or other first-quantized field equations from equation (13). (LLM, conf=0.78)

  Experimental — accuracy not yet measured

For a p-brane, the natural candidate for a local symmetry that replaces Sp(2,R) is the conformal group on the worldvolume SO(p+1,2).

“For a p-brane X M τ, σ 1 , . . ., σ p the natural candidate for a local symmetry that replaces Sp(2, R) is the conformal group on the worldvolume SO(p+1, 2).This was pointed out in [1].”

• cites [1] sourcePartly supported

  In §5 (Outlook), the paper proposes extending its worldline theory, viewed as conformal gravity, to p-brane worldsheets or worldvolumes. It does not identify SO(p+1,2) as the local symmetry that would replace Sp(2,R); elsewhere, it describes SO(d,2) as a global symmetry of the particle theory (§2, p. 3). (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The equation of motion of the gauge potential leads to the constraints, hence the target spacetime must have signature (d,2) as an outcome of the Sp(2,R) gauge symmetry.

“The equation of motion of the gauge potential leads to the constraints, hence the target spacetime must have signature (d, 2) as an outcome of the Sp(2, R) gauge symmetry.”

These constraints have non-trivial solutions only if the target spacetime has two timelike dimensions.

“These constraints have non-trivial solutions only if the target spacetime has two timelike dimensions.”

If the signature has three timelike directions, there are too many ghosts that cannot be removed by the Sp(2,R) gauge symmetry.

“If the signature has three timelike directions, then there are too many ghosts that cannot be removed by the Sp(2, R) gauge symmetry.”

For all other cases of gauge (super)symmetries, there is just enough gauge symmetry to remove one timelike and one spacelike direction from all SO(d,2) vectors and also remove half of the SO(d,2) spinor components to arrive at SO(d-1,1) spinors.

“For all other cases of gauge (super)symmetries mentioned above-or their generalizations to strings and branes-their content is as follows: there is just enough gauge symmetry to remove one timelike and one spacelike directions from all SO(d, 2) vectors and also remove half of the SO(d, 2) spinor components to arrive at SO(d -1, 1) spinors.”

In the special dimensions d=3,4,6, the action (15) reduces to the well-known action for the massless superparticle (16).

“When all the gauge fixing is done, in the 1T gauge that corresponds to the massless particle of figure 1, the action (15) reduces to the following form in d-dimensional superspace (x µ , θ a α ) with N supersymmetries a = 1, 2, . . .N”

The mass of the field as given by the Casimir of SO(4,2) is fully determined by the Casimir of SO(6).

“Thus, the mass of the field as given by the Casimir of SO(4, 2) is now fully determined by the Casimir of SO(6).”

There is a local Sp(2,R) symmetry.

“There is a local Sp(2, R) symmetry.”

There is local symmetry under the bosonic subgroup SO(d-k,2) × SO(k) × h_L ∈ G_L.

“There is local symmetry under the bosonic subgroup SO(d -k, 2) × SO(k) × h L ∈ G L .”

The local field interactions in d+2, as in (14), result in various forms of interaction in different 1T gauges.

“The local field interactions in d + 2, as in (14), result in various forms of interaction in different 1T gauges.”

There is a kappa-type local supersymmetry under part or all of the fermionic transformations in G_L.

“There is a kappa-type local supersymmetry under part or all of the fermionic transformations in G L .”

The 16 fermions provide Ramond-type vacua with 128 bosons and 128 fermions.

“First, the 16 fermions provide Ramond-type vacua with 128 bosons and 128 fermions.”

Because of the projectors, kappa supersymmetry can remove part but not all of the (τ) degrees of freedom.

“Because of the projectors kappa supersymmetry can remove part but not all of the (τ ) degrees of freedom.”

In each case one timelike dimension and one spacelike dimension is eliminated from each vector (X^M, P^M) by the gauge choice and the solution of the two constraints X^2 = X·P = 0.

“In each case one timelike dimension and one spacelike dimension is eliminated from each vector (X M , P M ) by the gauge choice and the solution of the two constraints X 2 = X • P = 0.”

The quantum toy 'M-model' is unitary.

“The quantum toy 'M-model' is unitary.”

The OSp(1|2) gauge singlet condition produces a new type of spinor field equation (and Lagrangian) which have a kappa-type symmetry that cuts down the number of spinor components by a factor of two.

“Interestingly, the OSp (1|2) gauge singlet condition produces a new type of spinor field equation (and Lagrangian) which have a kappa-type symmetry that cuts down the number of spinor components by a factor of two.”

The process also generates the Yang-Mills and gravitational equations with consistent interactions in d+2 dimensions.

“The process also generates the Yang-Mills and gravitational equations with consistent interactions in d + 2 dimensions.”

The action (or equations of motion) of the 1T dynamical systems emerge by gauge fixing the 2T Lagrangian (or its equations of motion).

“The action (or equations of motion) of those 1T dynamical systems emerge by gauge fixing this 2T Lagrangian (or its equations of motion).”

The physical spectrum of the toy M-model is 128 bosons and 128 fermions (states of 11D supergravity) in the presence of 2- and 5-brane charges with certain relations among them.

“The physical spectrum can then be easily determined: 128-bosons and 128-fermions (states of 11D supergravity) in the presence of 2-and 5-brane charges with certain relations among them.”

The fraction of (τ) degrees of freedom that can be removed ranges from 0 to 3/4 depending on the specific details of the couplings in (15).

“The fraction of that can be removed ranges from 0 to 3 4 depending on the specific details of the couplings in (15).”

The kinematic equations such as those in (13) and (11) are imposed as additional field equations which do not follow from the field theory Lagrangian.

“This is because the kinematic equations such as those in (13) and (11) are imposed as additional field equations which do not follow from the field theory Lagrangian.”

There is a global supersymmetry G_R under right multiplication of g → g g_R.

“There is a global supersymmetry G R under right multiplication of g → gg R .”

The 2T Lagrangian (14) reduces to the correct 1T field theory Lagrangian.

“Furthermore, the 2T Lagrangian ( 14) reduces to the correct 1T field theory Lagrangian.”

The gauge symmetry and constraints are such that d+2 is reduced to d dimensions for the entire brane for any p.

“The gauge symmetry and constraints are such that d +2 is reduced to d dimensions for the entire brane for any p.”

The SO(d,2) covariant quantization of the 2T worldline theory leads to a formulation of 2T interacting classical field theory for spinning fields.

“The SO(d, 2) covariant quantization of the 2T worldline theory leads to a formulation of 2T interacting classical field theory for spinning fields.”

The model in this gauge realizes a BPS representation (a short representation) of OSp(1|64), with 32 vanishing supercharges.

“The model in this gauge realizes a BPS representation (a short representation) of OSp(1|64), with 32 vanishing supercharges.”

The structure of the Lagrangian (15) guarantees local and global symmetries.

“The structure of the Lagrangian above guarantees the following local and global symmetries.”

• AssumptionThe authors state this as an assumption: The bosonic subgroup of G_L contains at least SO(d-k,2) × SO(k) for some k=0,...,d-2.

The remaining gauge choice and constraint P^2 = 0 is associated with the τ-reparametrization gauge symmetry.

“The remaining gauge choice and constraint P 2 = 0 is associated with the τ -reparametrization gauge symmetry, and it may be convenient to gauge fix it to complete the relation between τ and the 1T-dimension embedded in (d, 2) target spacetime X M (τ ).”

The kinematic equations correspond to the worldline theory constraints Q11=Q12=0.

“They correspond to the worldline theory constraints Q 11 = Q 12 = 0.”

The first-class constraints associated with the gauge symmetry ensure unitarity (no ghosts).

“The first-class constraints associated with the gauge symmetry ensure unitarity (no ghosts).”

Different embeddings of d dimensions x^μ inside d+2 dimensions X^M produce different dynamics (different Hamiltonians) in the chosen d dimensions.

“The different embeddings of d dimensions x µ inside d + 2 dimensions X M produce different dynamics (different Hamiltonians) in the chosen d dimensions.”

All 1T particle interactions with Yang-Mills, gravitational and other fields are included in the d+2 reformulation.

“All 1T particle interactions with Yang-Mills, gravitational and other fields are included in the d + 2 reformulation.”

Different 1T evolutions (different Hamiltonians) emerge from the same 2T-theory when gauge fixing is done with different embeddings of d dimensions inside d+2 dimensions.

“Different 1T evolutions (i.e.different Hamiltonians) emerge from the same 2T-theory when gauge fixing is done with different embeddings of d dimensions inside d + 2 dimensions.”

The gauge symmetry demands that neither fewer nor more timelike dimensions are permitted in the description of a single point particle.

“Thus, the gauge symmetry demands that neither fewer nor more timelike dimensions are permitted in the description of a single point particle (the many-body theory described by the corresponding field theory also must have exactly two timelike dimensions).”

Further generalizations of the gauge symmetries also remove ghost components from SO(d,2) tensors.

“Further generalizations of the gauge symmetries (see, e.g., the toy M-model in section 7.3) also remove ghost components from SO(d, 2) tensors (such as antisymmetric tensors associated with branes).”

Two-time physics originally developed from hints about two-timelike dimensions in the mathematical structure of M-theory.

“Although two-time physics [1][2][3][4][5][6][7][8][9][10][11] is currently best understood in simple everyday physics, it originally developed from hints about two-timelike dimensions in the mathematical structure of M-theory.”

For spinless particles the gauge symmetry is Sp(2,R) acting on position-momentum as a doublet.

“For spinless particles the gauge symmetry is the Sp(2, R) that acts on position-momentum (X M , P M ) as a doublet.”

Taking into account various dualities, 13 dimensions with (11,2) signature appeared more appealing because S-theory can unify type-IIA and type-IIB supersymmetric systems in 10 dimensions.

“Taking into account various dualities, 13 dimensions with (11,2) signature appeared more appealing because in that framework S-theory can unify type-IIA and type-IIB supersymmetric systems in 10 dimensions.”

The standard model of particle physics can be regarded as a gauge-fixed form of a 2T-theory in 4+2 dimensions.

“In particular, the standard model of particle physics can be regarded as a gauge-fixed form of a 2T-theory in 4 + 2 dimensions.”

A given 2T-physics system defined by an action in (d,2) dimensions displays a unification of a class of one-time dynamical systems in (d-1,1) dimensions.

“A given 2T-physics system defined by an action in (d, 2) dimensions X M , displays a unification of a class of one-time dynamical systems in (d -1, 1) dimensions x µ .”

In the 2T setting, the distinguished 1T which we call time is a gauge-dependent concept.

“Thus, in the 2T setting, the distinguished 1T which we call 'time' is a gaugedependent concept.”

2T-physics describes various 1T dynamical systems as different d-dimensional holographic views of the same 2T system in d+2 dimensions.

“2T-physics describes various 1T dynamical systems as different d-dimensional 'holographic' views of the same 2T system in d + 2 dimensions.”

Two-time physics arises from a gauge principle.

“Thus, two-time physics arises from a gauge principle.”

These facts provide evidence for a new type of higher-dimensional unification.

“These facts already provide evidence for a new type of higher-dimensional unification.”

The holography is due to gauge symmetries that tend to reduce the number of effective dimensions.

“The 'holography' is due to gauge symmetries that tend to reduce the number of effective dimensions.”

With spacetime supersymmetry the gauge symmetry is enlarged with a new version of kappa supersymmetry in d+2 dimensions.

“With spacetime supersymmetry the gauge symmetry is enlarged with a new version of kappa supersymmetry in d + 2 dimensions.”

A given 2T-action modulo gauge symmetries is equivalent to many 1T actions with different 1T dynamics.

“That is, a given 2T-action modulo gauge symmetries, is equivalent to many 1T actions with different 1T dynamics.”

The gauge symmetry relates the different d-dimensional 1T actions in the same class to each other in a way reminiscent of duality.

“The gauge symmetry relates the different d-dimensional 1T actions (in the same class) to each other in a way reminiscent of 'duality'.”

The extra one-spacelike and one-timelike dimensions (1,1) are interchangeable with gauge degrees of freedom.

“The extra one-spacelike and one-timelike dimensions (1, 1) are interchangeable with gauge degrees of freedom.”

For spinless particles the Sp(2,R) constraints are X^2 = P^2 = X·P = 0.

“For example for spinless particles the Sp(2, R) constraints are X 2 = P 2 = X • P = 0.”

For spinning particles the gauge symmetry is OSp(n|2) acting on super-phase space.

“For spinning particles the gauge symmetry is OSp(n|2) which acts on super-phase space (ψ M a , X M , P M ) in its fundamental representation (where a = 1, . . ., n).”

In the first-order formulation of any theory, the first term in the action has a global Sp(2,R) symmetry.

“This shows that the first term in the action has a global Sp(2, R) symmetry.”

In d=3,4,6, the massless superparticle action has the larger global supergroup symmetry G = OSp(N|4), SU(2,2|N), OSp(8*|N), respectively, that includes additional N hidden superconformal symmetries.

“In the special dimensions d = 3, 4, 6 it does indeed have the larger global supergroup symmetry G = OSp (N |4), SU (2, 2|N ), OSp (8 * |N), respectively, that includes the additional N hidden superconformal symmetries [5,29].”

• cites [5] Vongehr S, 1999Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [29] sourceNot checked — no full text of the source

The 2T point of view of ordinary 1T-physics has revealed a new subtle dimensional unification in d+2 dimensions.

“The 2T point of view of ordinary 1T-physics has revealed a new subtle dimensional unification in d + 2 dimensions.”

In this sense all the d+2 dimensions exist.

“In this sense all the d +2 dimensions exist.”

The standard model may be regarded as a 2T field theory in six dimensions which is reduced to four dimensions in a particular gauge.

“Then one sees that the standard model may be regarded as a 2T field theory in six dimensions which is reduced to four dimensions in a particular gauge.”

When the 2T-theory is further fixed to some kappa gauge and some 1T Sp(2,R) gauge, the d+2 supersymmetry reduces to the correct spacetime supersymmetry in d dimensions.

“When the 2T-theory is further fixed to some kappa gauge and some 1T Sp(2, R) gauge, the d + 2 supersymmetry (that has twice as many fermionic parameters) reduces to the correct spacetime supersymmetry in d dimensions.”

Such embeddings of time as a curve inside d+2 dimensions can be done in an infinite number of ways, thereby producing a class of related 1T dynamical systems from the same 2T-theory.

“Such embeddings of 'time' as a curve inside d + 2 dimensions can be done in an infinite number of ways, thereby producing a class of related 1T dynamical systems from the same 2T-theory.”

Each 1T-system can be considered as a d-dimensional 'holographic' image of the d+2 theory.

“Each 1T-system can be considered as a d-dimensional 'holographic' image of the d + 2 theory.”

The unique closure of the 64 supercharges that contains SO(11,2) is OSp(1|64).

“The unique closure of the 64 supercharges that contains SO (11,2) is OSp(1|64).”

The global symmetry SO(d,2) is observable in every holographic image.

“The global symmetry SO(d, 2) is observable in every holographic image.”

For 'M-theory' 2T-physics makes a general prediction independent of any details [6]: it requires that the explicit and hidden global symmetries of 'M-theory' must be described by the non-Abelian OSp(1|64) superalgebra.

“For 'M-theory' 2T-physics makes a general prediction independent of any details [6].It requires that the explicit and hidden global symmetries of 'M-theory' must be described by the non-Abelian OSp(1|64) superalgebra.”

• cites [6] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Various 1T actions in d dimensions are different holographic descriptions of the same 2T system in d+2 dimensions.

“One may say that various 1T actions in d dimensions are different 'holographic' descriptions of the same 2T system in d + 2 dimensions.”

The usual 11D superalgebra (1) with commuting momenta and commuting 2- and 5-brane charges is part of the non-Abelian OSp(1|64).

“As shown in [6], the usual 11D superalgebra (1) with commuting momenta and commuting 2-and 5-brane charges is part of the non-Abelian OSp(1|64).”

• cites [6] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The same 2T field theory in d+2 dimensions unifies many types of 1T field theories in d dimensions, such as those corresponding to figure 1.

“Therefore, the same 2T field theory in d + 2 dimensions unifies many types of 1T field theories in d dimensions, such as those that correspond to figure 1.”

The gauge-fixed action describes the entire AdS5×S5 supersymmetric Kaluza-Klein tower [8] previously derived in the context of compactified 10D type IIB supergravity.

“The gauge-fixed action describes the entire AdS 5 ×S 5 supersymmetric Kaluza-Klein tower [8] which was previously derived in the context of compactified 10D type IIB supergravity.”

• cites [8] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The resulting transformation becomes precisely the needed global spacetime supersymmetry in d+2 dimensions.

“The resulting transformation becomes precisely the needed global spacetime supersymmetry in d + 2 dimensions.”

It was shown that 1T tensionless and rigid branes on flat or AdS × S backgrounds could be obtained from the 2T p-brane by choosing certain 1T gauges for SO(p+1,2).

“It was shown that 1T tensionless and rigid branes on flat or AdS × S backgrounds could be obtained from the 2T p-brane by choosing certain 1T gauges for SO(p + 1, 2).”

• cites [4] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The uncontracted versions of the relevant AdS × S supersymmetries contain additional brane charges such that the closure is uniquely OSp(1|32) contained in OSp(1|64).

“The uncontracted versions of the relevant AdS × S supersymmetries contain additional brane charges such that the closure is uniquely OSp (1|32) contained in OSp(1|64) [6,31].”

• cites [6] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [31] Ooguri H, 1991, Nucl. Phys. sourceNot checked — no full text of the source

OSp(1|64) correctly contains the various supersymmetries of several corners of S-theory, F-theory and M-theory, including 13D, 12D, 11D, 10D type-IIA, 10D type-IIB, heterotic, type-I, and AdS × S supersymmetries.

“Further support for this point of view can be found in the fact that OSp(1|64) correctly contains the various supersymmetries of several corners of S-theory, F-theory and M-theory, including 13D, 12D, 11D, 10D type-IIA, 10D type-IIB, heterotic, type-I, and AdS × S supersymmetries.This has been demonstrated in [6].”

• cites [6] Bars I, 1998Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.