Information Preservation in Quantum Gravity

[Lawrence B. Crowell]

This is a continuation of the last post

With this definition of the gauge potential $A_\mu$and the vacuum potential $\Omega_\mu$ we examine gauge transformations by the infinitesimal change in the sectional basis $\delta {\vec e}~=~{\vec\epsilon}\times{\vec e}$, for $\vec\epsilon$ an infinitesimal vector displacement of vectors in $S^3$ of $SU(2)$. The vector symbol is for the vector part on $S^3$ For a general ${A^\prime}^k_\mu$ determined from $A^k_\mu~=~-B^k_\mu$ by a gauge covariant gauge displacement vector $\chi^k_\mu$ give the transformation

$$e_i\delta A^i_\mu~=~e_i\partial_\mu\epsilon^i,~e_i\delta{A^\prime}^i_\mu~=~D_\mu e^i$$

$$\delta B^i_\mu~=~-\delta B^i_\mu,~\delta\chi^i_\mu~=~-(\epsilon\times\chi_\mu)^i$$

which illustrates that the covariant gauge vector $\chi^i_\mu$ and the gauge potential are gauge independent. This covariant gauge vector is a chromodynamic-like field similar to a gluon, or the weak gauge bosons. These potentials determine the physical fields

$$F^i_{\mu\nu}~=~\partial_\nu A^i_\mu~-~\partial_\mu A^i_\nu,~e_iG^i_{\mu\nu}~=~-{\vec e}\cdot\partial_\nu{\vec e}~-~{\vec e}\cdot\partial_\mu{\vec e}~=~e_i(\partial_\nu B^i_\mu~-~\partial_\mu B^i_\nu)$$

$${F^\prime}^i_{\mu\nu}~=~F^i_{\mu\nu}~+~G^i_{\mu\nu}$$

This then indicate that the vector portions of gauge connections $A^i_\mu$ transform identically with the $SU(2)$ portion of the connection $B^i_\mu$.

The basis vector $\vec e$ defines a phase $U~=~e^{\phi\sigma\cdot\vec e}$, for $\sigma$ a vectors of Pauli matrices. A momentum for a particle in the $SU(4)$ space is $p_\mu~=~U\partial_\mu U$, which when defined in a basis with $cos(\phi)~=~0$ gives the Lagrangian

$${\cal L}~=~\frac{1}{2}p_\mu p^\mu~+~\frac{1}{16}Tr\big([p_\mu,~p_\nu]\big)^2$$
$$=~-\frac{1}{2}G^\prime_{\mu\nu}G^{\mu\nu}~-~B_\mu B^\mu,$$

for $G^\prime_{\mu\nu}~=~G_{\mu\nu}~+~B_\mu\wedge B_\nu$. This Skymre Lagrangian then leads to the dynamical equation

$${\vec e}\times\nabla^2 {\vec e}~-~(\partial_\mu G^{\mu\nu})\partial_\nu{\vec e}~=~~0$$

This differential equation is $\partial^\mu j^i_\mu~=~0$ for the term

$$j^i_\mu~=~({\vec e}\times \partial_\mu {\vec e})^i~-~G_{\mu\nu}e^i,$$

where the continuity equation defines a conservation of charge across flux tubes. With $({\vec e}\times\partial_\mu{\vec e})^i~=~\epsilon^{ijk}B^k_\mu e^j$ the current components are clearly of the form

$$j^1_\mu~=~B^1_\mu~+~{G_\mu}^\nu}(B^2_\nu e^3~-~B^3_\nu e^2),~j^2_\mu~=~B^2_\mu~+~{G_\mu}^\nu}(B^3_\nu e^1~-~B^1_\nu e^3),~j^3_\mu~=~B^3_\mu~+~{G_\mu}^\nu}(B^1_\nu e^2~-~B^2_\nu e^1)$$

The third of these equations may be removed by the gauge condition $D_\mu e^3~=~0$, which reduces the problem to two dimensions. The two continuity equations found from $\partial_\mu{\vec j}^\mu~=~0$ are then

$$\partial^\mu B^1_\mu~+~B^{3\mu}B^2_\mu~+~\partial^\mu{G_\mu}^\nu B^2_\nu~=~0$$
$$\partial^\mu B^2_\mu~-~B^{3\mu}B^1_\mu~-~\partial^\mu{G_\mu}^\nu B^1_\nu~=~0$$

This equation may be replaced by the simple substitution $\omega^{\pm}_\mu~=~(B^1_\mu~\pm~iB^2_\mu)/\sqrt{2}$, which defines creation and annihilation operators for the $SU(2)$ field. The differential equation may be written as

$$\omega^{\pm}\partial^\mu\omega^{\pm}_\mu~\mp~\omega^{\pm}B^{3\mu}\omega^\mp_\mu~=~\pm\omega^{\pm}\partial^\mu{G_\mu}^\nu \omega^{\mp}_\nu$$

which when integrated over $S^3$ the right hand side defines a charge according to a Chern Simons index

$$kQ~=~\pm\int d^3x\omega^{\pm}\partial^\mu{G_\mu}^\nu \omega^{\mp}_\nu~=~\pm\int d^3x\epsilon_{ijk}\epsilon^{\mu\nu\sigma}\omega^i_\mu\omega^j_\nu\omega^k_\sigma$$

which is also an index for the knot equation. This index is then a topological invariant for the knot topology of the "gluon-like" threads.

This gauge theory with gluon threads on the manifold for conformal gravity connects spin-gravity terms in a multiply connected manner, which might be interpreted as wormholes. The index derived defines a topological invariant for this multiple connectivity in conformal gravity embedded in $SU(4)\times SU(2)$. This structure will next be used to derive a form of noncommutative quantum gravity embedded in the heterotic group $E_6$, which in turn is embedded in $E_8$.

 

[Lawrence B. Crowell]

The gluon-like threads on the conformal space is define alternative assignments for how points are connected to each other. One connection between points in conformal gravity might be a path on the $spin(4,2)$ manifold of conformal gravity, while another connection is along the gluon-like flux tube. This may be considered from the perspective of a thickened $S^2~\subset~S^3$, where the knot equation is defined. The $S^2$ is a bloch sphere defined by the gauge condition, where every point on this sphere is associated with a copy of $spin(4,2)$. The Cartesian product of conformal gravity with the isposin group defines a fibration on the two sphere by $spin(4,2)$, every bit as much as a spinor fibration on the manifold of conformal gravity. Thus for the spinor field connection [tex]A_\mu^i[/itex] the multiple connectivity between two points may be given by the covariant

$$x^\mu~=~y^\mu~+~\Omega_i^{\mu\nu} A^i_\nu$$

The spinor group defines a set of noncommutative coordinates $y^\mu$on the space of conformal gravity due to the internal rotation on the Bloch sphere at each point on the conformal manifold

$$[y^\mu,~y^\nu]~=~i\Omega_i^{\mu\nu}e^i$$

where [tex]\Omega_i^{\mu\nu}e^i[/itex] is a symplectic matrix. The orientation of each Bloch sphere on the conformal manifold will deviate by the curvature on that manifold, as well as due to an internal gauge rotation. A general field over the set of noncommutative coordinates established by local interal rotations is then $\phi~\in~{\cal A}_\omega$. With an intertwiner or n-trad basis $E^i_\mu$ the coodinates on a Bloch sphere may be defined as $y^i~=~E^i_\mu y^\mu$ so that a covariant set of coordinates on the spinor space is

$$x^i~=~y^i~+~\Omega^{ij}_\mu A^\mu_j.$$

Either perspective is equally valid. This n-trad basis is then a frame where on $spin(4,2)$ for a given local basis on the Bloch sphere, or may equivalently define a frame on the Bloch sphere for a covariantly constant basis on $spin(4,2)$.

The noncommutative scalar field induced by internal spinor rotation defines a two dimensional space determined by a local patch on each Bloch sphere at every point on the conformal manifold. This space is then a tesselated space $R^2~=~{\bf U}_i^{\infty} N_i$, where each $N_i$ is a neighborhood on each $S^2$ around where two state vectors point in adjacent or neighboring Bloch spheres. This is a form of an atlas-chart construction which can be explicitely constructed. The transition function between the two charts, or Bloch wheres, defines on $R^2$ a space of connection terms, which is ${\cal A}_\omega$. For a general curved space, here the conformal space, the $\sigma_i$ directions will vary and thus define at each point on the $spin(4,2)$ a unique eigen-basis. This infinite set of of eigen-bases then is what the field is expanded upon in an orthogonal and complete set of states

$$\phi(y)~=~\sum_{m,n}|m\rangle\langle n|.$$

This defines ${\cal A}_\omega$ as a $C^*$ algebra who's generators act on an element $\phi(y)~\in~{\cal A}_\omega$ according to

$$e^{ik\cdot y}\phi(y)e^{-ik\cdot y}~=~\phi(y~+~k\cdot\Omega)$$

The coordinates $y_\mu$ define the adjoint action of the spinor group on the $spin(4,2)$ manifold according to

$$ad_{x_\mu}[\phi]~=~[x_\mu,~\phi(y)]~=~i\Omega^{ij}\frac{\partial x_\mu}{\partial y^i}\frac{\partial\phi}{\partial y^j},$$

where $V^i_\mu \partial_i~=~\Omega^{ij}\frac{\partial x_\mu}{\partial y^i}\frac{\partial_j}$ is a gravitational vielbein for a spacetime metric. The intertwiner $E^i_\mu$ between the two spaces defined by $SU(4)~\sim~spin(4,2)$ and $SU(2)$ defines the simplectic form $\Omega_{\mu\nu}$, which gives the inverse matrix $\langle B_{\mu\nu}\rangle~=~(\Omega^{-1})_{\mu\nu}$, which is the vacuum state for the gauge fields, as indicated above. The vielbein above also has its inverse analog $U^\mu_i \partial_\mu~=~\Omega^{\mu\nu}\frac{\partial x_i}{\partial y^\mu}\frac{\partial_\nu}$

The space $R^2$ is defined by the difference between state vectors on Bloch spheres within a chart. The Bloch sphere is defined by Hilbert state vectors modulo the phase, which is the projective Hilbert space $PH$. Hence the spaces $SU(4)$ and $SU(2)$ represent a double fibration over the projective Hilbert state space. Each these spaces defines a fibration on the projective Hilbert space. The general relativity portion of the conformal group is then a fibration on the state space which is a more complete version of the above discussion on Hawking radiation from an elementary consideration of the Berry phase. The spinor group determines another fibration on the projective Hilbert space, which determine currents or flux tubes between different point on the conformal manifold. Alternatively it defines noncommutative coordinates on the conformal space, which are the gauge parameterization of the quantization of the conformal gravity theory. This double fibration with the sympletic matrix $\Omega$ when it acts on a basis vector $\partial_{x}$ on either the manifold of conformal gravity or the space of $SU(2)$ defines the sympletic dual as $p~=~\omega\cdot\partial_{x}$. Thus the double fibration is analogous to the one-form $\omega~=~\gamma(dx~-~vdt)$, which can be the Lorentz boosts, or a Lagrangian form used in Finsler geometry. Thus the spaces $SU(4)$ and $SU(2)$ define the two fibrations over the projective Hilbert space according to the vacuum state, equivalently the symplectic form, for the field which defines a noncommuting basis over the spacetime. The term $ad_{x_\mu}[\phi]$ defines the symplectic variable conjugate to $x^\mu$, which describe a quantum fluctuation due to noncommutative variables.

In what will follow the noncommutative spacetime is then a coarse grained perspective on the manifold, which is a quantum phase transition. For the full group $E_6$, or even further $E_8$ the noncommutative basis transitions into a full heterotic gauge group action that has no scale dependency. But then at the symmetry breaking $E_6~\rightarrow~SU(4)\times SU(2)\times U(1)$ the universal scale invariance of the heterotic group is lost. In the full heterotic group $E_8$ there exist 16 scalar fields $4\times(2~+~{\bar 2})$, which are dilaton/Higgsian type of fields, of which 9 are associated with the conformal $SU(4)$ and the other six are associated with the breaking of the full heterotic $E_8$.

 

[Lawrence B. Crowell]

Gravitation and gauge fields, such as in the standard model, exhibit a different invariance with respect to vacuum energy rescaling. The Lagrangian for general relativity is $L~=~R~+~2\Lambda$, for $R$ the Ricci scalar. Thus under a rescaling of the cosmological constant the momentum energy tensor is changed by $T_{ab}~\rightarrow~T_{ab}~-~(c^4/8\pi G)\Lambda g_{ab}$. This then breaks the symmetry of the gauge field Lagrangian invariance under vacuum rescaling. The cosmological constant then introduces the global curvatures of a cosmology as an Einstein space directly into the gauge field, as well as the Dirac Lagrangians for the source fields or particles. The cosmological constant, with units of inverse length squared and the coupling constant $(c^4/8\pi G)~=~(\hbar c/8\pi)L_p^2$ give a unit of energy density

$$\rho~=~\frac{c^4}{8\pi G}\Lambda~=~(L_pL_c)^{-2},$$

where [itex]L_c[itex] is a length associated with the cosmological constant $L_c~=~\sqrt{1/\Lambda}~\sim~10^{27}cm$, or about the distance to the cosmological event horizon. This then means the vacuum energy density would be approximately $\rho~\sim~10^{12}cm^{-4}$, or approximately $10^{-46}GeV^4$.

This vacuum energy connects two scales, one which is the smallest scale whereby physics may well breakdown on smaller scales, and the second a cosmological scale, approximately where the cosmological event horizon lies $r~=~\sqrt{3/\Lambda}$. On scales larger than this cosmological scale quantum amplitudes across the cosmological event horizon are related by Bogoliubov transformations and the vacuum states for fields separated by this distance are not equivalent. Further, on this cosmological scale the homogeneity of time breaks down. As the metric components for the deSitter cosmology explicitely depend on time, such as $g_{rr}~=~exp(\sqrt{\Lambda/3}t)$, there is no way in which there can exist a Killing vector which acts to maintain energy, or the four components of the momentum four vector, holonomically constant. Consequently the constraints on physics according to the Coleman-Mandula theorem, where these symmetries permitted conserve energy, are locally restricted. They pertain within a scale $(r~>~L_p,~r~<<~L_c~=~10^{60})L_p$, where vacua structure separated by distances approaching the cosmological distance $L_c$ become unitarily inequivalent.

These two scales $L_p,~L_c$are comparable during the big bang. The early high energy universe, or on a scale where $E_8$ has been broken and spacetime was governed by $SU(4)\times SU(2)\times U(1)$. The gauge induced wormholes, governed by $SU(2)$, act strongly on the early universe by the shifting $T_{ab}~=~T_{ab}~-~\Lambda g_{ab}$, where the cosmological constant was large during this inflationary period. As one approaches $t~\rightarrow~0$ the two scales close $L_c~\rightarrow~L_p$. Under these conditions the classical spacetime structure fragments into a noncommutative froth as unitarily inequivalent vacua are compressed together into ever small spacetime volumes which become classically indefinable. The symplectic matrix $\Omega_{\mu\nu}$ defines the vacuum energy density in any local region as

[tex]
\rho~\simeq~|\Omega_{\mu\nu}|^{-2}~=~|\langle G_{\mu\nu}\rangle|^2~\sim~\frac{1}{(L_pL_c)^2},
[/itex]

where $G_{\mu\nu}$ is the field strength tensor of the $SU(2)$ gauge field. The expansion of the spacetime under the inflationary pressure of a large cosmological constant reduces the influence of the gauge field on the conformal spacetime on larger scales.

The very early universe after the breaking of the heterotic symmetry is defined as a system of gravity-like excitons which have multiple connections in a spin net and further are defined on vacua which are not unitarily equivalent. The expansion of the universe and the establishment of a stable cosmological horizon length by the Higgsian vacuum extends regions of unitary equivalence over vast distances $\sim~10^{10}$ light years. As the universe expands and ages it asymptotically approaches a deSitter spacetime configuration with a diminishing mass-energy content. Eventually Hawking-like radiation from the cosmological horizon will very slowly cause the cosmological horizon radius increase to infinity as the spacetime approaches a Minkowski spacetime as {itex]t~\rightarrow~\infty[/itex]. This final state of the universe is the AdS conformal infinity, which is shared by the dS cosmology, since $(SO(4,1),~SO(3,2))~\subset~SU(4)$.

The evolution of the universe is then a process which maps a set of inequivalent vacua, a purely quantum system of excitons, into a completely classical spacetime configuration with $\rho~=~0$. The universe is then a map between these voids. The complete symmetry of the universe is then some form of quantum error correction code which preserves the total quantum information through the process. The quantum error correction code is then a Golay or Goppa code, such as that defined by the Leech lattice $\Lambda_{24}$, which includes three $E_8$ heterotic groups in a modular system.

 

[Lawrence B. Crowell]

I have so far laid down an intermediate energy hypothesis for quantum gravity which stems from the breakdown of the heterotic group $E_6$ into a spinorial gauge-like field plus conformal gravity plus an abelian group. The spinorial gauge-like field is $SU(2)$ and induces a noncommutative spacetime basis on conformal gravity that is a spin-net. The non-commutative aspects of the spin-net then describe a holonomy for any enclosed path. The structure of this holonomy is comparatively simple, as seen in Wheeler's trick of rotating a book tied to a ribbon with the other end fixed twice or by $4\pi$ radian rotation. In general the parallel translation of a vector along a path connecting the two points $(x_0,~y_0,~z_0)$ to $(x_1,~y_1,~z_1)$ rotates that vector around the axis $\vec a$

[tex]
{\vec a}~=~\kappa {\vec x}_0\times{\vec x_1},
[/itex]

where $\kappa$ is a curvatures scalar related to the Gauss' fundamental forms. The parallel translation around a small parallelogram of area $A$ means that for small angle deviation and ${\vec x}_0\times{\vec x_1}~=~|x_0||x_1|sin(\theta)~\simeq~|x_0||x_1|\theta$ that this angle is given by

$$\theta~\simeq~2A\kappa,$$

which is the result for the Ricci curvature scalar on a small region of constant curvature.

In a similar light a spinor may be parallel translated. A particle with a spin $j\hbar$, where the Dirac unit of action is set to unity, is transformed by a unitary operator $U_j$, which is a group element of the $SU(2)$, which is a group of homomorphisms between spin states. For two points on the base manifold we may define a curve $C$, or in a sufficiently local small region an edgelink, which defines the amplitude for a spin state at the start and end point of this curve as

$$A_{C}(m_1,~m_2)~=~\langle j_1, m_1|U_j|j_2,~m_2\rangle,$$

for $m$ the eigenvalue of the $j_z$ spin operator. This amplitude defines the elements of the unitary $U_j$, $\langle j_1, m_1|U_j|j_2,~m_2\rangle~=~\langle j_1, m_1|(z|j_2,~m_2\rangle)$, for $z$ a complex number. This may then be used to define the amplitude for the rotation of a state $|j_3,~m_3\rangle$ as the output from a node $n$

$$A_n(m_1,~m_2,~m_2)~=~\langle j, m_1|(z|j,~m_2\rangle\otimes|j,~m_3\rangle).$$

This may be interpreted as the input of two spins $j_1,~j_2$ from two edgelinks into a node and the output state $j_3$. The complex number is evaluated from the unitary operators for the spins $j_1,~j_2,~j_3$ as

$$A_{n}(m_1,~m_2,~m_3)~=~(U_{j_3}\odot\langle j_3,~m_3|) z(U_{j_1}|j_1,~m_1\rangle\otimes U_{j_2}|j_2,~m_2\rangle)$$
$$~=~\langle j_3,~m_3| U_{j_3}^{-1}U_{j_3} z(|j_1,~m_1\rangle\otimes |j_2,~m_2\rangle)$$
$$~=~\langle j_3,~m_3| z(|j_1,~m_1>\otimes|j_2,~m_2\rangle) .$$

Then from the individual unitary spin operators the complex number $z$ is seen to define a map which reduces tensor products from the incoming states to the outgoing state, or equivalently $z$ is a Clebsch-Gordon coefficient.

In an interative manner these edgelinks may be combined to define a spin network, which also describes an underlying non-commutative structure to spacetime. Any edgelink between two points $x,~y$, $c_{x,y}$ carries a representation of the $SU(2)$ plus that of conformal gravity. These quantum numbers are $j,~q$ for spin and charge, here the charge being mass. The non-commutative structure of the spacetime is due to a disordered nonlocality or multiple connectivity. We might think of the $SU(2)$ as defining a wormhole that connects two points along another connected path. Any holonomy will is defined according to a loop with an ambiguity with respect to the ordering of points. Energetically this disordered nonlocality is a frustrated system, and can only remain within this ambiguity within the Heisenberg uncertainty in energy. Classically this ambiguity is forbidden.

Up to this point the subject of supersymmetry has been avoided. Supersymmetry introduces an underlying structure to spacetime. This is a graded algebraic system that defines a supermanifold with coordinates $y^\mu~=~x^\mu~+~i\theta^\alpha\sigma^\mu_{\alpha\dot\beta}{\bar\theta}^{\dot\beta}$. To the classical structure of the manifold is a spinorial term due to Grassmannian generators $\theta,~\bar\theta$. $N=2$ supersymmetry permits an $SU(2)$ symmetry to a manifold with the two supergenerators $Q^1_\alpha,~Q^2_\alpha$. In what follows some familiarity with supersymmetry is presumed.

The $N=1$ Grassmannian generators are $\theta^1,~{\bar\theta}^1,~\theta^2,~{\bar\theta}^2$. A supermanifold coordinate is then given by

$$y^\mu~=~x^\mu~+~i\theta_1^\alpha\sigma^\mu_{\alpha\dot\beta}{\bar\theta}_1^{\dot\beta}~+~i\theta_2^\alpha\sigma^\mu_{\alpha\dot\beta}{\bar\theta}_2^{\dot\beta}$$

The commutator between two super-coordinates is calculated to be

$$[y^\mu,~y^\nu]~=~\sigma^{\mu\nu}(\theta_1\theta_2 \sigma^\mu\cdot A_\mu~+~H.C.)$$

where $\{{\bar\theta}_1,~{\bar\theta}_2\}~=~\sigma^\mu\cdot A_\mu$, which is a gauge potential. This is similar to the symplectic form of the non-commutative coordinates previously derived. An anti-commutator between two supergenerators $Q_i^\alpha$ gives

[tex]
\{Q^i_\alpha,~{\bar Q}^j_{\bar\beta}\}~=~2\sigma^\mu_{\alpha{\bar\beta}}(\partial_\mu~+~A_\mu),
[/itex]

which is the gauge covariant $SU(2)$ operator.

This then links the internal gauge symmetry for the non-commutativity of spacetime is due to supersymmetry, or is a gauge action due to an $N~\ge~2$ supersymmetry. From here we have a link to supersymmetry types and their small groups. This will begin to connect us to the 24-cell and quantum error correction code systems.

 

[Lawrence B. Crowell]

For a black hole which emits a particle the back reaction $g~\rightarrow~~g'$ is a classical convenience. It is almost a Copenhagen-esque treatment of the black hole as a classical system which collapses the decoherent state of the emitted quanta (photons etc). Yet for a small enough black hole the metrics for the two black hole configurations differ enough so that the Killing vectors K and K' can't be regarded as equivalent under any sort of "nice" map. To do so is to say that the two metrics have a coordinate to coordinate relationship which violates the covariance principle of relativity. Thus there is no diffeomorphism which can be established to tie one metric to the other --- they pertain to different spaces.

In what follows there is a fairly extensive discussion on mathematics and some conjectures. This is induced by a physical argument that the Wick rotation from a Euclidean to a Lorentzian metric induces a breaking of general covariance between the configuration metrics and decoherence. This gives rise to an energy fluctuation with an associated time uncertainty. This time uncertainty is what gives rise to a macrotime with a Lorentzian signature. Once this mathematics is presented, the existence of knot topology is argued for with a braid group structure $PSL_2(Z)$. This leads to a connection between quantum states of gravity for metric configuration variables with the Lorentzian signature and a Fermi surface. The purpose of this here is to construct a one to one relationship between a Lorentzian spacetime structure and its Euclidean analogue. This will then permit a universal scaling of physics with imaginary time fluctuations $\delta t~=~\hbar/kT$.

So to first reiterate, let the manifolds $(M,~g)$ and $(M',~g')$ be related by a function $f$ so that $f:M~\rightarrow~M'$. For these two metric homeomorphic to each other we have that

f is bijective (injective & surjective),

f is continuous,

the inverse function f^{-1} is continuous (f is an open mapping).

If such a $f:M~\rightarrow~M'$ exists then $M$ and $M'$ are homeomorphic. Homeomorphisms define an equivalence relation between topological spaces, eg a coffee cup and donut are equivalent. Classes of topological spaces are homeomorphism classes.
Is this homeomorphism a diffeomorphism? A diffeomorphism is a homeomorphism that is $C^r$, $r~\ge~1$ differentiable. But as we have seen if the map between the metrics $g~\rightarrow~g'$ is differentiable then $\partial U/\partial x$ exists and this is (in coordinate terms)

$${{\partial U^a_b}\over{\partial x^c}}~=~{{\partial^2y^a}\over{\partial x^b \partial x^c}}$$

which is a non-covariant map between metrics. Thus the homeomorphism between the two spaces is not a diffeomorphism.

Consider a set of $R^4$s in the Euclideanized domain, in particular for disks $D^4$ on a Euclideanized DeSitter spacetime, exist as a set of all possible small $R^4$s homeomorphic to the disk. We might then consider them as classical configurations, eg non-holomorphic outcomes (state reductions) from the wave functional for spacetime $\Psi$, for quantum states. In the quantum tunnelling with $t~\rightarrow~it$ there is then a continuation of this homeomorphic structure into the Lorentzian domain of possible spaces. These spaces are similarly not identified to each other by diffeomorphisms. As such the path integral over all possible spacetime configurations is coarse grained into decoherent sets, where these sets are over fields that are diffeomorphic, but decoherence exists between topological spaces that are not diffeomorphic.

With two metrics $g$ and $g'$, with different Killing vectors we might find an estimate of the change in gravitational self-energy between the two as

$$\delta E_G ~= |\Gamma~-~\Gamma'|^2,$$

where the $\Gamma,~\Gamma'$ are connection terms on the two metric spaces. This energy uncertainty obeys $\delta E_g~\sim~\hbar/\delta T$. For a weak gravity field these terms will simply be $\Gamma~\sim~\nabla\Phi$, where $\Phi$ is the Newtonian gravitational potential. So we have as our approximation

$$\delta E_G ~= |\nabla(\Phi~-~\Phi')|^2,$$

where we might set $\Phi~-~\Phi'~=~f$, and consider this function in a more general setting. This energy functional on the unit ball $B$, or $B~homeo~B'$, defines the map or function $f:B~\rightarrow~R$,

$$E(f)~=~\int_B|\nabla f|^2 db$$

For the functional to be continuous, the vector space of all such functions need a well behaved topology. This is then well defined in a Euclidean metric, but no the Lorentzian case since the moduli is nonHausdorff. Also for 4-manifolds it can be the case that the ball $B$ has a homeomorphic identification with another such ball (small exotic R^4's) that is not diffeomorphic.

This leads to some conjectures on the relationship between nonHausdorff moduli space, which pertain to the Lorentzian spacetime, and strange properties of four dimensional manifolds determined by the intersection form for such manifolds. Physically this should provide a map between a superposition of states, each over a unique metric configuration variable, and a semiclassical state weighted around a classical state with a nonHausdorff moduli space. The inability to separate different moduli in the Lorentzian case has a relationship with the homeomorphic maps between configuration metrics in the Euclidean case. The terms in the moduli space are $\Lambda^1(M_l)/{\cal G}$ and the curvature terms $F~=~D\Gamma$ in the Euclidean case are in $\Lambda^2(M_e)$, and the intersection form ${\underline\omega}^4~=~F\wedge F$ exists in $\Lambda^4$ and $4\pi k~=~\int{\underline\omega}^4$ is a Chern class.

The set of 4-manifolds, which includes these strange "fake" spaces, have their homeomorphic structure determined by the embedding to two-disks or two-spheres in the space. In one dimension higher this structure leads to a knot topology. Whitney's theorem tells us we can embed a 2-dim space into a 5 dim space. We have a curious ambiguity about embedding 2-dim spaces in 4-dim spaces, for neither of Whitney's theorems apply. Yet we have all this duality/self-duality machinery, which suggests there is some embedding property. Suppose that we have a five dimensional space as $R^4\times[0,~\epsilon]$, where $\epsilon$ is some small number. So we have a thickened $R^4$. Now consider a disk $D^2$, where maybe there are "curvatures" from $F_{ab}$ or the Weyl curvature $C^{ab}_{cd}$. This cap might have been removed from some 2-dim space (call it $X^2$) we embed in our $R^4\times[0,~\epsilon]$. If this cap is a cross-cap (points identified with an $x~\rightarrow~-x$ for nonorientable structure), or with some double covering or ..., this then contains some topological information of $X^2$. We may then isolate all non-trivial topology of $X^2$ on a set of disks or caps. We know we can embed this $D^2$ in this 5-dim space by Whitney's theorem. Further, the dual curvatures $*F_{ab}$ or $*C^{ab}_{cd}$ for * = Hodge dual operator, will exist on a dual cap or disk $*D^2$ if we restrict to $R^4$ or assume curvature two-forms with direction in the 5-direction are approximately zero. Again Whitney tells us that we can embed this in our thickened $R^4$.

A theorem by Taubes tells us we can "collect" all the topological charge on a manifold given by the intersection form into a finite number of points and maintain the topology of the space. So as a result I am going to propose that all of the relevant curvature information is restricted to these disks with all of the curvature information. So we have a set of disks $\{D^2_i,~ i~=~1,~\dots,~n\}$ and their dual disks $\{*D^2_i,~ i~=~1,~\dots,~n\}$, with curvature information and respective duals.

Now for curvatures given be a connection one-form A we may evaluate on a disk *D^2 the Wilson line

$$W(C)~=~exp(i \int_C A),$$

for the curve $C$ on our disk $D^2$. The expectation of this Wilson curve is the partition function or path integral

$$\langle W(C)\rangle~=~\int D[A] W(C)exp(-iS),$$

where $S$ is the action determined by the curvature two-form $F~=~DA$. Thus we have there is the skein relationship for over-crossings and under-crossings of lines with

$$\alpha\langle W(L^+)\langle~-~\alpha^{-1}\langle W(L^-)\rangle~=~z\langle W(L^0)\rangle$$

$\alpha~=~1~-~2\pi i/kN$, $z~=~-2\pi i/k N$, and $L^+$ the over-crossing lines, $L^-$ the undercrossing lines and $L^0$ the noncrossing lines

This suggests that the Whitehead torsion is related to knot polynomials or Braid groups. In other words the Whitney trick works in 5-dim, and it appears that the cobordism is related to the knot polynomial or braid group, or that the Whitehead group $W(M)~=~B_n$ or $W(B_n)$, here restricted to braid groups. For $n~=~2$ the braid group is the cyclic group, and $W(Z)$ is trivial. For $n~>~2$ where

[tex]
s_i s_j~=~s_j s_i
[/itex]
[tex]
s_i s_{i+1} s_i~=~s_{i+1}s_i s_{i+1}~Yang-Baxter~equation
[/itex]

the braid group is the trefoil knot group for $n~=~3$ and for $n~>~2$ these groups are infinite nonabelian groups. It appears that in general $W(B_n)$ is trivial.

These crossings might be considered to be punctures through the disk $D^2$, and as such have homotopy content. The permutation of these punctures, given by the Braid group relationship, is a homotopy. This makes some contact with the idea that $W(M)~=~W(B_n)$, here in the case where we are looking at just one disk on the $X^2$. This appears to be a way of looking at what happens with $R^4~=~\lim_{e-->0}R^4\times [0,~\epsilon]$.

The modular braid group is $PSL_2(Z)$, which is an orbifold configuration on a hyperbolic group $SL_2(C)~=~SU(2)\times SU(1,~1)$, or $SO(2,1)$. This is the group for anyons. The first homotopy group of $SO(2,1)$ $\pi_1(SO(2,1)~=~Z$, or infinite cyclic. The braid group is a projective representation of the special orthogonal group SO(2,1), which are representations of anyons.

For later I will discuss the structure of the Fermi surface. This will connect with my prior discussion of a universal scaling principle with spin fields off a Fermi surface which give a Landau electron-like liquid.

 

[Lawrence B. Crowell]

This is a note on how a Bloch wave analysis of quantum states and fields may be used in quantum field theory and gravitation. This is an approach which uses some results of solid state physics. There have been some approaches of this nature in lattice gauge field theory and there have been some connections with ion traps and quantum gases.

The wave function in a crystalline solid will have a periodicity which matches the occurrence of ions in the lattice. The lattice defines a translation operator $T({\vec r})~=~exp({\vec k}\cdot{\vec r}$ with a periodicity relationship $T({\vec r}+{\vec a})~=~T({\vec r})$, for $|\vec a|$ the lattice distance. The wave function for a particle is then translated by this operator as

$$T({\vec r})\psi({\vec x})~=~\psi({\vec x}+{\vec r}).$$

The wave function also will exhibit local phase shifts according to a gauge group $\cal G$. The wave function transforms according to ${\cal G}:\psi(\vec x)~\rightarrow~\psi({\vec x})e^{i\theta}$. The action of the gauge group and the translation operator is to change the wave function by

$$\psi({\vec x})~\rightarrow~UT(\vec r)U^\dagger\psi(\vec x)~=~\psi({\vec x}+{\vec r})e^{i\theta}.$$

Consider the wave function $\psi({\vec x})~=~\psi_0exp(-i{\vec k}\cdot{\vec x})$. The action of the group operators $U$ and the translation operator $T(\vec r)$ is then to shift the position variable of the wave function in the position representation. The combined action of the group operator and the translation operator for small group actions $U~\simeq~ 1~+~i\theta$ and small translations $T({\vec r})~\simeq~1~+~i{\vec k}\cdot{\vec r}$ is

$$UT(\vec r)U^\dagger~\simeq~(1~+~i\theta)(1~+~i{\vec k}\cdot{\vec r})(1~-~i\theta)$$

$$=~1~-~[\theta,~{\vec k}\cdot{\vec r}]~=~1~-~{\vec k}\cdot [\theta,~{\vec r}]~-~[\theta,~{\vec k}]\cdot{\vec r}~+~O(\theta^2).$$

The action of this composite operator is to translate wave function from $\vec x$ to $\vec x'$ so that $x'~=~x~-~[\theta,~{\vec r}],$ which maybe written according to

$$x'~=~x~+~iU\nabla_kU^\dagger.$$

In the conjugate momentum representation for $T~=~T(\vec k)$ which translates the wave function on the reciprocal Brillioun lattice there is the conjugate result that

$$p'~=~p~+~iU\nabla_rU^\dagger.$$

The directional derivatives of the group elements define potential terms ${\vec A}~=~U\nabla_r U^\dagger$ and the conjugate potential $\vec C~=~U\nabla_kU^\dagger$, where in the first case this gives the standard gauge covariant definition of the momentum operator. This results in the $O(\hbar)$ commutation relationships

$$[x,~x']~=~i\nabla_{[k}C_{k']},~[p,~p']~=~i\nabla_{[r}A_{r']},~[x,~p]~=~i~+~i\nabla_{r}C_{p}~-~\nabla_{p}A_{r},$$

which is a quantum system with noncommutative coordinates. These noncommutative variables are defined by gauge potentials which are evaluated around a loops in position and momentum space, which are a form of Berry phase. The potential terms $A_r$ and $C_k$, which we can in general write as $A_z$ defines a quantum magnetic-like field $B_{z'}~=~(\nabla_{[z}A_{z'']})_{z'}$, which has a clear analogue with the Maxwellian result ${\vec B}~=~\nabla\times{\vec A}$. This quantal magnetic field is entirely due to the lattice structure of the space, which can be a lattice in solid state physics or a winding number for string on orbifolds. In the case of solid state physics this is related to quantum hall effects and magnetic flux quanta in periodic solids.

The Dirac equation is the square root of the Klein-Gordon equation and constitutes a quaterionic system. The Dirac algebra is a $32$-part algebra produced with the combination of the four-vector units $(i,~{\bf i},~{\bf j},~{\bf k})$ with the unit quaternions $(1,~{\bf e}_1,~{\bf e}_2,~{\bf e}_3)$. Physically the two sets denote spacetime and mass-charge parameters. These enter into the dynamics of a relativistic particle with the relativistic momentum-energy invariant interval

$$m^2~=~E^2~-~p^2.$$

This may then be factorized according to the quaternion elements with the additional phase term $e^{-i(et~-~{\bf p}\cdot{\bf x})}$, so this interval assumes the form

$$(\pm {\bf e}_3E~\pm~i{\bf e}_1{\bf p}~+~i{\bf e}_2 m)(\pm {\bf e}_3E~\pm~i{\bf e}_1{\bf p}~+~i{\bf e}_2 m)e^{-i(et~-~{\bf p}\cdot{\bf x})}~=~0.$$

By the standard quantization rule $E~\rightarrow~i\partial/\partial t$ and ${\bf p}~\rightarrow~-i\nabla$ the energy momentum interval is reproduced as

$$\Big(\pm i{\bf e}_3{\partial\over{\partial t}}~\pm~{\bf e}_1\nabla~+~i{\bf e}_2m\Big)(\pm {\bf e}_3E~\pm~i{\bf e}_1{\bf p}~+~i{\bf e}_2 m)e^{-i(et~-~{\bf p}\cdot{\bf x})}~=~0.$$

The wave function is then a quaternionic state vector

$$\psi~=~(\pm {\bf e}_3E~\pm~i{\bf e}_1{\bf p}~+~i{\bf e}_2 m)e^{-i(Et~-~{\bf p}\cdot{\bf x})}~=~0,$$

which obeys the invariant interval is the quantum wave equation

$$\Big(\pm i{\bf e}_3{\partial\over{\partial t}}~\pm~{\bf e}_1\nabla~+~i{\bf e}_2m\Big)\psi~=~0.$$

where the basis vectors define the Dirac matrices. It is clear that for a massless system that for the right hand side with the phase term boosted that the Dirac equation is equivalent to

$$(\pm {\bf e}_3E~\pm~i{\bf e}_1{\bf p})(\pm {\bf e}_3E'~\pm~i{\bf e}_1{\bf p}')e^{-i(Et~-~{\bf p}\cdot{\bf x})}~=~0,$$

for ${\bf p}'~=~{\bf p}~+~{\bf A}_r$. The Dirac equation then predicts that the momentum vector and energy evolves according to

$${{\partial {\bf p}}\over{\partial t}}~+~\nabla_r E~+~[E,~{\bf p}']~=~0$$

$${{\partial E}\over{\partial t}}~-~\nabla_r {\bf p}~+~[{\bf p},~{\bf p}']~=~0$$

The first of these differential equations is a modified Hamilton equation due to the addition of a quantum correction term, and the second term is a noncommutative or quantum corrected version of the invariant interval $m^2~=~E^2~-~p^2$. The commutator is determined by the action of the translation operator upon the group elements. In the case where the group contains the Lorentzian group the gauge potential will in general be $A_r~=~A_r~+~\omega\wedge A_r$, for $\omega$ the gravitational connection term. The quantum correction due to the noncommutative coordinates will then be of the form

$$[{\bf p},~{\bf p}']_\mu~=~(\nabla_{[r}A_{r']})_\mu~+~R_{\mu\nu\alpha\beta}U^\nu p^\alpha p^\beta$$

for both differential equations above when $\mu~=~0$, and $\mu~\in~\{1,~2,~3\}$.

 

[Lawrence B. Crowell]

The S-matrix is a topological approach to quantum field theory. It is the jumping off point for the Veneziano amplitude and the basis for string theory. It is based on the notion of an order for the relationship between different quantum fields, which in a more standard QFT approach is a time ordered product. The physical S-matrix obeys unitarity , and in the case of an ordering the vertices are defined so they do not exchange position “freely.” Any change in the ordering of vertices are done according to some discrete symmetry, such as transitions between S-T-U amplitudes.

The ordered S-matrix is constructed so that each vertex, or particle, has a neighbor. In a linear chain for instance a general state is an S-matrix channel of the form

$$|\phi\rangle=~|p_1,~\dots,~p_i,~\dots,~p_j,~\dots,~p_n\rangle$$

This state or S-matrix channel is related to but distinction from the channel

$$|\phi\rangle=~|p_1,~\dots,~p_j,~\dots,~p_i,~\dots,~p_n\rangle$$

The best way to see this is that the particles or vertices $p_i$ and $p_j$ have exchanged their neighbors, and a certain "relationship” structure to the amplitude has been fundamentally changed. The S-matrix is written according to $S~=~1~-~2\pi T$, so that given two states or channels $|p_1,~\dots,~p_n\rangle$ and $|q_1,~\dots,~q_n\rangle$ will be related to each other by the S-matrix as

$$\langle p_1,~\dots,~p_n|S|q_1,~\dots,~q_n\rangle~=~\langle p_1,~\dots,~p_n|(1~-~2\pi T)|q_1,~\dots,~q_n\rangle$$
$$~=~\langle p_1,~\dots,~p_n|q_1,~\dots,~q_n\rangle~-~2\pi\langle p_1,~\dots,~p_n|T|q_1,~\dots,~q_n\rangle.$$

For the $\langle-|$ as the in channel and $|-\rangle$ as the out channel then it is clear that $p_n$ and $q_1$ are neighbors, plus as neighbors through the T-matrix. This then eliminates an open vertex in the chain. The vertices or particles $p_1$ and $q_n$ are the open elements in the chain and define the “anchor” for the chain, and are thus defined as neighbors in this manner. Hence this process defines a complete linear chain, which is similar in its structure to a gauge-“Moose,” which is a cycle of gauge fields on a compactified space, such as a Calabi-Yau space.

Each of these elements $p_i$ defines a particle or vertex completely according to some set of quantum numbers. Thus each $p_i$ is defined by a vector space $V$, which is physically some Hilbert space. The linear chain here is then an ordering on a total Hilbert space ${\cal H}~=~\otimes_i V_i$. Since this construction is based upon the relationship between a $p_i$ and $p_{i+1}$ there is then some natural bilinear operation of the form

$$[-,~-]:V\times V~\rightarrow~V.$$

This defines some product structure for the change in position of any of these elements. In order for this bilinear operation to describe physical states it must obey the Jacobi identity. This then requires that the vector space by k-equipped so that the bilinear operation is defined according to an isomorphism on the vector space ${\cal H}~=~k\times V$, where the modulus $|k|$ is the number of elements in the chain. We then have that the isomorphism

$$Y:{\cal H}\times{\cal H}~\rightarrow~{\cal H}\times{\cal H}$$
$$Y\big((x,~p)\otimes(y,~q)\big)~=~(x,~p)\otimes(y,~q)~+~(1,~0)\otimes(0,~[p,~q]).$$

The application of $Y\otimes id$ on ${\cal H}\times{\cal H}\times{\cal H}$ then gives

$$Y\otimes id\big((x,~p)\otimes(y,~q)\otimes(z,~r)\big)~=~(x,~p)\otimes(y,~q)\otimes(z,~r)~+~$$
$$(1,~0)\otimes(0,~[[p,~q],~r]~+~[[q,~r],~p]~+~[[r,~p],~q]).$$

This isomorphism on the three spaces is the Yang-Baxter equation, which is satisfied if the permuted double commutator sum vanishes, which is the Jacobi equation. The elements $p,~q,~r$ could be the momentum operators $D~+~iA$, and the Jacob identity the conservation law

$$[[D_a,~D_b],~D_c]~=~\epsilon_{abcd}D_eF^{de}~=~0.$$

The Yang-Baxter equation is satisfied by the following commutative diagram:

The Yang-Baxter relationship is then defined in the S-matrix by the following observation. The optical theorem with $S~=~1~-~2\pi T$ and the projection of the density matrix according to

$$\rho^\prime~=~S\rho S^\dagger~=~\rho~+~2\pi i[T,~\rho].$$

The neighborhood rule tells us that the commutator is then between elements of the $|-\rangle$ and the $\langle-|$ with regards to the transition or T-matrix, and is thus an example of a neighbor exchange rule.

The Yang-Baxter equation describes braids, which are in this case links between nodes, vertices or particles in the chain. These links are the exchanged with each other just as a braid can connect links at different points in an array. We consider the four nodes with the braids indicated in figure 2. The composition of these braids then leads to the following rules. Every braid in the braid group $B_4$ is a composition of a number of these braids and their inverses. These three braids are generator of the group $B_4$. An braid is read from left to right; whenever a crossing of strands i and i + 1 occurs, $s_i$ or $s_i^{-1}$ is written down, if strand i moves under or over strand i + 1 respectively. Upon reaching the right hand end, the braid has been written as a product of the σ's and their inverses.
It is clear that

$$s_1s_3~=~s_3s_1$$

in addition there are the relations which are not quite as obvious:

$$s_1s_2s_1~=~s_2s_1s_2,~s_2s_3s_2~=~s_3s_2s_3$$

These last two are specific examples of the Yang-Baxter equation for braids, which is in general $s_is_{i+1}s_i~=~s_{i+1}s_is_{i+1}$.

{\bf Associahedra and Configuration Spaces of the S-matrix}

The Yang-Baxter equation describes braids, which are compositions of paths. In general the theory here needs to be extended to compositions of loops. The S-matrix acts upon a loop composed of $\langle-|$ and $|-\rangle$ to define the composition of two loops $\langle-|-\rangle$ with $2\pi i\langle-|T|-\rangle$. Homotopy is the mathematical theory for loop topology. For a topological space $(X,~p)$ the loop space $\Omega X$ is defined by the continuous map

$$\phi:[0,~1]~\rightarrow~X,$$

with the compact open-set topology on the endpoints $\phi(0)~=~\phi(1)~=~p$. Here the vertex or particle $p$ is considered to be the base point of the map. The composition or multiplication of points is given by the rule,

$$\pi_1:\Omega X\times \Omega X~\rightarrow~\Omega X,$$

as the composition of two such maps parameterized by $t~\in~[0,~1]$ by

$$\pi_1(\phi,~\psi)(t)~=~\phi\cdot\psi(t)~=~\left\{\matrix{\phi(2t),& 0,~\le~t~\le~1/2\cr\psi(2t~-~1),& 1 /2~\le~t~\le~1}\right\}$$

The interval $[0,~1]$ is the interval which maps two loops and then labelled as $K^2$. Similarly the composition of three loops may be defined a second homotopy group according to the map between two compositions of three maps

$$\pi_2(\phi\cdot\psi)\cdot\chi~\rightarrow~\phi\cdot(\psi\cdot\chi),$$

given by the map

$$\pi_2:K^3\times(\Omega X)^3~\rightarrow~\Omega X,$$

with $K^2~=~[0,~1]^2$

$$\pi_2(0,~-,~-,~-)~=~\pi_1(\pi_1(-,~-),~-),~\pi_2(1,~-,~-,~-)~=~\pi_1(-,~\pi_1(-,~-))$$

This map or composition may be written for two parameters $t,~u$ as

$$\pi_2(\phi,~\psi,~\chi)(t,~u)~=~$$
$$\left\{\matrix{\phi((4~-~2u)t),& 0,~\le~t~\le~1/(1~-~2u)\cr\psi(4t~-~(1~+~u)),& 1 /(4~-~2u)~\le~t~\le~1/4~+~1/(4~-~2u)\cr \chi((1~+~u)(2t~-~(4~-~u)/(4~-~2u)),& 1/ 4~+~1/(4~-~2u)~\le~t~1}\right\}$$

Of course this can be extended further! In this case the parameter space is $K^4$ and $\partial K^4$, for $K^4$ a two dimensional space. This defines the 3-homotopy or compositions of two dimensional surfaces. Homotopy parameterized by $\partial K^3$ is constructed from compositions of $\pi_1$ and $\pi_3$, while $\pi_2$ parameterized by $K^4$ is constructed from four continuous maps. These are a set of maps

$$\pi_3:K^4\times(\Omega X)^4~\rightarrow\Omega X,~\otimes_i\prod_i\partial K^4~\rightarrow~\Omega X$$

where the second map is composition of the lower homotopy groups on the boundary of the pentagon $K^4$, and the product is a sum over the indices on $\pi_i$ which equal 3. This results in the homotopy groups associated with the pentagon $K^4$ and its boundary $\partial K^4~\simeq~S^1$ as seen in figure 3

This of course may be extended to arbitrary homotopy groups according to

$$\pi_{n-1}:K^n\times(\Omega X)^n~\rightarrow~\Omega X,$$

which defines a polytope $K^5$. This defines the vertices according to compositions of $\pi_1$, lines according to compositions of $\pi_2$ with two $\pi_1$ groups, faces according to compositions of $\pi_3$ with a $\pi_1$, or a $\pi_2$ and a $\pi_2$, and the volumes as given by $\pi_4$. The homotopy elements are computed, or found, rather tediously. The vertices of the polytope are

$$\pi_1(\pi_1(\pi_1(\pi_1(-,-),-),-),-),~\pi_1(\pi_1(\pi_1(-,\pi_1(-,-)),-),-),$$
$$\pi_1(\pi_1(-,\pi_1(\pi_1(-,-),-)),-),~\pi_1(-,\pi_1(\pi_1(\pi_1(-,-),-),-)),$$
$$\pi_1(\pi_1(-,\pi_1(-,\pi_1(-,-))),-),~\pi_1(-,\pi_1(-,\pi_1(\pi_1(-,-),-))),$$
$$\pi_1(-,\pi_1(\pi_1(-\pi_1(-,-)),-)),~\pi_1(-,\pi_1(-,\pi_1(-\pi_1(-,-)))),$$
$$\pi_1(\pi_1(\pi_1(-,-),(\pi_1(-,-)),-),~\pi_1(\pi_1(-,-),(\pi_1(\pi_1(-,-),-)))$$
$$\pi_1(\pi_1(-,-),\pi_1(-,\pi_1(-,-))),~\pi_1(\pi_1(-,-),(-,\pi_1(\pi_1(-,-)))$$
$$\pi_1(-,\pi_1(-,\pi_1(\pi_1(-,-),-)),~\pi_1(-,\pi_1(\pi_1(-,-),\pi_1(-,-)))$$

The edgelinks of the polytope are similarly

$$\pi_2(\pi_1(\pi_1(-,-),-),-,-),~\pi_2(\pi_1(-,\pi_1(-,-)),-,-),~\pi_2(-,\pi_1(\pi_1(-,-),-),-),$$
$$\pi_2(-,\pi_1(-,\pi_1(-,-),-),~\pi_2(\pi_1(-,-),\pi_1(-,-),-),~\pi_2(-,\pi_1(-,-),\pi_1(-,-)),$$
$$\pi_2(\pi_1(-,-),(-,\pi_1(-,-))),~\pi_2(-,\pi_1(-,\pi_1(-,-),-),~ \pi_2(-,-,\pi_1(\pi_1(-,-),-)),$$
$$\pi_2(-,-,\pi_1(-,\pi_1(-,-))),~\pi_1(-,\pi_1(-,\pi_2(-,-,-)),~\pi_1(\pi_2(-,-,-),\pi_1(-,-)),$$
$$\pi_1(\pi_1(-,-),\pi_2(-,-,-)),~\pi_1(\pi_2(-,-,-),\pi_1(-,-)),~~\pi_1(\pi_1(-,\pi_2(-,-,-)),-)$$
$$\pi_1(\pi_2(\pi_1(-,-),-,-),-),~\pi_1(\pi_2(-,\pi_1(-,-),-),-),~\pi_1(\pi_2(-,-,\pi_1(-,-)),-)$$
$$\pi_1(-,\pi_2(-,-,\pi_1(-,-))),~\pi_1(-,\pi_2(-,\pi_1(-,-),-))),~\pi_1(-,\pi_2(\pi_1(-,-),-,-))$$

Finally the terms corresponding to the faces or plaquettes of the polytope are

$$\pi_3(\pi_1(-,-),-,-,-),~\pi_3(-,\pi_1(-,-),-,-),~\pi_3(-,-,\pi_1(-,-),-)$$
$$\pi_3(-,-,-,\pi_1(-,-)),~\pi_1(\pi_3(-,-,-,-),-),~\pi_1(-,\pi_3(-,-,-,-))$$
$$\pi_2(\pi_2(-,-,-),-,-),~\pi_2(-,\pi_2(-,-,-),-),~\pi_2(-,-,\pi_2(-,-,-))$$

Finally the body of the polytope is given by $\pi_4(-,-,-,-,-)$ There are $14$ vertices, $21$ edgelinks and $9$ plaquettes, which gives the Euler characteristic $\chi~=~V~-~E~+~P$ $=~14~-~21~+~9$ $=~2$, which is the topology of a sphere and a proper polytope. The plaquettes of the polytope indicate the structure of polytope, where there are $6$ faces composed of $\pi_3$ and $\pi_1$ and $3$ composed of two $\pi_2$’s. This polytope is illustrated below in figure 4

This is a regular polytope, which is Stasheff’s $K^5$ associahedron. Each $K^n$ is a generalization of the pentagon. A full generalization of the theory of associahedra is given by Jean-Louis Loday at:

http://www.claymath.org/programs/outreach/academy/LectureNotes05/Loday.pdf

The number of nodes in the $n-1$-associahedron is equivalent to the number of binary trees with $n$ nodes, which is the Catalan number $C_n$.

The associahedron is the basic tool in the study of homotopy associative Hopf spaces.

Loday provides the following method for associahedron construction. Let $A_n$ be the set of planar binary trees with $n+1$ leaves. Define $a_n$ as the number of leaves to the left of the $i^{th}$ vertex and $b_i$ as the number of leaves to the right of the $i^{th}$ vertex. For $t~\in~A_n$, define

$$M(t)~=~(a_1b_1,~a_2b_2,~\dots,~a_nb_n)$$

The $n-1$-associahedron is then defined as the convex hull of $M(t)$.

The above construction of homotopy groups is an association system of elements. The homotopy groups are associative, but the operand they define determines associators and commutators. The pentagon determines a system of commutators and associators according to diagram 5

The associahedra then define operads as a sequence of topological spaces $A_{[0]},~\dots,A_{[n]}$ with an elements $e~\in~A_{[1]}$ and a multiplication map

$$\mu_s:A_{[k]}\times A_{[n_1]}\times A_{[n_{k-1}]}~\rightarrow~A_{[n]}$$

for the order preserving map $s:[n]~\rightarrow~[k]$ and fibration $s^{-1}(k)~\simeq~[n_l]$. These obey associativity and unitary conditions on sequences of order preserving maps. This is then a generalized construction of the S-matrix according to operad compositions illustrated in figure 5


This figure gives an ordered tree-leaf configuration of order elements. The S-matrix may then be arranged according to such more general orderings and sequences determined by Catalan numbers. These combinatorial relationships will then define generalized amplitude and vertex operators such as the quark gluon interaction in figure 6 determined by the root vectors of the field theoretic group.

 

[Lawrence B. Crowell]

{\bf Associativity, Octonions and Skyrme Quark-like Models}

The octonions are generally nonassociative, and the associator $(ab)c~-~a(bc)~=~[a,~b,~c]$ leads to uncertainty in the definition of Jacobi identities. However here an identification is found of the associator with a new Chern-Simons Lagrangian that determines the dynamics due to the nonassociative structure of the octonions. This further indicates that octonionic field theory has a moduli space construction with connections to the geometry of knots. The mathematics of this moduli space are only introduced in the most elementary fashion, and this is then presented as a new frontier of mathematical research.

We consider the nonassociative bosonic fields $\phi_\alpha,~\phi_\beta,~\phi_\gamma$ and their commutators. The commutator $[\phi_\alpha\phi_\beta,~ \phi_\gamma]$ is

$$[\phi_\alpha\phi_\beta,~ \phi_\gamma]~=~(\phi_\alpha\phi_\beta)\phi_\gamma~-~\phi_\gamma(\phi_\alpha\phi_\beta)$$
$$~=~(\phi_\alpha\phi_\beta)\phi_\gamma~-~(\phi_\gamma\phi_\alpha)\phi_\beta~+~[\phi_\gamma,~ \phi_\alpha,~ \phi_\beta]$$
$$~=~(\phi_\alpha\phi_\beta)\phi_\gamma~-~(\phi_\alpha\phi_\gamma)\phi_\beta~-~[\phi_\gamma,~ \phi_\alpha]\phi_\beta~+~[\phi_\gamma,~ \phi_\alpha,~ \phi_\beta]$$
$$~=~\phi_\alpha(\phi_\beta\phi_\gamma)~-~(\phi_\alpha\phi_\gamma)\phi_\beta~+~[\phi_\alpha,~ \phi_\beta,~ \phi_\gamma]~-~[\phi_\gamma,~ \phi_\alpha]\phi_\beta~+~[\phi_\gamma,~ \phi_\alpha,~ \phi_\beta]$$
$$~=~\phi_\alpha(\phi_\gamma\phi_\beta)~-~(\phi_\alpha\phi_\gamma)\phi_\beta~+~\phi_\alpha[\phi_\beta,~ \phi_\gamma]~+~$$
$$[\phi_\alpha,~ \phi_\beta,~ \phi_\gamma]~-~[\phi_\gamma,~ \phi_\alpha]\phi_\beta~+~[\phi_\gamma,~ \phi_\alpha,~ \phi_\beta]$$

and so

$$~=~\phi_\alpha[\phi_\beta,~\phi_\gamma]~+~[\phi_\alpha,~\phi_\gamma]\phi_\beta~-~[\phi_\alpha,~ \phi_\beta,~ \phi_\gamma]~+~ [\phi_\gamma,~\phi_\alpha,~ \phi_\beta]~+~ [\phi_\alpha,~ \phi_\gamma,~ \phi_\beta]$$

The double commutator is

$$[\phi_\alpha,~[\phi_\beta,~\phi_\gamma]]~=~2\phi_\alpha[\phi_\beta,~\phi_\gamma]~+~[\phi_\alpha,~\phi_\beta]\phi_\gamma~-~[\phi_\alpha,~\phi_\gamma]\phi_\beta~-~$$
$$[\phi_\alpha,~ \phi_{[\beta},~ \phi_{\gamma]}]~+~ [\phi_{[\gamma},~\phi_\alpha,~ \phi_{\beta]}~+~ [\phi_\alpha,~ \phi_{[\gamma},~ \phi_{\beta]}],$$

and that the Jacobi identity is of the form

$$[\phi_\alpha,~[\phi_\beta,~\phi_\gamma]]~+~ [\phi_\beta,~[\phi_\gamma,~\phi_\alpha]]~+~ [\phi_\gamma,~[\phi_\alpha,~\phi_\beta]]~=~$$
$$[\phi_\alpha,~\phi_{[\beta},~\phi_{\gamma]}]~+~[\phi_\beta,~\phi_{[\gamma},~\phi_{\alpha]}]~+~[\phi_\gamma,~\phi_{[\alpha},~\phi_{\beta]}],$$

where the cyclicity of the associator has been used. Here the subscripts $[$ and $]$ indicate anticommutation of these indices, and the associator $(~)$ are defined by the commutator.

If the fields $\phi_{[\alpha}\phi_{\beta]}~=~F_{\alpha\beta}$, where these field $\phi_\alpha$ are thought of as the differential operator ${\cal D}_\alpha$ and the $F_{\alpha\beta}$ are components of the field strength tensor, then the above Jacobi identity is reduced to

$$Cyc[\phi_\alpha,~[\phi_\beta,~\phi_\gamma]]~=~\phi_\alpha F_{\beta\gamma}~+~\phi_{[\beta}F_{\gamma]}\alpha~+~\phi_{[\gamma}F_{\alpha\beta]}~-~$$
$$(\phi_\alpha\phi_\beta)\phi_\gamma~-~(\phi_\beta\phi_\gamma)\phi_\alpha~-~(\phi_\gamma\phi_\alpha)\phi_\beta$$

The nonassociativity of octonions means this involves a \lq\lq torsion\rq\rq. Using permutation symmetry of the associator an alternate Jacobi identity is

$$Cyc[[\phi_\alpha,~\phi_\beta],~\phi_\gamma]~=~ F_{\alpha\beta}\phi_\gamma~+~ F_{\beta\gamma}\phi_\alpha~+~ F_{\gamma\alpha}\phi_\beta ~-~$$
$$\phi_{[\alpha}(\phi_\beta]\phi_\gamma)~-~\phi_{[\beta}(\phi_{\gamma]}\phi_\alpha)~-~\phi_{[\gamma}(\phi_{\alpha]}\phi_\beta).$$

The Jacobi identity is then imposed as a topological requirement that $d^2~=~0$, where the fields $\phi_\alpha$ are coboundary operators. With the antisymmetry of the associator we have the difference between these two permuting commutators by the Jacobi identity requires that

$$Cyc[\phi_\alpha,~[\phi_\beta,~\phi_\gamma]]~+~ Cyc[\gamma,~[\phi_\alpha,~\phi_\beta]]~+~ Cyc[[\phi_\gamma,~\phi_\alpha],~\phi_\beta]~=~0,$$

which means that the associator may be identified as

$$[\phi_{[\alpha},~\phi_{\beta]},~\phi_\gamma]~=~{1\over 2}[\phi_\alpha,~ F_{\beta\gamma}]$$

This indicates that the associator $[\phi_\alpha,~\phi_\beta,~\phi_\gamma]$ is then a determinant of a part of the Chern-Simons Lagrangian

$${\cal L}_{cs}~=~{\cal L}^0_{cs}~+~ {\bf\Lambda}_{[\phi_\alpha,~\phi_\beta,~\phi_\gamma]}~=~\int (A\wedge dA~-~{2\over 3}A\wedge A\wedge A)~+~\int{[\phi_\alpha,~\phi_\beta,~\phi_\gamma]}$$

Here ${\cal L}^0_{cs}$ pertains to dynamics of with the seven associated subspaces, and ${\bf\Lambda}_{[\phi_\alpha,~\phi_\beta,~\phi_\gamma]}$ involves dynamics from nonassociative action of coboundary operators. This is an interesting find that has connections to knot theory. Here the Chern-Simons Lagrangian from the associator determines the dynamics involved with products of connection terms that are nonassociative.

{\bf Knot Theory, Hopf fibration and Octonions}

Consider $\cal M$ to be out $7$ dimensional manifold. We then assign a vector space to this by $Z({\cal M})$. Consider the vectors $X$ and $Y$ where $XY$ is the Fano plane product. There are then a number of possible properties of this vector space

$$Z({\cal M}^*) ~=~ Z({\cal M})^* ~duality$$
$$Z(XY) ~=~ Z(X)\otimes Z(Y)$$

and in the latter case if $X$ and $Y$ are generalized to disjoint subsets of $S^7$ then

$$Z(S_1\vee S_2)~ =~ Z(S_1)\otimes Z(S_2),$$

where "$\vee$" is a product analogous to the wedge product between differential forms. So let $V_1$, $V_2$, $V_3$ be disjoint subsets of $S^7$, and let the boundaries of these be

$$\partial V_1~=~S_1~+~S_2$$
$$\partial V_2~=~S_2~+~S_3$$
$$\partial V_3~=~S_3~+~S_4$$

Then $Z(V_1)$ is a homomorphism $Z(S_1)~\rightarrow~Z(S_2)$ and $Z(V_2)$ is a homomorphism $Z(S_2)~\rightarrow~Z(S_3)$ etc. We then have that

$$Z(V_1 \cup V_2) ~=~ Z(V_1)Z(V_2) \in Hom(Z(S_1), ~Z(S_2)).$$

This leads to an interesting result

$$Z((V_1 \cup V_2) \cup V_3) ~=~ (Z(V_1)Z(V_2))Z(V_3) ~\in~ Hom(Hom(Z(S_1), ~Z(S_2)), ~Z(S_3)))$$
and
$$Z(V_1 \cup (V_2 \cup V_3)) ~=~ Z(V_1)(Z(V_2)Z(V_3))~\in~ Hom(Z(S_1), ~Hom(Z(S_2), ~Z(S_3))),$$

and nonassociativity means that

$$Z((V_1 \cup V_2) \cup V_3) ~\ne~ Z(V_1 \cup (V_2 \cup V_3))$$
and
$$Hom(Hom(Z(S_1), ~Z(S_2)), ~Z(S_3)))~\ne~Hom(Z(S_1), ~Hom(Z(S_2),~ Z(S_3)))$$

This leads to some interesting generalizations of the constructions used in topology. This generalizes the traditional atlas-chart construction with transition functions. This construction is a bit more complex than usual, for one has to consider the overlap of 3 charts to get associative and nonassociative parts. There will then be transition functions for elements. We have the duality

$$\langle e_i,~ \omega_j\rangle ~=~ \delta_{ij}$$

and differential forms are covariant $d~\rightarrow~d ~+~ iA$. The space of connections and the moment map $\{A\}~\rightarrow~ F$ for gauge fields is the moduli space $A/{\cal G}$, $\cal G$ = Lie group.

These correspond to the set of gauge connections that are equivalent by a gauge choice. The two-form for the curvature $F ~=~ dA$ is then the set of fields. We have $dF ~=~ 0$, and so $F$ is closed and this defines a conservation law. Let $a$ be a 1-form on $A/{\cal G}$, so that $B ~=~ da$ and $dB ~=~ 0$. One could have $a~ =~ 0$ for a simply connected region. However, for multiply connected $A/{\cal G}$ and may be a closed/exact form. This is the Chern-Simons form. For an abelian theory this leads to $a ~=~ A\wedge dA$,

$$L(A) ~= \int A \wedge dA,$$
and for the nonabelian theory
$$L(A) ~=~ \int(A \wedge dA ~+~ {2\over 3}A \wedge A \wedge A) = \int A \wedge DA$$

The integrand is the Chern-Simons form and the integral is the Hopf invariant. This integral then enters into the path integral

$$Z[A] ~=~ \int {\cal D}[A] exp(iL(A)).$$

{\bf Discussion}

We have illustrated a connection between the S-matrix, associahedra and knot theory. The underlying nonassociativity of quantum states, and their corresponding associative homotopy groups, illustrates that a more general method for ordering states is possible with an underlying nonassociativity. The result is a field theory which is similar to a Skyrme model. This also can produce a generalized system for computing vertex operators. The Chern-Simon’s Lagrangian results in actions for coherent states, which under a product of all vacuum configurations gives a beta function for a four-point vertex function. In a similar manner the four point vertex function corresponds to ordered arrangements of four nodes. All possible combinations of these elements is a binomial distribution. However, the tree leaf ordering on these combinations according to a system of preferential attachments. This will define an Euler Beta function, which is the Veneziano amplitude result.


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