There is a deep connection between the U-duality groups of M-theory and the embedding of the 11-dimensions in the extended superspace which under the gauge and diffeomorphism group actions, induces a continuous \({E_{d(d)}}\) symmetry. Here, I will relate the F-theory action to that of M-theory in the context of the F-theory/M-theory duality with an \({\rm{SL}}\left( N \right) \times {\mathbb{R}^ + }\) representation. Recall that F-theory is a one-time theory, so let us start with how to make a space-like brane time-like in M-theory. Keeping in mind that the total action of M-theory is given by:

\[\begin{array}{*{20}{l}}{{S_M} = \frac{1}{{{k^9}}}\int\limits_{world - vol} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} T_p^{10}d\Omega {{\left( {{\phi _{Inst}}} \right)}^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right)}\\{ + \sum\limits_{Dp} {D_\mu ^S} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\mu ^S} {e^{ - H_3^b}}/{S^{Total}}}\end{array}\]

as I showed here, with \({T_p} \sim {\alpha ^\dagger }\frac{{p + 1}}{2}\) the D-p-brane world-volume tension, and the Yang-Mills field strength being:

\[{F_{\mu \nu }} = {\partial _\mu }A_\mu ^H - {\partial _\nu }\bar A_\mu ^H + \left[ {A_\mu ^H,\Upsilon _{2\kappa }^i(\cos \varphi )} \right]\]

and by a Paton-Chern-Simons factor, we get:

\[\left[ {A_\mu ^H,A_\nu ^H} \right] = \sum\limits_{k = 1}^N {A_\mu ^{H,ac}} A_\nu ^{H,cb} - A_\nu ^{H,ac}A_\mu ^{H,cb}\]

\({\phi _{Inst}}\) the instanton field, with:

\[{e^{ - {\phi _{Inst}}{g_{\mu \nu }}}} = {e^{ - 2{\phi _{Inst}}\left( {{g_{\mu \nu }} - 1} \right)}}\]

and \({g_{\mu \nu }} = {e^{{{\left( {{\phi _{Inst}}} \right)}^2}}}\).

Space-like branes are a class of time-dependent solutions of M-theory with topological defects localized in (P + 1)-dimensional space-like surfaces and exist at a moment in time, and are time-like super-tachyonic kink solutions of unstable D(P + 1)-branes in string theory and provide the topology of the throat-bulk. Let us start with a Dp-Dp pair Lagrangian, fixing the boundary of the string field theory superspace, so that the action is:

\[S = {\mkern 1mu} - 2{T_{D9}}\int {{d^{10}}} x{e^{ - \pi {{\left| T \right|}^2}}}F\left( {X + \sqrt Y } \right)F\left( {X - \sqrt Y } \right)\]

with

\[\left\{ {\begin{array}{*{20}{c}}{X \equiv {\partial _\mu }T{\partial ^\mu }\bar T}\\{Y \equiv {{\left( {{\partial _\mu }T} \right)}^2}{{\left( {{\partial ^\nu }\bar T} \right)}^2}}\end{array}} \right.\quad p = 9\]

and

\[T = {T_{cl(st)}}(x) = x + \sum\limits_{cl{{(st)}_x}} {\int_{cl{{(st)}_x}} {{e^{\tilde T(x)}}} } \gg 0\]

A Teichmuller BPS D(P+1)-brane 2-D reduction gives us the throat action:

\[S = {\mkern 1mu} - \int {{d^{p + 2}}} xV(T)\sqrt {1 + {{\left( {{\partial _{\mu T}}} \right)}^2}} \]

The post M-Theory, Kaluza-Klein Splitting, U-Duality and F-Theory appeared first on George Shiber.

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as I showed here, with the D-p-brane world-volume tension, and the Yang-Mills field strength being:

and by a Paton-Chern-Simons factor, we get:

the instanton field, with:

and .

Space-like branes are a class of time-dependent solutions of M-theory with topological defects localized in (P + 1)-dimensional space-like surfaces and exist at a moment in time, and are time-like super-tachyonic kink solutions of unstable D(P + 1)-branes in string theory and provide the topology of the throat-bulk. Let us start with a Dp-Dp pair Lagrangian, fixing the boundary of the string field theory superspace, so that the action is:

with

and

A Teichmuller BPS D(P+1)-brane 2-D reduction gives us the throat action:

with , , the metaplectic D-field whose potential achieves its maximum at and asymptotes to zero (closed string vacuum) at large . Note now, the action above gives the known exponentially super-decreasing pressure at late-times while being consistent with the string-theory calculation, where is interpreted as an exponential function of .

Since the energy:

is conserved, one gets the homogeneous solution

When D-fields approach their minimum, , their time-dependence simplifies to . Hence, the location of a static domain wall is determined by the equation where is the semi-classical solution of the domain wall, so time-dependent D-field solutions are analogously characterized by and the S-brane is found wherever . So, from

it follows that we must choose the Sp-brane field solution to be the space-like p+1-dimensional space . So now, we are in a position to deform the S-brane worldvolume as given by analyzing Heisenberg fluctuations of D-fields around semi-classical solutions given above,

Substituting this into

while keeping terms quadratic in , one gets the Heisenberg fluctuation action

with

being the key to time-like transformation, with and the time-dependent mass is

The factor in front of in the Heisenberg fluctuation action diverges at late time hence the Heisenberg fluctuation is governed by the Carrollian bulk-metric and ceases to propagate, which is what we expect. Now, since

breaks translation invariance along the time direction, there is a zero mode on the defect S-brane, which gives us

with depending only on the coordinates along the Sp-brane. By substituting into the fluctuation action, the mass term in

cancels with the contribution from the term . Hence, the effective action for a massless displacement field is

with the constant depending only on the energy , and hence, the S-brane effective action for a Euclidean world-volume to lowest order has been determined. Now, one naturally expects gauge fields on the S-branes, just like on D-branes. So, to proceed, first note that the constant gauge field strength appears in the S-field action only through the overall Born-Infeld factor

and the open string metric

used for contracting the indices of the derivatives. Since the equations of motion for the gauge fields are also satisfied in the time-dependent homogeneous Sp-background, the open string metric satisfies

So, allowing such an introduction of dynamical gauge fields, while also preserving the Sp-equations of motion, essentially entails that we cannot turn on EM fields on a Euclidean worldvolume and the dependence on the zero mode in the Sp-action ought to then be

with

giving us world-volume spacetime continuity and can be fixed by the global Lorentz invariance in the world-volume. The condition that the Lorentz super-boost preserves the open string symplectic metric is

which allows us to define the Lorentz boost as

hence deriving

Now, integration over in

and including the dependence, we obtain the Dirac-Born-Infeld S-brane action

Note however the above Dirac-Born-Infeld S-brane action differs from the usual D-brane action in two deep respects: first, the action is defined on a Euclidean world-volume, and second the kinetic term of the transverse scalar field has a wrong sign since it represents time translation. Covariantizing the Dirac-Born-Infeld S-brane action reduces the Lagrangian to with the induced metric on the brane. It differs from the usual DBI lagrangian only by a factor of , and therefore has the same equations of motion. Finally, I must show that this transversality has no D-brane charge at future infinity. Take the Ramond-Ramond coupling for an S-brane to be the same as that for a D-brane. So, the coupling of RR fields to the particular S-brane above is

Transforming into the embedding time , it follows that

hence the D-brane charge of this solution shrinks to zero at future infinity due to

Deep point is that in the T-dual picture by compactifying , becomes a spatial coordinate, and the S-brane solution

implies that although by definition S-branes are spacelike objects, they are however constructed using the open string D-field and hence governed by the open string metric and have time-like holographic embedding on the brane-bulk.

On the extended super-coordinates , we define super-diffeomorphisms that in higher rank groups yield a unified description of 4-D diffeomorphisms along with the p-form gauge transformations and provide a unified description of part of the local symmetries of IIB-string-theory and 11-D supergravity.

Generators of generalized diffeomorphisms act on vectors locally on via the associated fiber Lie derivative of weight and differs from the standard Lie derivative by a Calabi-tensor and the only non-vanishing components are :

where the universal weight term is given by:

with:

and gauge parameters, in our context, being .

Then, the transformation rules follow as such:

Contrasted with standard Lorentzian geometry, our Lie algebra of generalised diffeomorphisms involves an -bracket given by:

satisfying the closure condition:

The exceptional-algebraic diffeomorphism symmetry of the action is given by:

where is covariant under internal diffeomorphisms and given by:

and our gauge field transforms under generalised diffeomorphisms as:

and the generalised diffeomorphisms metric is:

For simplicity, let’s restrict our analysis to 2 with no loss of generality. Our theory is determined by an external metric and a generalised metric that parametrises and on , decouples as:

Hence, we can define by:

thus allowing extra degrees of freedom. Naturally, we have a ramified hierarchy of gauge fields mirror dual to the tensor ramified hierarchy of gauged supergravities. We can determine the exceptional action now in general form:

where the covariantised external Ricci scalar is:

the scalar kinetic generalized term is given by:

the gauge terms are given by:

with the topological Chern-Simons term, which is gauge invariant in 10+2 dimensions, is given by:

and our scalar potential:

and the equation of motion for the field is:

U-duality entails that field theory is equivalent to 11-D and 10-D IIB supergravity under the exceptional Kaluza-Klein splitting, and the relation between M-theory and F-theory is explicitly expressed by:

for exceptional F-theory, and:

The correspondences can be written as such:

M-theory has the section condition . The fields functionally depend on which are associated with the coordinates of 11-D supergravity in a 9+2 Kaluza-Klein splitting. Extra degrees of freedom come from the spacetime metric, with the Kaluza-Klein field:

and the gauge fields:

with:

and:

In the IIB case, we have and the coordinate-dependence is on and become the coordinates of 10-D type IIB supergravity in a 9+1 Kaluza-Klein split. The spacetime metric contribution comes from and the Kaluza-Klein field is hence:

which parametrise the external metric and the components of the generalised metric as:

It is clear now that the Kaluza-Klein field is identical -component-wise to the gauge field and the parametrisation of in terms of the axio-dilaton

is given by:

The gauge dualities are:

which establish not just an equivalence between -EFT and 11-D supergravity and 10-D type IIB supergravity, but also with F-theory:

which “is what results when KK-compactifying M-theory on an elliptic fibration (which yields type IIA superstring theory compactified on a circle–fiber bundle) followed by T-duality with respect to one of the two cycles of the elliptic fiber. The result is (uncompactified) type IIB superstring theory with axio-dilaton given by the moduli of the original elliptic fibration, see below. Or rather, this is type IIB string theory with some non-perturbative effects included, reducing to perturbative string theory in the Sen limit. With a full description of M-theory available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation. For instance while the modular structure group of the elliptic fibration in principle encodes (necessarily non-perturbative) S-duality effects, it is presently not actually known in full detail how this affects the full theory, notably the proper charge quantization law of the 3-form fluxes, see atS-duality – Cohomological nature of the fields under S-dualityfor more on that.”

Hence, in our context, F-theory is a 12-D lift of IIB supergravity yielding a geometric interpretation on the -duality symmetry. By KK-compactifying M-theory on an elliptic fibration, we naturally have an M-theory/F-theory duality, and the -EFT although a manifest 12-D theory, however reduces to 11-D and type IIB supergravity field-theories due to topological features of worldvolumes of sevenbranes monodromy. James Halverson has an excellent exposition on that here.

Next, we must connect the one-time property of F-theory to M-theory’s U-duality, which at face-value, seems highly problematic.

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]]>Let us see how the Yukawa couplings among 4-D fermionic fields can be derived from the F-theory superpotential and relate them to the tree-level superpotential. This is of utmost importance since D7/D3-brane-phenomenology of 4-D F-theory can be promoted to M-theory in light of the F/M-theory duality and the compactness of Calabi-Yau 4-folds. Start with a Kähler coordinate expansion of \(\gamma \) which gives us, after inserting it in:

\[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

the following:

\[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]

which is the exact 7-brane superpotential for F-theory and the integrand is independent of \(\lambda \), entailing that the F-term conditions are purely topological and in no need for \(\alpha '\)-corrections.

However, the D-term in:

\[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]

is in need of \(\alpha '\)-corrections, since it is evaluable as:

\[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\lambda P\left[ J \right]} \right.} \wedge F - \frac{{i\lambda }}{6}{\iota _\Phi }{\iota _\Phi }{J^3} + \\\frac{{i{\lambda ^3}}}{2}{\iota _\Phi }{\iota _\Phi }J \wedge F \wedge F - {\rm{P}}\left[ {J \wedge B} \right] \wedge F\\\left. { + i{\lambda ^2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge B} \right) \wedge \frac{{i\lambda }}{2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge {B^2}} \right)} \right\}\end{array}\]

and the non-Abelian D-term has the form:

\[D = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {\tilde A\left( {\tilde T} \right)/\tilde A\left( {\tilde N} \right)} \]

With \({Y_4}\) our target Calabi-Yau 4-fold and Lie algebra \(G\), for:

\[\left[ {{{D'}_i}} \right] \in {H^2}\left( {{Y_4}} \right)\]

we have:

\[\int_{{Y_4}} {\left[ {{{D'}_i}} \right]} \wedge \left[ {{{D'}_j}} \right] \wedge \tilde \omega = - {C_{ij}}\int_S {\tilde \omega } \]

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the following:

which is the exact 7-brane superpotential for F-theory and the integrand is independent of , entailing that the F-term conditions are purely topological and in no need for -corrections.

However, the D-term in:

is in need of -corrections, since it is evaluable as:

and the non-Abelian D-term has the form:

With our target Calabi-Yau 4-fold and Lie algebra , for:

we have:

with and the Cartan matrix of , effectively reflecting the F/M-theory duality.

In the local patch on the C-manifold, one takes the flat-space-Kähler-form as having the form:

Then, we decompose the Kähler-background B-field as:

with:

thus giving us:

with the Abelian pull-back to determined by:

where locally the Higgs field is given by:

with a matrix in the complexified adjoint representation of and its Hermitian conjugate. Thus, I could derive:

with:

Hence we have:

Now: realize that is a zero-form and does not have transverse-legs to , and thus the pull-back has a trivial action. So, after solving:

the D-term equation amounts to with:

and with the -field vanishing on the sheave of the C-manifold, one gets a reduction to:

which yields a non-Abelian -corrected Chern-Simons action of a stack of D7-branes with both terms at leading order in , entailing that for matrix algebras:

they are the matrix products in the fundamental representation of

and so the **-corrections on D-terms** with the gauge flux F diagonalization yields

Deep upshot: the -corrections are given entirely by the abelian pull-back of the Kähler-form to

And this has a deep physical interpretation which can be extracted from the energy-momentum tensor and D-term of Q-clouds.

In the special case that is of interest, the Yukawa couplings among 4-d matter fields can be derived from the superpotential:

with the F-theory characteristic scale, and with dynamical dependence on the D-term:

Our equations of motion that follow from the superpotential and the D-term are given by:

which are the F-term equations, and the following holds for our fundamental form on :

which is the D-term equation.

In the bosonic case, to derive the equations of motion, define:

and expand the F-term equations and the D-term equation to first order in the fluctuations . Thus we find:

with the following relations:

and locally, we have the Kähler form:

Hence, our equations:

admit zero mode solutions that are localized on fermionic curves which are determined by the background of which in the absence of fluxes depends holomorphically on the complex coordinates of . So, a nontrivial VEV with the property that its rank changes at curves implies that instead of a single there are intersecting surfaces:

Now, at any point on , splits to times due to the 7-branes wrapping the , and at there are additional commuting generators whose associated fluctuations give rise to matter localized on as implied by solving the equations of motion. At point where the matter curves intersect, there is bi-uplifts to . Locally, a worldvolume flux is included, entailing that a hypercharge generator exists that breaks to the group.

A sketch of the proof:

Take

such that:

with the mass parameter; thus we have a VEV breaking of to at generic points in and so the group is enhanced to:

at curves:

The generators determined by:

with:

commute with when . Inducing chirality involves including the flux:

Under the holomorphic gauge such that:

solutions to the equations of motion are derived by gauge transformations, noting that equations:

and:

satisfy:

Hence, the following equations of motion:

admit an F-theory zero-mode local model and the Yukawa couplings tree-level superpotential:

includes the trilinear term:

leading to 4-d couplings – given by an integral of the zero mode wavefunctions – among the zero modes of and .

Solving the D-term equation:

we get a description of an F-theory-GUT model in the vicinity of a single point by computing the down-like Yukawa couplings:

or the up-like Yukawa couplings:

The proof, and the structure of the argument, generalizes to F-theory models with with differing , the most interesting cases being:

and

all of which explicitly reflect the ‘no-two-time’ property of F-theory.

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]]>Any adequate account of how micro-causality and quantum coherence can explain the emergent-property of spacetime and how the Wheeler-DeWitt problem of time can be solved must incorporate a theory of how the Lindblad master equation solves the Markov quantum fluctuation problem as well as showing how the quantum Jarzynski-Hatano-Sasa relation can be homologically defined globally for both, Minkowski space and Friedmann-Robertson-Walker generalized Cartan space-times. This is a step towards those goals. Consider a wave-function \(\left| {{\psi _t}^{S,m,c}} \right\rangle \) and the entropic quantum entanglement relation of the total system consisting of 'S', 'm' and the quantum-time measuring clock 'c' subject to Heisenberg's UP. It follows then that the probability that any given initial state \(\left| {\psi _t^{S,m,c}} \right\rangle \) evolves for a time \(T\) that undergoes \(N\) jumps during intervals \(\delta t\) centered at times \({t_1},...,{t_N}\) is given by:

\[\begin{array}{l}{\left( {2\delta t{\kappa ^2}/G} \right)^N}{\rm{Tr}}\left\{ {{e^{ - i{{\tilde H}_{eff}}\left( {T - {t_N}} \right)}}} \right. \cdot \\\hat a{e^{ - i{{\hat H}_{eff}}}}\left( {{t_N} - {t_{N - 1}}} \right)\hat a...\,\hat a{e^{ - i{{\hat H}_{eff}}t}}\\ \times \left| {\psi _t^{S,m,c}} \right\rangle \left\langle {\psi _t^{S,m,c}} \right|{e^{i{{\tilde H}^\dagger }_{eff}{t_1}}}{{\hat a}^\dagger }...\,\left. {{{\hat a}^\dagger }{e^{i{{\tilde H}^\dagger }_{eff}\left( {T - {t_N}} \right)}}} \right\}\end{array}\]

So, the master equation:

\[\begin{array}{l}{{\dot \rho }_{00}} = - i\left[ {{{\hat H}_0},{\rho _{00}}} \right] + \frac{{2{\kappa ^2}}}{G}\hat a{\rho _{00}}{{\hat a}^\dagger }\\ - \frac{{{\kappa ^2}}}{G}{{\hat a}^\dagger }\hat a{\rho _{00}} - \frac{{{\kappa ^2}}}{G}{\rho _{00}}{{\hat a}^\dagger }\hat a\end{array}\]

is valid iff the Markovian approximation is faithful and valid only on time-scales longer than \(1/{\Gamma _1}\), hence the jump occurs during an interval \(\delta t \sim 1/{\Gamma _1}\) centered on \({t_i}\). Therefore, with the Hamiltonian:

\[{\hat H_I} = \kappa \left( {{{\hat a}^\dagger } \otimes \hat b + \hat a \otimes {{\hat b}^\dagger }} \right)\]

where \(\left( {\hat a,\hat b} \right);\left( {{{\hat a}^\dagger },{{\hat b}^\dagger }} \right)\) are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

\[\begin{array}{c}\dot \rho = - i\left[ {\hat H,\rho } \right] + {\Gamma _1}\hat b\rho {{\hat b}^\dagger } - \frac{{{\Gamma _1}}}{2}{{\hat b}^\dagger }\hat b\rho \\ - \frac{{{\Gamma _1}}}{2}\rho {{\hat b}^\dagger }\hat b + {\Gamma _2}{\sigma _z}\rho {\sigma _z} - {\Gamma _2}\rho \\ \equiv L_s^L\rho \end{array}\]

where the Pauli operator \({\sigma _z}\) acts on the output mode and \(L_s^L\) is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

\[\rho ({t_2}) = \exp \left\{ {L_s^L\left( {{t_2} - {t_1}} \right)} \right\}\rho ({t_1})\]

and so the evolution of the density matrix \({\rho _t}\) is given by the Lindblad master equation:

\[\begin{array}{l}{\partial _t}{\rho _t} = - i\left[ {{H_t},{\rho _t}} \right] + \sum\limits_{i = 1}^I {\left( {{V_i}{\rho _t}V_i^\dagger } \right.} \\\left. { - \frac{1}{2}V_i^\dagger {V_i}{\rho _t} - \frac{1}{2}{\rho _t}V_i^\dagger {V_i}} \right)\end{array}\]

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So, the master equation:

is valid *iff* the Markovian approximation is faithful and valid only on time-scales longer than , hence the jump occurs during an interval centered on . Therefore, with the Hamiltonian:

where are the lowering/raising operators for the system and output mode respectively, it follows that the total system satisfies the master equation:

where the Pauli operator acts on the output mode and is the Liouville superoperator. Given that it is a linear equation, it has a solution given as:

and so the evolution of the density matrix is given by the Lindblad master equation:

where

is the conservative part and is the time-dependent Hamiltonian of the system and the other terms refer to the bath of the interactive system and reflect the effect of measurements, and are the Kraus-operators, not necessarily hermitians and are typically explicitly dependent on time. The Kraus number depends on the bath. In the case where the system is a closed one, the Kraus operators vanish identically and the Lindblad master equation reduces to the quantum version of the Liouville equation, giving us:

with the Lindbladian superoperator acting on the density matrix and determines its dynamics. The associated space of operators is equipped with a Hilbert-Schmidt scalar product:

with the hermitian conjugate of . We now define a pair of adjoint superoperators and as follows:

Hence, we have:

with the trace-conservation property:

The solve quantum Master equation:

one typically introduces an evolution superoperator defined implicitly by:

where is the initial-time-density-matrix, and the superoperator evolution is given by:

And in this time-ordered exponential, time is monotonically increasing from left to right.

To prove:

note that it is true at since is the identity operator. Thus, from:

one finds that:

holds, and leads to:

entailing that it satisfies the Lindblad equation:

with initial condition . Now, for the evolution operator, one writes an expression for multi-time correlations for distinct observables. For:

the time-ordered correlation is:

and can be evaluated in the Heisenberg representation formalism by using the full Hamiltonian of the system plus its environment. Since the total density matrix factorizes at each observation time and the weak Lindblad Master equation coupling assumption holds in that formalism, the time-ordered two-time correlation function satisfies an evolution equation which is the dual to:

our proof is complete.

Now note that in:

the operator represents the initial density matrix of the system and the superoperator acts on all terms to its right.

Thus, we have the crucial equation:

which for systems prepared in a thermal state at:

with the Boltzmann-constant, we have:

For closed systems, one has:

which holds for any Markovian process weakly coupled with a thermal bath at provided the bath satisfies the KMS condition. Here, we shall consider generally, far from equilibrium cases, where is not given by the canonical Gibbs-Boltzmann formula.

**Deriving the Jarzynski-Hatano-Sasa identity for quantum Markovian dynamics**

Even though the density-matrix does not obey the Lindblad equation, it is a solution of the deformation-evolution equation:

Now, let us define non-stationarity via the operator:

Define the modified superoperator as such:

where acts by multiplication on the left. Such a superoperator corresponds to the auxiliary dynamics:

and yields a modified evolution superoperator via:

Given:

we can derive:

with solution:

Now, for any observable , such a solution gives us:

One can derive a quantum variant of the Jarzynski-Hatano-Sasa relation by connecting the auxiliary evolution superoperator to the initial evolution superoperator . In order to do that, we need to prove an extension of the Feynman-Kac formula: write the Dyson-Schwinger expansion of , with a perturbation of the Lindbladian :

where acts on all the terms to its right. Now, insert the Dyson-Schwinger expansion into the r.h.s. of:

and we get:

Reformulate the trace within the scope of the integrals as a multi-time correlation via:

and we get:

and by linearity and the relation , we get a reduction to:

where the terms inside the brackets are summable as a time-ordered exponential:

*This is an extension of the Feynman-Kac formula for quantum Markov semi-groups.*

and by non-commutativity of the operator algebra, the Feynman-Kac exponential is replaced by a time-ordered exponential. Hence, one gets:

and is a quantum extension of the classical Jarzynski-Hatano-Sasa identity

If we set , the above identity reduces to:

given that holds, and is a quantum measurement number-raising and book-keeping formula for correlation functions.

Now, from a first order expansion of:

we can deduce a generalized fluctuation-dissipation theorem valid in the Heisenberg-vicinity of a quantum non-equilibrium steady state.

The case of a closed isolated system determined by a time-dependent Hamiltonian, the Lindbladian reduces to the Liouville operator:

with unitary evolution. For a closed system, the evolution superoperator acts on observables as follows:

with:

The image of X under the superoperator operator defines the Heisenberg operator with representing the Heisenberg operator:

Since the superoperator is multiplicative, the r.h.s. of:

for multi-time correlations can be evaluated and one gets:

Hence, for a closed system the quantum Jarzynski-Hatano-Sasa relation is:

From multiplicativity and:

we have:

and given that we have , we can derive:

Now, from , we get the Kurchan-Tasaki quantum Jarzynski relation for closed systems. Moreover, since we have the commutation relation:

we can derive:

and for the critical case where holds, the Hänggi-Talkner quantum-Jarzynski relation for closed systems reduces to:

For open systems, which are of more foundational interest, it follows from:

that the generalized fluctuation-dissipation theorem is valid in the vicinity of any quantum non-equilibrium steady state. To see that, take a perturbation of the Lindbladian of the form:

with

time-dependent perturbations. The density matrix satisfying:

is given by:

with satisfying:

and the non-stationary operator becomes:

By differentiating, we get:

where:

is taken with respect to the unperturbed density matrix .

*Lindbladian time-reversal dynamics*

Time reversal on the states of the Hilbert space in quantum mechanics is implemented by an anti-linear anti-unitary operator satisfying:

for spin-0 particles without a magnetic field, is the complex conjugation operator: that is, by time reversal, the Schrödinger wave-function becomes . In the scenario where there is a magnetic field, time-inversion must be augmented by requiring that the reversed system evolves with vector potential . Time reversal of Hilbert space observables is hence implemented by a superoperator that acts on an operator as such:

Hence, as promised, is multiplicative, anti-unitary, and satisfies:

We are finally in a position to define time-reversal for a quantum Markov process. Take a constant Lindbladian lying in a steady state with density-matrix . Note that the superoperator that determines the reversed process is given by:

and the micro-reversibility condition is:

yielding the finite-time formula:

which, given two arbitrary observables , , is equivalent to:

Hence, we can see that the stationary density matrix associated with the time-reversed dynamics is given by:

given that:

We have therefore the following crucial Lindbladian:

From this Lindbladian equation and , we obtain:

and:

thus connecting the Lindbladian distribution of the time-reversed system with that of the classical system.

Now, applying:

to the time-reversed system, we find that the evolution superoperator of the time-reversed system is given by:

with multi-time correlations:

Continuing with our proof, let be a scalar such that and define two -deformed superoperators, that act on an observable X as follows:

Def.:

and:

Now, the superoperators interpolate between and when varies from to . Likewise, is an interpolation from to . The corresponding -deformed evolution superoperators are given by:

and:

Crucially, they satisfy the following duality relation that lies at the heart of the quantum fluctuation theorem:

hence, we can derive the following for the unitary operator:

and satisfies:

So we can write the operator as:

Using the duality above to any pair of observables and , and the multiplicative property and anti-unitary of , our proof is finalized by the following relation:

This is precisely the equation that axiomatically captures the essence of the quantum fluctuation theorem that explains quantum decoherence and undergirds the solution to the measurement problem, up to a Yukawa-Higgs S-matrix coupling constant. To see this, express it in terms of quantum density-matrix stochasticity via the expectation-operator:

with the asymmetric reversed operator given by:

and is closed with Hamiltonian

and the evolution operator satisfies:

by the Boltzmann law, the above equation in the Heisenberg representation is:

whose set of solutions is the set of solutions to the quantum decoherence equation describing a wave-function collapse.

Bonus:

This is a conditional proof of the reality of the wave-function

The post The Lindblad Master Equation, Feynman-Kac Formula, and the Measurement Problem appeared first on George Shiber.

]]>I will derive a crucial property of loop quantum cosmology it shares with string/M-theory and asymptotically free quantum gravity theory, namely, that the associated Wigner-Moyal-Groenewold operator-formalism entails that the Holst-Barbero-Immirzi 4-spinfold has the property of spacetime uncertainty that I derived for string/M-theory, an essential property if loop quantum gravity is to be a valid quantum gravity theory. As I showed, in 4-D spacetime, the general relativistic starting point for canonical loop quantum gravity is given by:

\[\begin{array}{l}{S_{4{\rm{D}}}}\left[ {e',\omega } \right] = \int_{\tilde M} {\left( {\frac{1}{2}} \right.} {\rm{tr}}\left( {e \wedge e \wedge F} \right)\\\left. { + \frac{1}{\gamma }{\rm{tr}}\left( {e \wedge e \wedge * F} \right)} \right)\end{array}\]

where the dynamical variables are the tetrad one-form fields:

\[{e^I} = e_\mu ^I{\rm{d}}{x^\mu }\]

and the \(SL\left( {2,\mathbb{C}} \right)\)-valued connection \(\omega _\mu ^{IJ}\) whose curvature is:

\[F = {\rm{d}}\omega + \omega \wedge '\omega \]

and is a connection on the holonomy-flux algebra for a homogeneous isotropic Friedmann–Lemaître–Robertson–Walker 'space'

Hence, we have the two-form:

\[\begin{array}{l}{F^{IJ}} = \left( {{{\not \partial }_\mu }} \right.\omega _\nu ^{IJ} - {{\not \partial }_\nu }\omega _\mu ^{IJ} + \omega _\mu ^{IK}{\omega _\nu }{K^J}\\\left. { - \omega _\nu ^{IK}{\omega _\mu }{K^J}} \right){\rm{d}}{x^\mu } \wedge '{\rm{d}}{x^\nu }\end{array}\]

with:

\[ * {F^{IJ}} = \frac{1}{2}{\varepsilon ^{IJ}}_{KL}{F^{KL}}\]

and \({\rm{Tr}}\) is the Killing form on the Lie algebra \(SL\left( {2,\mathbb{C}} \right)\):

\[{\rm{Tr}}\left( {e \wedge e \wedge F} \right) = {\varepsilon _{IJKL}}{e^I} \wedge {e^J}{F^{KL}}\]

with

\[{\varepsilon _{IJKL}}\]

the totally antisymmetric tensor given by:

\[{\varepsilon ^{0123}} = + 1\]

Now, I can write down the Holst action more informatively:

\[\begin{array}{*{20}{l}}{{S_{4D}}\left[ {e,\omega } \right] = \int_{{{\tilde M}_4}} {{{\rm{d}}^4}} x{\varepsilon ^{\mu \nu \rho \sigma }}\left( {\frac{1}{2}} \right.{\varepsilon _{IJKL}}}\\{e_\mu ^Ie_\nu ^JF_{\rho \sigma }^{KL}\left. { + \frac{1}{\gamma }e_\mu ^Ie_\nu ^J{F_{\rho \sigma }}_{IJ}} \right)}\end{array}\]

and from the Ashtekar variables, our action is:

\[{{S_H} = \int {{d^3}} x\left\{ {{{\tilde E}^a}_B\dot A_a^B - \frac{1}{2}{\omega _{aBC}}{\varepsilon ^{BCD}}{t^a}{G_D} - {N^a}{C_a} - NH} \right\}}\]

\[{\left\{ {A_a^B\left( x \right),\tilde E_A^b\left( y \right)} \right\} = \delta _a^b\delta _A^B\delta \left( {x,y} \right)}\]

The post Loop Quantum Cosmology and the Wigner-Moyal-Groenewold Phase Space appeared first on George Shiber.

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where the dynamical variables are the tetrad one-form fields:

and the -valued connection whose curvature is:

and is a connection on the holonomy-flux algebra for a homogeneous isotropic Friedmann–Lemaître–Robertson–Walker ‘space’

Hence, we have the two-form:

with:

and is the Killing form on the Lie algebra :

with

the totally antisymmetric tensor given by:

Now, I can write down the Holst action more informatively:

and from the Ashtekar variables, our action is:

with the Gaussian constraint:

the diffeomorphism constraint:

and our Hamiltonian is given by:

The LQC Wigner-Moyal-Groenewold operator is the unique operator with the following properties:

the LQC Holst-cylindrical functions and the LQC characteristic function, the same as the Fourier transform of the the quasi probability density function of the group characters.

It immediately follows from Fourier phase space symplecticity that the LQC Wigner-Moyal-Groenewold operator satisfies the following relation:

and for the LQC characteristic function, we have:

noting that any connection is gauge and diffeomorphism invariant in homogeneous isotropic space.

We can now define the holonomy-flux algebra for homogeneous isotropic Friedmann–Lemaître–Robertson–Walker space model via:

where the holonomy and the flux operators act as:

The Hilbert space basis is given by the connection-lifter LQG spin-networks:

with the configuration variable corresponding to the connection, the number of the Fourier fiducial cell repetition, and satisfy:

and:

with a constant satisfying:

Let us derive now the Wigner function and show that it satisfies the property that when integrated by one variable it reduces to the distribution density of the other variable. Define it as:

with:

For the distribution density function to be definable, the mutual quasi distribution function of and the following two equalities should be true:

Hence, when integrating with respect to one variable it becomes the distribution density of the other one. The above equalities hold since our measures and satisfy:

and:

with the Bohr dual space and a Kronecker delta.

Now, the characters of the compactified line are the functions , hence the Fourier transform of the function on is given by:

which is an isomorphism of:

and comprise the basis of:

We need to prove the above equalities. First, substitute the expression:

of and the expression:

for into:

giving us:

with , and since integration with respect to is just a sum as is discrete, we have:

Now, using:

and integrating with respect to , we can derive:

Given , it follows that summation by makes the terms with equal to zero and the terms with equal to one and all terms with and vanish from the sum. Hence, by using:

we derive:

and to prove the equality:

we substitute the expression:

of and the expression:

for into it, yielding:

Now, integration with measure given:

yields:

Hence, only the terms with remain in:

and since integration with respect to gives us:

And, from the last two equalities, it follows that:

then after substituting it into:

we can derive:

Hence, the integrals with respect to and are equal to one, yielding:

So, from:

and:

it follows that is a LQC Wigner function in variables , , completing the proof.

Next, we need to prove that the first momentum has the following property:

with . Start with substituting the red area:

into the left-hand-side of:

yielding:

By using the red-area expression below

for , we can deduce:

repeating the step above, integrating by can be replaced by summing over :

and integrating by yields:

thus since the sum over equals one, we have:

substituting into it and using the definition of the LQC cylindrical functions:

we get:

Proving that the first momentum has the desired property:

Now, we need to show that the LQC Wigner-Moyal-Groenewold operator has the following property:

We substitute from:

to obtain:

and by utilizing from our definition:

it reduces to:

The same integration rules applied above go through now as well, yielding:

Combining the terms in the exponents and using the LQC cylindrical function above, we have:

Hence, we derived the LQC characteristic function as a Fourier transform of :

Now, we must prove that the following operator:

is a Wigner-Moyal-Groenewold operator, where:

We start by substituting

into:

yielding:

and expanding the exponents into the Taylor series allows us to derive:

amounting to a proof that:

is a Wigner-Moyal-Groenewold operator for a homogeneous and isotropic space whose connection-form is gauge and diffeomorphism invariant, and by the symplecticity of the associated LQG holonomy-flux algebra, the Holst-Barbero-Immirzi 4-spinfold has the property of space-time uncertainty:

with:

The post Loop Quantum Cosmology and the Wigner-Moyal-Groenewold Phase Space appeared first on George Shiber.

]]>Among the many truly remarkable properties of M-theory, that it is a unified theory of all interactions, including quantum gravity, and gives a completely well-defined analytic S-matrix satisfying all the axioms for a physically acceptable theory entailing Lorentz invariance, macro-causality and unitarity is perhaps the deepest, and to boot, the only quantum gravity paradigm that has that essential feature. Here, I will discuss some key aspects of nonlocality and space-time uncertainty in string theory. Let us start with an action smoothly interpolating between the area preserving Schild action and the fully reparametrization invariant Nambu–Goto action:

\[I\left[ {\Phi ,X} \right] \equiv \frac{{{\mu _0}}}{2}\int\limits_\Sigma {{d^2}} \sigma \left[ {\frac{{\det \left( {{\gamma _{mn}}} \right)}}{{\Phi \left( \sigma \right)}} + \Phi \left( \sigma \right)} \right]\]

where \(\Phi \left( \sigma \right)\) is an auxiliary world-sheet field, \({\gamma _{mn}} \equiv {\eta _{\mu \nu }}{\partial _m}{X^\mu }{\partial _n}{X^\nu }\) the induced metric on the string Euclidean world-sheet \({x^\mu } = {X^\mu }\left( \sigma \right)\), and \({\mu _0} \equiv 1/2\pi \alpha '\) is the string tension. Combining, we get the Nambu-Goto-Schild action:

\[{S_{ngs}} = - \int\limits_\Sigma {{d^2}} \xi \left\{ {\frac{1}{e}\left[ { - \frac{1}{{2{{\left( {4\pi \alpha '} \right)}^2}}}{{\left( {{\varepsilon ^{ab}}{\partial _a}{X^\mu }{\partial _b}{X^\nu }} \right)}^2}} \right] + e} \right\}\]

And to make the Nambu-Goto-Schild action quadratic in space-time coordinates, we use the Virasoro constraint and an auxiliary field that transforms as a world-sheet scalar and as an anti-symmetric tensor with respect to the space-time indices:

\[\left\{ {\begin{array}{*{20}{c}}{{b_{\mu \nu }}\left( \xi \right)}\\{{P^2} + \frac{1}{{4\pi \alpha '}}{{\hat X}^2} = 0,\;P \cdot \hat X = 0}\end{array}} \right.\]

to yield:

Before proceeding, let us get some clarity.

The post Space-Time Uncertainty and Non-Locality in String-Theory appeared first on George Shiber.

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where is an auxiliary world-sheet field, the induced metric on the string Euclidean world-sheet , and is the string tension. Combining, we get the Nambu-Goto-Schild action:

And to make the Nambu-Goto-Schild action quadratic in space-time coordinates, we use the Virasoro constraint and an auxiliary field that transforms as a world-sheet scalar and as an anti-symmetric tensor with respect to the space-time indices:

to yield:

Before proceeding, let us get some clarity.

Recalling the relations:

and the world-manifold Poisson Bracket:

Hence, the action:

is reparametrization invariant only if the auxiliary field transforms as a world-sheet scalar density:

By implementing reparametrization invariance, can be transformed to unity and the Schild action can be recovered as a gauge-fixed form:

Thus, by solving in terms of from:

one recovers, on-shell, the Nambu–Goto action:

The inverse equivalence relation can be deduced by starting from the Schild action:

and we lift the world–sheet coordinates to the role of dynamical variables via the reparametrization :

By a -variation, we get the field equation:

Hence:

which allows us to regain the Nambu–Goto action.

**Space-Time Uncertainty**

Note that the fully reparametrization invariant Nambu–Goto action:

is a special case of the general two-parameter family of p-brane actions:

Thus, the key notion is the geometric structure of the p-brane world volume and string world-sheet topological embedding. The whole notion then, given Witten’s results on supersymmetric quantum mechanics, for space-time uncertainty relation comes from a simple analogy concerning the nature of string quantum mechanics. The central necessary condition of string perturbation theory is world-sheet conformal invariance, and one of the key insights of string theory as a unified theory is due to conformal invariance. The elimination of the ultraviolet divergences in the presence of gravity is essentially due to modular invariance, and that is part of conformal symmetry. From the viewpoint of generic two-dimensional field theory, conformal invariance forces us to choose a very narrow class of all possible two-dimensional field theories corresponding to the fixed points of the Wilsonian renormalization group. In the final formulation of quantum mechanics, the quantization condition is replaced by the more universal framework of Hilbert spaces and the corresponding operator algebras representations. This analogy suggests the importance of reinterpreting the conformal invariance requirement by promoting it to a universal form that ultimately can be formulated in a way that does not depend on perturbative methods.

Modular invariance can be expressed as the string-reciprocity-relation of the extremal length which is a conformally invariant notion of length corresponding to families of curves on Riemann surfaces. If we take some finite region and a set of arcs on , the extremal length of is defined by:

in the conformal gauge. Since any Riemann surface can be composed of a set of quadrilaterals pasted along the boundaries, it is sufficient to consider the extremal length for an arbitrary quadrilateral segment . Let the two pairs of opposite sides of be . Take to be the set of all connected sets of arcs joining . The set of arcs joining is the conjugate set of arcs, denoted by . Hence we have two extremal lengths, and : then the reciprocity relation is given by:

To appreciate how the reciprocity of the extremal length reflects target space-time, consider the Polyakov amplitude for the mapping from the rectangle on a Riemann surface to a rectangular region in space-time with the side lengths with the boundary condition:

Then the amplitude contains the factor:

multiplied by a power pre-factor. Hence, the quantum fluctuations of two space-time lengths satisfy an uncertainty relation:

Recalling that all and the only legitimate observables allowed in string theory is the on-shell S-matrix. So, it is natural take the above UR as setting an absolute limit, in some averaged sense, on the measurability of space-time lengths in string theory, since conformal invariance must be valid to all orders of string perturbation theory and the random nature of boundaries generally contributes to further fuzziness on the space-time lengths. This reciprocity relation reflects one of the most fundamental duality relations in string-amplitudes between ultraviolet and infrared structures. Since in the Minkowski metric one of the lengths is always dominantly time-like, the following uncertainty relation on the space-time lengths:

is a universal characterization of the short-distance space-time structure of string theory.

**D-branes and Yang-Mills theories**

More mathematical evidence for the validity of the above UR in string theory derives from its effectiveness for D-branes. Noting that effective Yang-Mills theories for the low-velocity D-p-branes predict that the characteristic spatial transverse to D-p-branes and temporal scales of D-p-brane scattering oppositely scale with respect to the string coupling, that is:

And though the case p = 3 is special in that the effective Yang-Mills theory is conformally invariant, the conformal transformation property actually necessitates the above space-time string theory uncertainty relation. One can easily derive the above characteristic scales in a Yang-Mills free-way. Recall that the characteristic scales:

of D-particle-D-particle scattering are a direct consequence of the space-time uncertainty relation and the quantum mechanical Heisenberg relation, given the fact that the mass of the D-particle is proportional to . All these properties are natural from the viewpoint of open string theories where the uncertainty relation (STUR):

must be valid.

A deep philosophical question is that the STUR is independent of the string coupling , it appears at first that it does not take into account gravity. So the natural question is: what is its relation to the Planck scale which is the characteristic scale of quantum gravity? In string theory, the existence of gravity is a crucial consequence of the string world-sheet conformal invariance. This is due to the possibility of deforming the background space-time by a linearized gravitational wave. However, in perturbation theory, the coupling strength of the gravitational wave is an independent parameter determined by the vacuum expectation value of the dilaton. Thus, the string coupling cannot be a fundamental constant which appears in the universal non-perturbative property of string theory. So in order to take into account the Planck length for the space-time uncertainty relation (STUR), one must put that information by hand. However, I will show that by combining the Planck scale with the space-time uncertainly relation (STUR), one can derive the M-theory scale without invoking D-branes or membranes at all.

We start by reinterpreting the meaning of the Planck length using the stringy space-time uncertainties by considering the limitation of the notion of classical space-time as the background against the possible formation of virtual black holes in the short-distance regime. To probe the space-time structure in the time direction to order , the Heisenberg uncertainty relation implies that the uncertainty with respect to the energy of order is necessarily induced. Further requiring that the structure of the background space-time is not influenced by this amount of fluctuation, then the spatial scale to be probed cannot be smaller than the Schwarzschild radius associated with the energy fluctuation. Hence we have:

in D-spacetime dimensions, setting the black-hole uncertainty relation for the characteristic gravitational scales in the form:

in D = 10 dimensional string theory, which reflects limitations only for observers at asymptotic infinity with respect to spatial and temporal resolutions below which the classical space-time structure without the formation of micro-black-holes can no longer be applied. In contrast to this, the space-time uncertainty relation sets the more fundamental limitation below which the space-time geometry itself loses its meaning. Thus, the most important characteristic scale associated with the existence of gravitation in string theory corresponds to the point of their crossover. The critical crossover scales are determined by:

And what is truly miraculous is that the spatial critical scale coincides exactly with the M-theory scale

**Space-Time Noncommutativity**

The validity of the uncertainty relation (STUR):

entails the existence of a noncommutative space-time structure underlying string theory. In fact, the relation:

is a string-theoretic version of the Wigner representation:

of the density matrix corresponding to the Gaussian wave packet in ‘particle’ quantum mechanics, implying that the space-time of string theory is quasi-morphic to the classical phase space in particle quantum mechanics!

Here’s how to exhibit the space-time noncommutativity of string quantum mechanics in a manifest way.

We start from a modified version of the Nambu-Goto-Schild action:

Note that the conformal invariance of string theory is now hidden in the standard Virasoro constraint:

and hence does not explicitly involve the string world-sheet auxiliary field and the world-sheet metric. It is crucial also to note that the Hamiltonian constraint comes from the following equation:

for the auxiliary field . Moreover, causality in string theory is embodied in the time-like nature of the area-element:

Hence, to reformulate our action such that it becomes quadratic in the space-time coordinates, one introduces another auxiliary field that transforms as a world-sheet scalar and simultaneously as an antisymmetric tensor with respect to the space-time indices:

allowing us to derive:

which plays the role of Lagrange multiplier for the causality condition in string-theory: that is, the requirement of conformal invariance is thus essentially reinterpreted as the condition that the world-sheet field be time-like. We then quantize this action by regarding the -field as an external field, and given that the action is first-order with respect to the world-sheet time derivative, the system has second class constraints:

implying that the space and the time become manifestly noncommutative as evidenced by the fact that the center-of-mass time:

and the spatial extension defined by:

satisfy:

That the expression:

is interpretable as the measure of spatial extension of strings can be demonstrated by recalling that in the semi-classical approximation the -field is just proportional to the area element of the world-sheet of strings:

and is derivable by taking the variation of the action with respect to the -field.

We have thus reformulated string theory in such a way that the noncommutativity between spatial extension and time is manifest and naturally conforms to the general property:

of space-time uncertainty relation derived on the basis of the string world-sheet conformal symmetry, and a paradigmatic Moyal-product argument in light of the fully reparametrization invariant Nambu–Goto action:

entails that space-time uncertainty and noncommutativity are a special case of Witten-deformation from classical space-time geometry to quantum and stringy geometry, implying non-locality. Finding a unique characterization of a stringy Witten-deformation of space-time geometry is however extremely mathematically challenging.

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]]>Multi-scalar field cosmology is essential for solving the Wheeler-DeWitt equation in the context of quantum gravity. Here, I will test MFI with supersymmetric quantum mechanics based on Witten's axiomatic approach. One can axiomatize multi-scalar field theory by the following conditions: 1) a Lagrangian containing up to second order derivatives of the fields, and 2) field equations that contain up to second order derivatives of the fields obeying:

with:

\[{{{\bar X}_{ij}} = \frac{1}{2}{\partial _a}{\pi _i}{\partial ^a}{\pi _j}}\]

and \(\frac{{\partial {A^{{i_1}...{i_m}}}}}{{\partial {X_{kl}}}}\) symmetric in all of its indices \({i_1}...{i_m},k,l\)

With the multi-field action in D dimensions having the form:

\[S = \int {{d^D}} x\hat L\left( {{\pi _i},{\partial _a}{\pi _j},{\partial _b}{\partial _c}{\pi _k}} \right)\]

whose Euler-Lagrange equations are given by:

\[\frac{{\partial \hat L}}{{\partial {\pi _i}}} - {\partial _a}\left( {\frac{{\partial \hat L}}{{\partial {\pi _{ia}}}}} \right) + {\partial _a}{\partial _b}\left( {\frac{{\partial \hat L}}{{\partial {\pi _{iab}}}}} \right) = 0\]

with a fourth derivatives constraint:

\[\frac{{\partial \hat L}}{{\partial {\pi _{icd}}\partial {\pi _{iab}}}}{\pi _{i,abcd}}\]

Thus, the universal multi-field action is:

for the multi-fields \(\left( {\sigma ,\phi } \right)\), hence the corresponding field equations:

\[\begin{array}{l}{G_{\alpha \beta }} + {g_{\alpha \beta }}\Lambda = + \frac{1}{2}\left( {{\nabla _\alpha }\phi {\nabla _\beta }\phi - \frac{1}{2}{g_{\alpha \beta }}{g^{\mu \nu }}{\nabla _\mu }\phi {\nabla _\nu }\phi } \right)\\ + \frac{1}{2}\left( {{\nabla _\alpha }\sigma {\nabla _\beta }\sigma - \frac{1}{2}{g_{\alpha \beta }}{g^{\mu \nu }}{\nabla _\mu }\sigma {\nabla _\nu }\sigma } \right)\\ - \frac{1}{2}{g_{\alpha \beta }}V\left( {\phi ,\sigma } \right) - 8\pi G{{\rm T}_{\alpha \beta }}\end{array}\]

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]]>with:

and symmetric in all of its indices

With the multi-field action in D dimensions having the form:

whose Euler-Lagrange equations are given by:

with a fourth derivatives constraint:

Thus, the universal multi-field action is:

for the multi-fields , hence the corresponding field equations:

with:

with:

the energy density, the pressure, and the velocity, satisfying .

The multi-scalar field cosmological paradigm requires the two canonical fields , the action of a universe based on such fields, the cosmological term contribution, and matter as a perfect fluid content, and is given by:

Our metric has the form:

with a 3 x 3 diagonal matrix,

and is a scalar and are one-forms that characterize each cosmological Bianchi type model, and obey the form:

and are structure constants of the corresponding model. Hence, in Misner’s parametrization, we get:

with the anisotropic conditions:

So the lagrangian density above can be written as:

with overdot denotes time derivative, with the re-scaling:

And the momenta are defined as:

and:

Thus, our Hamiltonian density is given by:

and by use of the above covariant derivative, we have:

with:

and:

and the density solution:

**The Wheeler-DeWitt equation**

Hence, our first approximation of the Wheeler-DeWitt equation is:

with:

the d’Alambertian in the coordinates:

with the potential that couples to the wave-function and gives the whole quantum dynamics by the following equation:

with:

Using the following ansatz for the wavefunction:

Hence, our Wheeler–DeWitt equation equation is:

with:

and:

where is the superpotential function, and is the probability amplitude.

Let us now utilize the mathematics of supersymmetric quantum mechanics to probe the Wheeler–DeWitt equation and the superpotential via Witten’s formalism of finding the supersymmetric supercharges operators and that produce a super-Hamiltonian , where the Wheeler–DeWitt equation can be derived as the bosonic sector of this super-Hamiltonian in the superspace. The right method to supersymmetrize a bosonic Lagrangian is to consider the true supersymmetry transformation in the superfield scheme into the bosonic Lagrangian, then the fermionic terms will emerge in a natural way.

In this Witten-method, our supercharges for the 3-D case are:

where is defined implicitly by the following equation:

and the super-algebra for the variables is given by:

Under the representation:

the superspace Hamiltonian takes the form:

with:

being the standard Wheeler–DeWitt equation, the 3-D d’Alambertian in the coordinates with , and and represent the anticommutator and the commutator respectively. The supercharges and the super-Hamiltonian satisfy the following algebra:

Hence, our supersymmetric physical states are selected by the constraints:

which reduces the problem of finding supersymmetric ground states because the energy is known a priori and the factorization of:

into

yields a first-order equation for the ground state wave-function due to the sovability of the bosonic Hamiltonians and normalization just means that supersymmetry is quantum mechanically unbroken.

In the 3-D Grassmannian variable-representation, the wave-function has the following decomposition:

and with the ansatz:

introduced into

and

where is the superpotential function obtained as a solution for the Einstein-Hamilton-Jacobi equation, the following identity:

yields the master equation for the auxiliary function :

with:

and with the following ansatz:

we get the second master equation in the form:

allowing us to get the reduction to:

and the equations for the functions are:

with solutions:

with the integration constants.

To solve our equation:

we need to write it as a homogeneous linear equation of second degree:

and we do this by introducing into it the ansatz:

This way, we obtain a wave-like equation:

with:

and:

The following wave-like ansatz:

suffices to solve, yielding a condition on the function:

where the following conditions hold:

Allowing us to construct the following term:

satisfying:

Now, for:

we must consider the two cases, one with taking the constant into account and one without.

For and with our superpotential. In this situation,

gives the following equation:

with vector solutions:

For and with our superpotential. In this situation, we need to separate the following two independent equations:

where:

entails that is a vector of null measure, and:

entails:

Now, when we use the superpotential function:

we get the following structural relations:

for , where the constants are:

Hence, supersymmetric quantum mechanics puts the exact constraints on the family of potential fields corresponding to the inflaton exponential Hubble-Fredholm integral.

In the scenario where both equations have no null solution, the solution for the function has the following structure:

thus and reduce to:

Our method above was used to obtain supersymmetric quantum solutions for all cosmological bianchi class A models in Sáez-Ballester theory.

From our superpotential function above, it follows that the only form for in which our equations are fulfilled, is when the functions and have exponential behavior. So in a supersymmetric way, the calculation by means of the Grassmannian variables of given by:

is:

where is implicitly defined by:

with the standard algebra for the Grassmannian numbers . The integration rules over these numbers are given by:

with:

and we get:

Thus, Grassmannian integration yields:

thus supersymmetric quantum mechanics yields the required probability density:

giving us a supersymmetric quantum canonical quantization of the multi-scalar field cosmology of the anisotropic Bianchi type I model and the exact supersymmetric quantum solutions to the Wheeler-DeWitt equation are derived under the ansatz to the wave function:

which is central for solving the Einstein-Hamilton-Jacobi equation.

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