\[\begin{array}{c}S = \frac{1}{{16\pi }}\int {{d^4}} x\sqrt { - g} \left\{ {\Phi \Re + } \right.\\\left. {\Phi \Re + \frac{{\tilde \omega \left( \Phi \right)}}{\Phi }{g^{\mu \nu }}\Phi {,_\mu }\Phi {,_\nu } - \tilde V\left( \Phi \right)} \right\}\end{array}\]

\(\Re \) the Ricci scalar, \({\tilde \omega \left( \Phi \right)}\) a function of the scalar field \(\Phi \) and \({\tilde V\left( \Phi \right)}\) is a scalar potential. The action in terms of the redefined field \(\varphi = - {\rm{In}}\left( {G\Phi } \right)\) reduces to:

\[S = \int {{d^4}} x\sqrt { - g} \left\{ {{e^{ - \varphi }}\left[ {\frac{\Re }{{16\pi G}} + \omega \left( \varphi \right){g^{\mu \nu }}{\varphi _{,\mu }}{\varphi _{,\nu }}} \right] - V\left( \varphi \right)} \right\}\]

Recalling that the Weyl-integrable spacetime action in this frame is given by:

\[S = \int {{d^4}} x\sqrt { - g} \left( {\Re + \xi {\nabla ^w}_\alpha {\phi ^\alpha } + {e^{ - 2\phi }}{L_m}} \right)\]

with the Weylian connection and Riemannian curvature satisfying:

\[{\nabla ^w}_\alpha {\phi ^{,\alpha }} = {\phi ^{,\alpha .}}_{,\alpha } - 2{\phi ^\alpha }{\phi _\alpha }\]

\[\Re = \bar \Re - 3{\diamondsuit ^{d'Ale}}\phi + \frac{3}{2}{\phi _\alpha }{\phi ^\alpha }\]

as well as the Palatini variational properties of \(\omega \left( \varphi \right)\) and \(V\left( \varphi \right)\):

\[\left\{ {\begin{array}{*{20}{c}}{\omega \left( \varphi \right) = {{\left( {16\pi G} \right)}^{ - 1}}\tilde \omega \left[ {\varphi \left( \Phi \right)} \right]}\\{V\left( \varphi \right) = {{\left( {16\pi G} \right)}^{ - 1}}\tilde V\left( {\varphi \left( \Phi \right)} \right)}\end{array}} \right.\]

thus yielding the affine connection:

\[{\nabla ^w}_\mu {g_{\alpha \beta }} = {\varphi _{,\mu }}{g_{\alpha \beta }}\]

which is the characteristic non-metricity condition for a Weyl-integrable geometry

The post Weyl-Integrable Geometry, General Relativity, and the Scalar-Tensor Duality appeared first on George Shiber.

]]>

the Ricci scalar, a function of the scalar field and is a scalar potential. The action in terms of the redefined field reduces to:

Recalling that the Weyl-integrable spacetime action in this frame is given by:

with the Weylian connection and Riemannian curvature satisfying:

as well as the Palatini variational properties of and :

thus yielding the affine connection:

which is the characteristic non-metricity condition for a Weyl-integrable geometry and the Weyl connection is invariant under the transformations:

and:

with an analytical function of the spacetime coordinates. Crucial to note, we have a strengthened conformal symmetry since the action:

is invariant under the diffeomorfism group action as well as under the above Weyl transformations. The Weyl transformations of the kinetic term in the action above yield:

which implies that our action is not invariant under Weyl transformations. To rectify, we introduce the gauge covariant derivative:

with the gauge vector field and the Yukawa coupling constant. Hence, an action for a scalar-tensor theory of gravity on Weyl-integrable geometrical background is given by:

and by the Weyl transformations, the action requires that the vector field , , and the scalar potential obey the following transformation rules:

and

The dynamics of the gauge vector field allows a modification of the action:

as such:

with the field strength of the gauge boson field being . Hence, our WIG field equations are given by:

and the energy-momentum tensor associated to the gauge field is:

with the Einstein tensor computed with the Weyl-integrable connection and the Weylian D’Alambertian operator. Taking the trace and solving our WIG field equations, we get:

Now, introducing the redefined vector field , the Weyl transformations become:

hence, reformulating the kinetic term in the action:

and factoring the above Weyl transformations, one finds terms of the form: , thus it follows that our field is massive for:

and so when the background geometry is the Weyl-integrable one, it can be interpreted as a massive vector field describing massive photons in a Brans-Dicke scalar-tensor theory of gravity.

Now recall that the non-metricity condition:

is invariant under the Weyl transformations:

and:

which entails that we can interpret the Weyl transformations as a Weyl-Frame scalar/tensor bi-mapping:

One has the structure of a differential manifold endowed with a metric tensor and the other as one with an affine connection and a scalar field such that the Weyl-integrable spacetime geometry is satisfied.

Now let us consider the central and interesting case where the non-metricity condition for a Weyl-integrable geometry:

is diffeomorphically equivalent to the Riemann metricity condition:

and 'name' the corresponding structure the Einstein-Riemann frame. Notable in this frame is that geodesics are Weyl invariant. Hence, in the Einstein-Riemann frame, our action **A** reduces to:

with:

and is the Riemannian covariant derivative, and . The following transformation:

in the Einstein-Riemann frame reduces to:

given our **ER** action and that the background spacetime geometry is Riemannian, a Weyl scalar field satisfies the symmetry: , thus the gauge vector field obeys the following transformation rule:

with: . Hence, the transformation above entails that the gauge field can be interpreted as a massless gauge vector field associated with the EM-field. A potential-scalar substitution allows the **ER**** **action to be extended by the addition of a source for the EM-field, which has the form:

with the corresponding Noether current.

Our new field equations now take the following form:

and:

and so we have a theory of gravito-electromagnetism on a Riemann geometrical background where the EM-field interacts with the scalar part of gravity described by the scalar field , and given the cosmological principle, the scalar-tensor duality allows us to extend to cosmological models with field equations:

In light of the cosmological principle, the above field equations become:

and the pivotal role of the scalar-tensor duality derives from the gauge symmetries:

of the 3+1D Weyl-Lagrangian:

which entail that the cosmological principle allows us to have a vector field on cosmological scales. Crucially then as one moves back in time, the relevance to quantum gravity is clear: with a Weyl-integrable geometrical background, the Palatini variational principle replaces the Riemannian one for a class of scalar-tensor theories that analytically admit renormalizability and whose renormalization group action on the Weyl-integrable metaplectic group fiber bundle yields a larger diffeomorphism group and the Einstein-Riemann frame action is symplectomorphic to an action of general relativity with a non-canonical scalar field interacting with an electromagnetic field. Hence, from the **EER**-action:

one can derive the actions of both, the Jordan and the Einstein frames.

Part two of this post will delve into the implications of:

for inflationary cosmology.

The post Weyl-Integrable Geometry, General Relativity, and the Scalar-Tensor Duality appeared first on George Shiber.

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

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

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

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:

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

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

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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:

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

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.

]]>

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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]]>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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]]>\[{B^s} = \frac{1}{2}{B^s}_{ab}d{y^a} \wedge d{y^b}\]

one quantizes spacetime via its quantum-phase Poisson algebraic structure:

\[{\theta ^{ab}} \equiv {\left( {{B^{ - 1}}} \right)^{ab}}\]

Hence, we have, for: \(f,g \in {\mathbb{C}^\infty }\left( M \right)\), the following:

Thus, the noncommutative algebra of operators is isomorphic to the deformed algebra of functions defined by the Weyl-Moyal product:

We then define a noncommutative space \({\mathbb{R}^{2n}}\) via the commutation relation:

\[{\left[ {{y^a},{y^b}} \right]_{{ * _{wm}}}} = i{\theta ^{ab}}\]

that allows us to interpret it as a noncommutative phase-space with Poisson structure given by \({\theta ^{ab}}\).

Now, any field \(\hat \phi \in {{\rm A}_\theta }\) can be expanded in terms of the complete-operator-basis:

\[\left\{ {\begin{array}{*{20}{c}}{{{\rm A}_\theta } = \left\{ {\left| m \right\rangle \left\langle n \right|,n,m = 0,...} \right\}}\\{\hat \phi \left( {x,y} \right)\sum\limits_{n,m} {{M_{mn}}\left| m \right\rangle \left\langle n \right|} }\end{array}} \right.\]

with \({{\rm A}_\theta }\) a C*-algebra, that is:

\[{e^{ik \cdot y}}{ * _{wm}}f\left( y \right){ * _{wm}}{e^{ - ik \cdot y}} = f\left( {y + \theta \cdot k} \right)\]

in infinitesimal form:

\[{\left[ {{y^a},f} \right]_{{ * _{wm}}}} = i{\theta ^{ab}}{\not \partial _b}f\]

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]]>

one quantizes spacetime via its quantum-phase Poisson algebraic structure:

Hence, we have, for: , the following:

Thus, the noncommutative algebra of operators is isomorphic to the deformed algebra of functions defined by the Weyl-Moyal product:

We then define a noncommutative space via the commutation relation:

that allows us to interpret it as a noncommutative phase-space with Poisson structure given by .

Now, any field can be expanded in terms of the complete-operator-basis:

with a C*-algebra, that is:

in infinitesimal form:

The coordinates in a gauge-theoretic setting will get promoted to the covariant coordinates defined by:

and thus:

get covariantized as:

and from

we get a quantum-geometry relation:

constituting an orthonormal frame and defining vielbeins of a gravitational metric.

Since Darboux's theorem in symplectic geometry is equivalent to the equivalence-principle in general relativity, it follows that the induced D-manifold defined by the metaplectic generalized quantum geometry continuously interpolates between a symplectic geometry and a Riemannian geometry.

To justify Born's thesis, let us simplify and work on a 4-D manifold and without loss of generality, a scalar field defined on it. The Hamiltonian in the Fourier representation is:

with , and the phase-space for each mode is:

with Poisson bracket and total field phase-space and with no loss of content, we can take: . Our symplectic form is given by the area form

locally:

and the Hamiltonian vector field associated to a smooth function can be locally written as:

and for any smooth curve , the Hamilton-equation for the Hamiltonian is:

Standard definition: with a diffeomorphism of a smooth manifold onto itself, the canonical lift of to the cotangent bundle is the transpose of the vector bundles isomorphism

So, letting be the canonical lift of to the cotangent bundle, we have for all :

thus allowing us to derive:

and being the non-linearity scale and the covering angular coordinates being: and . The field variables are parametrizable in terms of and as such:

with a dimensional constant. Our symplectic form reduces to:

with corresponding Poisson bracket:

and the spin-coordinates are:

satisfying:

with the 's spanning the corresponding Lie algebra:

To derive a Hamiltonian that is globally well-defined, has the minimum and the correct linearized limit, one applies the analogy with a spin in a constant magnetic field and we postulate that the Hamiltonian has the form:

with:

and:

Our Hamiltonian:

is hence recovered in the limit , up to an energy spectrum shift . It follows from the brackets:

that the Hamiltonian equations:

describing phase-space trajectories with parameters :

in the limit allow recovery of the classical expressions:

By Darboux's theorem and:

we get the Lie algebra commutator:

And for quantum states supported on field-values

one can expand in terms of to derive the deformed commutation relation:

which is, and that's where phase-space non-linearity comes in, the analytic dual to the **generalized uncertainty principle**:

And furthermore, with the expansion at , one can expand and in terms of the creation and annihilation operators:

and:

Thus, by Darboux's theorem, and generate a -deformed oscillator algebra:

with the deformation parameter:

Hence, the quantized Hamiltonian:

with the -ordering-symmetry, yields the energy eigenvalues:

with the eigenstates:

with the zero-th order of the expansion, and the coefficients are given as:

And the key relevance of this analysis to quantum geometry, and by extension, quantum gravity, is that the vacuum energy:

gets shifted by:

hence, the phase space of values of a given quantum field isisomorphicto a nontrivial metaplectic manifold

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