Continuing from my last post where I discussed the triangular interplay between string-string duality, string field theory, and the action of Dp/M5-branes, here I shall discuss Stueckelberg string fields and derive the BRST invariance of the Landau-Stueckelberg action. Recalling that the action of M-theory in the Witten gauge is:

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

with \(k\) the kappa symmetry term, \({g_{mn}}\) the metric on \({M^{11}}\), and \({x^m}\) the corresponding coordinates with \({B_{mnp}}\) an antisymmetric 3-tensor. Hence, the worldvolume \({M^3}\) is:

\[R \times {S^1} \times {S^1}/{Z_2}\]

and the worldsheet action:

\[{S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}\]

being the sum of three terms:

\[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\partial _i}{x^m}{\partial _j}{x^n}\]

\[{S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\partial _i}{x^I}{\partial _j}{x^m}A_m^I\]

\[{S_{\,\bmod \,}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\partial _i}{x^J}{\partial _j}{x^I}\]

and the index I = 1, ... , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action \({S_{\bmod }}\) has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

\[\frac{{SO\left( {19,3} \right)}}{{SO\left( {19} \right) \times SO\left( 3 \right)}}\]

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]]>

with the kappa symmetry term, the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:

and the worldsheet action:

being the sum of three terms:

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

Something deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, andnota three-dimensional theory

which is precisely the long wavelength limit behavior of the **open** membrane:

the gauge fields are defined in terms of fields that live on 10-dimensional boundaries ofM-theory

In the **closed** membrane case:

the gauge fields are defined in terms of11-dimensional fields

which brought us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

with:

for and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

with:

and the momentum operators are given by:

Since open string end-points are topologically glued to Dp-branes, open strings must have inequivalent quantum states and thus, the string field has to carry the gauge group indices of :

where are the generators of the SU(N) group, with . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

which yields the field theory action:

BRST-invariantly as:

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

with the dilaton, the curvature of a (p + 1)–form gauge field:

where the two–index NS/NS tensor and the dual six-index heterotic five–brane tensor are given by:

and

Now we have the general form of a 10-D p-brane solution:

with:

and:

with

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

is:

Hence, the M2-brane solution is:

squaring the field strength gives the following M5-brane solution:

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

From the string-string duality above and , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

with the WZ term:

and the dual 6–form potential:

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

Let us consider the gauge covariantization of the proper-time gauge and the Ramond-Ramond gauge discussed above. The action for the covariant bosonic open string field theory is implicitly defined by the BRST operator :

with respect to the BPZ conjugation-derived inner product, where the string field has the following Fock space expansion:

where the following holds:

for the bosonic case, and:

are the associated space-time fields. We can now write the action as:

and is invariant under the gauge transformation:

with the gauge parameter being a Grassmann string field of

given as:

In terms of and , the gauge transformation is expressible as:

It follows then that:

is gauge invariant. Hence, in terms of , the action:

becomes:

Note that the gauge invariance of each of:

and

entails that the string field, by conformal gauge theory, has the following form:

and we also have:

Thus, our action becomes a sum of two gauge invariant terms:

and

Now, crucially:

is equivalent to a gauge invariant action of massless vector field

and by the metaplecticity of:

it follows that gauge transformation up to level N = 1 is expandible in terms of a gauge parameter as:

We perform now a Virasoro reparametrization of the evolving string surface as a transformation:

with

the wave-function of the string, which in string field theory, must be interpreted as a functional , giving us the functional action:

where the inner product is defined in terms of integrals over the whole string configuration space and the string field kinetic energy operator. By reparametrization invariance, we can derive the following:

and the following relations can be easily checked:

Now, what makes unique contrastively to is its invariance under an large group of extra symmetries in addition to reparametrization implicitly expressed by:

specifically, given by shifts:

and is a type of meta-gauge symmetry acting directly on the metaplectic phase space. Let us study some properties of this metaplectic gauge group as well as .

We expand in terms of the eigenstates of the mass operator:

where the state is annihilated by all . Then the string field functional can be written as:

The kinetic gauge of is given by the action of on new string functionals. At first order we get the following equation of motion:

and since satisfies:

it follows that:

has the property of linearized Yang-Mills gauge invariance and the following transformation laws can be derived:

where is the Chan-Paton field term.

With a 0-level state and an eigenstate of with eigenvalue , we have the following definition for the contravariant form

Now we must define:

satisfying:

with the 0-th projection operator:

Combining, we have at n-th-mass level a Klein-Gordon equation:

which is gauge-invariant, thus the existence of .

Adding the Stueckelberg string fields to the fundamental string field , we have the local Stueckelberg action:

We now define N-th-projection operators:

along with:

with

Now, since we have:

it follows that the Stueckelberg action:

is equivalent to the kinetic metaplectic gauge field action:

Now we introduce a new gauge:

At N=1 gauge field level, we thus have:

which is a hybrid Landau-Stueckelberg gauge. So for

this fixes the gauge invariance of the Stueckelberg action above.

Note that under this condition, since the following holds:

any string field:

satisfies the Landau-Stueckelberg gauge condition given:

with:

This in turn entails that:

holds.

The action for this gauge condition is then:

with:

with the odd/even Grassmann string fields and the projection operator defined implicitly by:

Putting all together, it follows that the action is BRST invariant.

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]]>The D=6 string-string duality, crucial for allowing the interchanging of the roles of 4-D spacetime and string-world-sheet loop expansion, entails that there is an equivalence between the K-3 membrane action and the \({T^3} \times {S^1}/{Z^2}\) orbifold action. Here are some thoughts and reflections.

In the bosonic sector, the membrane action is:

\[\begin{array}{*{20}{l}}{S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^m}{\partial _j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^I}{\partial _j}{x^J} + {\varepsilon ^{ij}}{\partial _i}{x^J}{\partial _j}{x^m}\left. {A_m^J(x)} \right\}}\end{array}\]

where:

\[\begin{array}{*{20}{l}}{{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{\partial _i}{x^m}{\partial _j}{x^n}} } \right.} + }\\{\frac{1}{6}{\varepsilon ^{ijk}}{\partial _i}{x^m}{\partial _j}{x^n}{\partial _k}{x^p}\left. {{B_{mnp}}} \right)}\end{array}\]

Recall I derived the total action:

\[\begin{array}{l}{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x\,d\,\Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot \\\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{INST}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} \end{array}\]

which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of \({D^{SuSy}}\) via Green's-functions, yielding the on-shell action of M-theory in the Witten gauge:

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

with \(k\) the kappa symmetry term. With \({g_{mn}}\) the metric on \({M^{11}}\), and \({x^m}\) the corresponding coordinates with \({B_{mnp}}\) an antisymmetric 3-tensor. Hence, the worldvolume \({M^3}\) is:

\[R \times {S^1} \times {S^1}/{Z_2}\]

The bosonic sector lives on the boundary of the open membrane: two copies of \(R \times {S^1}\), which naturally couple to the U(1) connections \({A^J}\).

Now, double dimensional reduction of the twisted supermembrane on:

\[{M^{10}} \times {S^1}/{Z_2}\]

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]]>In the bosonic sector, the membrane action is:

where:

Recall I derived the total action:

which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of via Green’s-functions, yielding the on-shell action of M-theory in the Witten gauge:

with the kappa symmetry term. With the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:

The bosonic sector lives on the boundary of the open membrane: two copies of , which naturally couple to the U(1) connections .

Now, double dimensional reduction of the twisted supermembrane on:

of

entails that the bosonic sector is that of the heterotic string:

with gauge group indices I = 1, … , 16.

It gets interesting when we consider:

with dimension:

since the worldsheet action:

is now just a sum of three terms:

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

Now, something fundamentally deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, andnota three-dimensional theory

This is precisely the long wavelength limit behavior of the **open** membrane:

the gauge fields are defined in terms of fields which live on 10-dimensional boundaries ofM-theory

In the **closed** membrane case:

the gauge fields are defined in terms of11-dimensional fields

Hence, the gauge fields of the closed membrane must be defined over M3 and not over its boundary, unlike the closed membrane, whose action on is:

where is with the spacetime being .

Hence, the closed membrane action on reduces to:

with:

and

and since surfaces have no one-cycles, it follows that the three-form potential that appears in of the action:

can be expanded in terms of the cocycles of .

For the 22 2-cocycles of , one can decompose in a similar way for the two-form potential:

with I = 1, …, 22 labeling the two-cycles of . So after insertion into , we can derive:

Applying reparametrization invariance, one can set:

where is a worldvolume coordinate, and now one performs a dimensional reduction of:

Here are the key propositions relevant to the** membrane/string duality** of the low energy theory in D=7.

- the kinetic terms for the gauge fields in D=7 supergravity are:

derived by a split of the 4-4 field strength , of the 11-dimensional supergravity action:

from the following term:

- Membrane/string duality in D=7 requires the existence of a point in the moduli space of where all the 22 gauge fields are enhanced via U(1) gauging: this is key to preserving kappa symmetry. Thus, at the point in the moduli space when the 22 two-cycles vanish the following holds:

- Hence, dimensional reduction yields:

So, the S-duality map:

takes:

to:

and is equivalent to the term in:

So, the above map acts on the induced metric on the worldvolume. It follows then that the term in in:

yields, after a double dimensional reduction of , the following:

with:

which yields an equivalence between:

and

Thus, the S-duality map that takes to also takes to the dimensionally reduced .

To achieve the matching of gauge sectors of the closed and open membrane, we must generate the gauge fields of the closed membrane before dimensionally reducing the theory, as opposed to the gauge fields of the open membrane, which are always generated within the two-dimensional theory. This explains the origin of strong-weak duality in string theory. The strong coupling limit of type IIA string is 11-dimensional supergravity which is believed to arise as the long wavelength limit of supermembrane theory. So, gauge fields present in the 3-dimensional theory will be strongly interacting, and will continue to be strongly interacting after dimensional reduction to a two-dimensional theory. However, the open membrane has its gauge fields appearing in two dimensional theories, which are therefore weakly interacting.

So, we must consider the spacetime part of the action for the closed membrane:

The term:

can be dimensionally reduced to:

which is equivalent to the first term in:

and the term:

maps to:

with and members of

Now, since the term is topological, and S-duality of the seven dimensional space entails:

then one can reduce:

to:

with the Hodge dual and in turn, allows us to further reduce to:

Therefore the b-term in the spacetimestring actionis a direct consequence of thedualityof the seven dimensional duality between 3- and 4-forms, and so the dimensional reduction of yields the term , and this is tantamount to mapping the closed membrane action on to the open membrane action on , thus D=6 string-string duality follows and both theories will have the samespacetime supersymmetrysince they have the same massless spectra

This naturally brings us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

with:

for and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

with:

and the momentum operators are given by:

Since open string end-points are topologically glued to Dp-branes, open strings must have inequivalent quantum states and thus, the string field has to carry the gauge group indices of :

where are the generators of the SU(N) group, with . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

which yields the field theory action:

BRST-invariantly as:

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

with the dilaton, the curvature of a (p + 1)–form gauge field:

where the two–index NS/NS tensor and the dual six-index heterotic five–brane tensor are given by:

and

Now we have the general form of a 10-D p-brane solution:

with:

and:

with

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

is:

Hence, the M2-brane solution is:

squaring the field strength gives the following M5-brane solution:

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

From the string-string duality above and , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

with the WZ term:

and the dual 6–form potential:

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

for IIA:

and for IIB:

Since the Killing spinor is given by:

where is a constant spinor.

End of proof.

Hence, the triangular interplay between string-string duality, string-field theory, and the action of Dp/M5-branes establishes a duality between 4-D spacetime and string-world-sheet loop expansion, entailing the equivalence between the K-3 membrane action and the orbifold action. Here is a classic by Edward Witten et al. on why that is important.

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]]>String/M-[F]-theory remains by far the most promising - only? - theoretical paradigm for both, grand unification and quantization of general relativity. With the Dp-action given by:

\[S_p^D = - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{{D_{\mu \nu }}L}}{{{\partial _{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

for contextualization, note that a necessary condition for the world-sheet Dirac propagator \({\delta ^{\left( 2 \right)}}\left( {{\sigma _i} - {\sigma _j}} \right)\):

\[S = i\int {{d^2}} {\sigma _1}{d^2}{\sigma _2}\sum\limits_{i,j = + , - } {{\psi _i}\left( {{\sigma _1}} \right)} {A_{ij}}\left( {{\sigma _1},{\sigma _2}} \right)\psi \left( {{\sigma _2}} \right)\]

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

\[{S_\eta } = \frac{1}{\beta }\sum\limits_{\frac{{i2\pi }}{\beta }} {{{\left( {i\frac{{2n + 1}}{\beta }} \right)}^\pi }} W + \alpha '{R_{\left( 2 \right)}}\Phi \]

with

\[W \equiv {h^{mn}}{\partial _m}{X^a}{\partial _n}{X^b}{g_{ab}}\left( X \right)\]

and \(\beta \) the bosonic frequency, be analytically summable. The string world-sheet is given by:

\[{S_{ws}} = \frac{1}{{4\pi \alpha '}}\int\limits_{c + o} {d\tilde \sigma } d\tau '\sqrt h \left( {W + \alpha '{R_{\left( 2 \right)}}\Phi } \right)\]

A major problem is that by the Heisenberg's uncertainty principle:

\[\left( {\Delta A/} \right)\left( {\left| {\frac{{d\left\langle A \right\rangle }}{{dt}}} \right|} \right)\left( {\Delta H} \right) \ge \hbar /2\]

the string time-parameter on the world sheet \({\sigma _t}\) with Feynman propagator in Euclidean signature being:

\[\begin{array}{c}G\left( {x,y} \right) = \int_0^\infty {d{\sigma _t}} G\left( {x,y;{\sigma _t}} \right)\\ = \int {\frac{{{d^D}p}}{{{{\left( {2\pi } \right)}^D}}}} \exp \left[ {ip \cdot \left( {y - x} \right)} \right]\frac{2}{{{p^2} + {m^2}}}\end{array}\]

violates

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]]>

for contextualization, note that a necessary condition for the world-sheet Dirac propagator :

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

with

and the bosonic frequency, be analytically summable. The string world-sheet is given by:

A major problem is that by the Heisenberg’s uncertainty principle:

the string time-parameter on the world sheet with Feynman propagator in Euclidean signature being:

violates the integrability condition for the action:

and hence, in light of the principle of superposition, as a function of runs into the Riemann-Lebesgue Lemma problem, given that the Fourier transform of :

is non-convergent, with real, since the quantization of spacetime is an anti-smoothing dynamical breaking of the Ricci scalar . Hence we get,

which is incoherent. To see this, note that the anti-smoothing of spacetime implies that cannot be recovered from via

and that implies that the gravitonic wave-propagation travels in spacetime at infinite speed, given

and by wave-particle duality and the violation of special relativity, the graviton provably cannot exist. Or, by quantum tunnelling and the fact that gravitons self-gravitate, we have the instantaneous collapse of spacetime to a zero-dimensional singularity. Pick your poison.

A solution is to integrate over orbibolds and derive the Lagrangian of N=1 supergravity by orbifoidal D-11 and D-10 SUGRA-Barbero coupled actions. Let us see how this works. One must begin by giving a description of the field contents and the degrees of freedom, which will turn out to be a crucial number. At first, note that in D=11, SUGRA has a simple action: using exterior algebraic notation for the anti-symmetric tensor fields , with the field strength , it is surprisingly:

with the Newtonian constant in 11 dimensions. By dimensional reduction, the Type IIA can be derived from (1). Note that there are D=10 supergravity theories with only SUSY which couple to D=10 super-Yang-Mills theory. We still do not have a workable Type IIB theory since it involves an antisymmetric field with a self-dual field strength. Nonetheless, one may still derive an action that involves both dualities of . Then, by imposing the self-duality as a supplementary equation, we get:

with field strengths: , , , , , with the self-duality condition .

Note that the above action arises from the string low-energy limit and:

naturally yields the NS-NS sector of the theory, while:

is derivable from the RR sector of the theory. Now, Type IIB supergravity theory is invariant under the non-compact symmetry group and the key is that this symmetry is not manifest in:

To make it so, one must redefine fields, from the string metric in (2) to the Einstein metric , along with a complexification of the tensor fields:

Now, the action is easily seen to be:

the metric and fields are invariant under the symmetry of Type IIB supergravity. The axionic dilaton field varies with a Möbius super-transformation:

and , self-rotate under the Möbius super-transformation, and can most clearly be visualized as a complex 3-form field :

The SUSY transformation for Type IIB supergravity on the fermion fields are of the following form, via Seiberg–Witten analysis, without a need for bosonic transformation laws, with the dilaton and the gravitino :

It is crucial to realize that in the context, the SUSY transformation parameter essentially has charge, implying that has necessarily, given unitarity, and has .

The geometry of superstring theory in the Ramond-Neveu-Schwarz setting is given by the bosonic world-sheet fields and the fermionic world-sheet fields , with expressing chirality**,** and , must be functions of local world-sheet coordinates . Both and vectorially transform under the irreducible representation of the Lorentz group. By using Gliozzi-Scherk-Olive holographic projections, the spacetime supersymmetric derivative can act on (1) and (2) above. It is more informative to work with orientable strings. Type II and heterotic string theories are perfectly suited in this context. Field interactions in second quantization arise from the orbifoidal splitting and joining of the world-sheets, and causality is maintained. Moreover, the genus for orientable world-sheets equals the number of Witten-handles. The world-sheet bosonic field naturally gives rise to a non-linear sigma model:

with being the square root of the Planck length, being the world-sheet metric, and being the Gaussian curvature. The world-sheet fermionic field axially gives rise to a world-sheet supersymmetric completion of the sigma model. It suffices to give its form on a flat world-sheet metric with a vanishing world-sheet gravitino field:

and being the Riemann tensor for the metric . Now the all too important SUSY-covariant derivatives can be derived:

with the Levi-Civita connections for . Now one is in a position to solve the Seiberg-Lebesgue problem via the functional integral over all and by integrating over all world-sheet metrics and world-sheet gravitini fields via the amplitude:

This clearly solves the Feynman propagation problem. And the upshot is that the vacuum expectation value of the dilaton field is , and the string vacuum expectation value of the string amplitude is** **given by the Euler number of the world-sheet :

with the genus, is a number that counts the orbifoidal puntures in spacetime as the string world-sheet propagates under quantum fluctuations. Thus, a genus world-sheet with no boundary gets a multiplicative contribution: , thus deriving another truly remarkable identity, , representing the closed string coupling constant which lives on a Dp-brane, and hence by the supersymmetry action and the Witten Index:

where is the fermion number, and the trace is over all bound and continuum states of the supersymmetric-Hamiltonian, and PBC being the periodic boundary conditions on both the fermionic and bosonic fields**, **we get a finite, causal SUGRA action in D=11/ D=10, thus solving the Seiberg-Lebesgue problem.

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]]>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:

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

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:

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

The post F-Theory, the D-Term Equation and Representation Theory appeared first on George Shiber.

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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.

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