The Quantum Master Equation and Supersymmetric-Field-Cosmology

Supersymmetric-field theory remains the best new physics beyond the SM and also “builds a bridge between the low energy phenomenology and high-energy fundamental physics“. Moreover, I will show here that SuSy can incorporate aspects of loop quantum gravity to shed deep light on fundamental issues in quantum cosmology. Before getting deep, recall that the central action for supersymmetric field theory is:

    \[\begin{array}{l}{S_0} = \sum\limits_\nu {\int {d\phi } } \left[ {{{\tilde D}^2}} \right.\left\{ {\Omega _i^\dagger } \right.\left( \wp \right)\nabla _a^2{\Omega ^j}\left( \wp \right)\\{\left. {\left. { + \,\omega _i^a\left( \wp \right)\omega _a^i} \right\}} \right]_{\left| {_G} \right.}}\end{array}\]

with \left| {_G} \right. the Grassmannian variable and total action is given by:

    \[\begin{array}{c}{S_T} = {S_0} + \sum\limits_\nu {\int {d\phi } } \left[ {{{\tilde D}^2}} \right.\left\{ {{B_i}} \right.\left( \wp \right){{\tilde D}^a}\Gamma _a^i\left( \wp \right)\\ + \,{{\bar c}_i}\left( \wp \right){{\tilde D}^a}{\nabla _a}{c^i}{\left. {\left. {\left( \wp \right)} \right\}} \right]_{\left| {_G} \right.}}\end{array}\]

and the super-field/anti-super-field theory expansion in the Landau type gauge is given by:

    \[\begin{array}{l}{Z_L}\left[ 0 \right] = \int {\not D} M{e^{ - {W_L}\left[ {\Phi ,{\Phi ^ * }} \right]}} = \\\int {\not DM\exp \left[ { - \left( {{S_0}} \right.} \right.} + \sum\limits_\nu {\int {d\phi } } \left[ {\Gamma _{1i}^{a * }} \right.\\ \cdot {\nabla _a}{c^i} + c_{1i}^ * f_{kj}^i{c^k}{c^j} + \bar c_{1i}^ * \left. {\left. {{{\left. {{B^i}} \right]}_{\left| {_G} \right.}}} \right)} \right]\end{array}\]

with the super-covariant derivatives of

    \[\left\{ {\begin{array}{*{20}{c}}{{\Omega ^i}\left( \wp \right)}\\{{\Omega ^{i\dagger }}\left( \wp \right)}\end{array}} \right.\]

are defined by:

    \[{\nabla _a}{\Omega ^i}\left( \wp \right) = {\tilde D_a}{\Omega ^i}\left( \wp \right) - if_{kj}^i\Gamma _a^k\left( \wp \right){\Omega ^j}\left( \wp \right)\]


    \[{\nabla _a}{\Omega ^{i\dagger }}\left( \wp \right) = {\tilde D_a}{\Omega ^{i\dagger }}\left( \wp \right) - if_{kj}^i{\Omega ^{k\dagger }}\left( \wp \right)\Gamma _a^i\left( \wp \right)\]

and the super-derivative is:

    \[{\tilde D_a} = {\not \partial _a} + K_a^b{\theta _b}\]

where the Ashtekar-Barbero connection K_a^i, for a, b = 1, 2, 3, is given in terms of co-triads e_a^i:

    \[K_a^i(x) = {K_{ab}}(x){e^{bi}}(x)\]

and e_a^i satisfying

    \[E_i^a = \left| {\det \left( {e_a^i} \right)} \right|e_a^i(x)\]

and the extrinsic curvature {K_{ab}} being:

    \[{K_{ab}} = \frac{1}{{2{N_{lapse}}}}\left( {{{\dot h}_{ab}} - {\nabla _a}N{{_b^{shift}}^\prime } - {\nabla _b}N{{_a^{shift}}^ * }} \right)\]

For definitions of terms, see my ‘SuperSymmetric Field Theory and the Quantum Master Equation‘ post. Some philosophical points are in order first.

In this, part 1, I will take the analysis up to but not inclusive of deriving a supersymmetric generalization of group field cosmology and derive the group field cosmology action and the corresponding equation of motion. Aside: I highly recommend this book.

First, although loop quantum gravity is provably not an adequate quantum gravity theory nor a successful unification paradigm, some of its mathematical-physics insights can, via super-group-field theory, shed light, in conjunction with supersymmetric-field-cosmology, on a GUT quantum cosmology theory. LQG theory is background independent quantization of gravity (though not successful) where the Hamiltonian constraint is given via the Ashtekar-Barbero connection and densitized triad mentioned above. Hence, the curvature of the connection is determined by the holonomy around a loop whose corresponding Hilbert space is a space of spin networks and the spin foam arises analytically from the time evolution of these spin networks, implying that it is a second quantized formalism. One, as usual, gets via a third quantization of LQG the corresponding group field theory that is homeomorphic to a quotiented orfibold in supersymmetric field theoryRead this for a great exposition and background for this technical post.

Crucial is to realize that in loop quantum gravity, the holonomies of the connection are given by operators. Given the Ashtekar-Barbero connection, the metric is:

    \[d{s^2} = - {N^2}\left( t \right)d{t^2} + {a^2}\left( t \right){\delta _{ab}}d{x^a}d{x^b}\]

with a\left( t \right) the scalar factor and N\left( t \right) the lapse function discussed in my post here. With the crucial Barbero-Immirzi parameter \gamma and the Levi-Civita connection {\left( {\omega _0^i} \right)_a}, we have:

    \[A_a^i = \gamma {\left( {\omega _0^i} \right)_a}\]

and A_a^i is completely determined in loop quantum cosmology by:

    \[c = \pm \Upsilon _0^{1/3}{N^ - }\gamma a'\]

with a' the time derivative of a. The operators:

    \[\left\{ {\begin{array}{*{20}{c}}{p = \pm \,{a^2}\Upsilon _0^{1/3}}\\{{e^{i\mu c}}}\end{array}} \right.\]

c being the conjugate to p, \mu a function of p. Thus,

    \[\mu = {\left| p \right|^{ - 1/2}}\]

is a basis of the eigenstates of the volume operator \Upsilon and we have

    \[\Upsilon \left| {{\nu _\Upsilon }} \right\rangle = 2\pi \gamma G\left| {{\nu _\Upsilon }} \right|\left| {{\nu _\Upsilon }} \right\rangle \]

with the dimensions of length being

    \[{\nu _\Upsilon } = \pm \,{a^2}{\Upsilon _0}/2\pi \gamma G\]

Now, the curvature of A_a^i is given via the holonomy around a loop and the area of such a loop cannot be smaller than a fixed minimum area since the smallest eigenvalue of the area operator in loop quantum gravity is nonzero! Hence, in Planck units, the Hamiltonian constraint for a homogeneous isotropic universe with a massless scalar field \phi is given as:

    \[{K^2}\Phi \left( {{\nu _\Upsilon },\phi } \right) = \left[ {{E^2} - \not \partial _\phi ^2} \right]\Phi \left( {{\nu _\Upsilon },\phi } \right) = 0\]

where {E^2} is defined as:

    \[\begin{array}{l}{E^2}\Phi \left( {{\nu _\Upsilon },\phi } \right) = - {\left[ {B\left( {{\nu _\Upsilon }} \right)} \right]^{ - 1}}{C^ + }\left( {{\nu _\Upsilon }} \right)\\\Phi \left( {{\nu _\Upsilon } + 4,\phi } \right) - {\left[ {B\left( {{\nu _\Upsilon }} \right)} \right]^{ - 1}}{C^0}\left( {{\nu _\Upsilon }} \right)\Phi \left( {{\nu _\Upsilon },\phi } \right)\\ - {\left[ {B\left( {{\nu _\Upsilon }} \right)} \right]^{ - 1}}{C^ - }\left( {{\nu _\Upsilon }} \right)\Phi \left( {{\nu _\Upsilon },\phi } \right)\end{array}\]

with {\nu _\Upsilon } = 4 and for the gauge-choice in loop quantum cosmology, we have as usual:

    \[{C^ - }\left( {{\nu _\Upsilon }} \right) = {C^ + }\left( {{\nu _\Upsilon } - 4} \right)\]

Setting N = 1, the canonical quantization scheme yields:

    \[{C^ + }\left( {{\nu _\Upsilon }} \right) = \frac{1}{{12\gamma \sqrt {2\sqrt 3 } }}\left| {{\nu _\Upsilon } + 2} \right|\left| {{\nu _\Upsilon } + 1} \right| - \left| {{\nu _\Upsilon } + 3} \right|\]

    \[{C^0}\left( {{\nu _\Upsilon }} \right) = - {C^ + }\left( {{\nu _\Upsilon }} \right) - {C^ + }\left( {{\nu _\Upsilon } - 4} \right)\]

    \[B\left( {{\nu _\Upsilon }} \right) = \frac{{3\sqrt 2 }}{{8\sqrt {\sqrt 3 \pi \gamma G} }}\left| {{\nu _\Upsilon }} \right|{\left| {{{\left| {{\nu _\Upsilon } + 1} \right|}^{1/3}} - {{\left| {{\nu _\Upsilon } + 1} \right|}^{1/3}}} \right|^3}\]

and with N = {a^3}, solvably, we have:

    \[\left\{ {\begin{array}{*{20}{c}}{{C^ + }\left( {{\nu _\Upsilon }} \right) = \frac{{\sqrt 3 }}{{8\gamma }}\left( {{\nu _\Upsilon } + 2} \right)}\\{{C^0}\left( {{\nu _\Upsilon }} \right) = \frac{{\sqrt 3 }}{{4\gamma }}{\nu _\Upsilon }}\\{B\left( {{\nu _\Upsilon }} \right) = \frac{1}{{{\nu _\Upsilon }}}}\end{array}} \right.\]

and philosophically deep, as it is a fatal flaw, though not for the purposes of our