A Proposal to Solve Finite N Matrix Theory: Reduced Model Related to Quantum Cosmology

Round 1

Reviewer 1 Report

The goal of this paper is to apply some of the insights the authors
have gained in their earlier studies of supersymmetric quantum
cosmology to supersymmetric matrix theory and to the exploration of
its (supersymmetric) groundstate. To this aim they employ a number
of truncations which amount to a drastic simplification of the problem (e.g.
reducing the dimension of the fermionic Fock space from 2^{8(N^2-1)} for the
full theory to merely eight components). In my view this is a bit
of an over-simplification which misses crucial features of the original
problem: for instance, in (21) we would not get a normalizable wave function
if a,b and d were not kept constant but also integrated over.
Importantly, a similar truncation with gauge group SU(2) was already
formulated and studied long ago in section 4 of ref. [10]; subsequent work

J. Fröhlich and J. Hoppe,
"On zero mass ground states in supermembrane matrix models",
Commun.Math.Phys. 191 (1998) 613-626, hep-th/9701119 [hep-th]

then showed rigorously that there is *no* normalizable ground state. In
fact, the the general expectation is that no normalizable ground state
exists except for the maximal theory in D=11 (corresponding to N=4 SYM). At
the very least, the authors should clarify the relation of these important
insights to their present work.

As for references: the result mentioned in the first sentence of
the abstract is entirely due to Ref.[10], which precedes the work of
[1,2] by nine years (apart from presenting mathematically precise results
instead of mere speculations), but no reference is given. This should also
be rectified.

Author Response

We acknowledge valuable comments and criticism of Reviewer 1. Please find attached our response to all comments made. Hope we have addressed her/his suggestions properly.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper presents several specific solutions of matrix models in lower
dimensions and compares them with results from quantum cosmology. The
comparison is instructive and the paper may be publishable, but at present it
contains several conceptual and mathematical mistakes.

1. In Section 4, the authors introduce restrictions and special choices that
lead to their solutions as "models." However, since they do not discuss the
consistency of restricted equations or Hamiltonians, they should better be
referred to as "sets of solutions" in the headings of subsections 4.1, 4.2 and
4.3. If models are introduced in a strict sense, the authors would have to
show that the conditions (11) together with corresponding conditions on the
momenta imply consistent constraints and Hamiltonians that maintain the
required commutation relations. Since some of the constraints are second
class, setting fields equal to one another or two constants and their momenta equal to zero, the consistency of this reduction as a new model is not guaranteed. However, the conditions (11) may well be used to find specific solutions within the original matrix model.

2. Equation (23) is not fully correct because it does not take into account
all the conditions found by imposing both Q_1=0 and Q_2=0. The bottom line on page 6 shows that these two conditions imply D_4=D_1, D_7=-D_6 and
D_8=-D_5. There are therefore only five free constants in (23).

3. The relationship between the choice (22) and FRW cosmology should be explained in more detail. Is this relationship based on a similarity of the
corresponding Hamiltonians? If so, how do the authors interpret the conceptual difference that the matrix model Hamiltonian is not a constraint, unlike the cosmological Hamiltonian?

4. The inner product (24) appears to be arbitrary and should be justified in more detail.

Author Response

We acknowledge valuable comments and criticism of Reviewer 2. Please find attached our response to all comments made. Hope we have addressed her/his suggestions properly.

Author Response File: Author Response.pdf