Optimized Fock space in the large N limit of quartic interactions in Matrix Models

We consider the problem of quantization of the bosonic membrane via the large $N$ limit of its matrix regularizations $H_N$ in Fock space. We prove that there exists a choice of the Fock space frequency such that $ H_N$ can be written as a sum of a non-interacting Hamiltonian $H_{0,N}$ and the original normal ordered quartic potential. Using this decomposition we obtain upper and lower bounds for the ground state energy in the planar limit, we study a perturbative expansion about the spectrum of $H_{0,N}$, and show that the spectral gap remains finite at $N=\infty$ at least up to the second order. We also apply the method to the $U(N)$-invariant anharmonic oscillator, and demonstrate that our bounds agree with the exact result of Brezin et al.

(1. 4) It has been shown by Goldstone and Hoppe [2,3] that the full field-theoretic Hamiltonian (1.3) admits a regularization procedure, where the classical phase-space variables x i (ϕ), p j (ϕ) are replaced by n-dimensional matrices, the Poisson bracket by the matrix commutator and integrals over Σ by the matrix trace. The original volume-preserving diffeomorphisms symmetry of Σ, represented by (1.2), is recovered in the n → ∞ limit from the SU (n) invariance of its matrix regularizations. The family of n-dimensional matrix models constructed in this way reads H N = T r( P 2 ) − (2πn) 2 n d i<j T r([X i , X j ] 2 ), (1.5) with the SU (n) invariance constraints i T r(T a [T b , T c ]), we can rewrite H N (and the constraints) in terms of d(n 2 − 1) canonical pairs p ia , x ia (X i = x ia T a , P i = p ia T a ) as a finite sum over the matrix modes, cp. (1.3), In contrast to string theory, the Hamiltonian of the membrane with its quartic interaction makes the problem of quantisation rather difficult. One approach, which was proposed in the literature many years ago [2,3], is to take advantage of the symmetry preserving matrix regularizations (1.5)/(1.7), quantize it for finite N and then take the limit N → ∞. While it has been proved that all finite N expressions are well defined Schrödinger operators 1 with purely discrete spectrum [4,5], it seems that almost nothing is known about the large N limit (apart from the case d = 1, where the quartic potential vanishes [7]). In particular, it is interesting to ask whether the spectrum remains purely discrete in the limit and how it scales with N and d. In this paper we study the large N behaviour of the canonically quantized family of Hamiltonians (1.7) based on a Fock space description in order to approach those questions. According to common knowledge, in order to quantize a system with many degrees of freedom one should rescale the quartic interaction by the suitable power 2 of n to make the quadratic part competitive at large n. This can be realized as a rescaling of the classical phase-space variables preserving the canonical Poisson-commutation relations (and the form of the constraints), i.e. x ia → N −α x ia , p ia → N α p ia , which leads to the following classical energy where by abuse of notation we denote the rescaled operator N −2α H N by the same symbol H N and γ := 6α. The canonically quantized expression corresponding to (1.9 ) is a priori ill-defined in the large n limit due to possible divergences coming from the infinite vacuum energy and therefore it needs to be renormalized by subtracting a multiple of the identity operator. This is a rather common property of quantum models with many degrees of freedom, so in order to see the general pattern let us consider a whole class of Hamiltonians parametrized by a sequence of real tensors c  (1.10) where J N is the index set having the form of a cartesian product of two discrete sets J N = D×K n , with |D| = d = const, |K n | = N ր ∞. For instance, for the Membrane Matrix Models (MMM) (1.7) we have N = n 2 − 1, K n = {1, ..., n 2 − 1}, D = {1, ..., d}, ω 0I = 0 and It is known that a Fock space description of local Hamiltonians of the form p 2 + q 2 + V (x) is rather inconvenient due to the occurence of terms containing only annihilation or only creation operators, which implies that the so-called single trace sector is not invariant under the action of the Hamiltonian and the true vacuum is not a simple gaussian in the variables x I (see [9] for a review). Therefore the problem of finding eigenvalues becomes in most cases extremely difficult due to the rapidly increasing with N number of relevant degrees of freedom. We will however show that despite all of that, one can get certain spectral properties of local Hamiltonians (1.10) using Fock space methods, such as upper and lower bounds for the spectrum in the planar limit, as well as a qualitative picture of the subtle interplay between the quadratic terms and the interaction leading to a redefinition of the vacuum energy and mass. The tools described in this paper are of high relevance especially for multi matrix-models where other methods based on the diagonalisation of the matrix degrees of freedom and integrating out the "angle" variables are not directly applicable, like for the Membrane Matrix Models (MMM).
The rest of the paper is organized as follows. In the next section we will introduce the notion of optimized Fock space providing a very convenient decomposition of the quantum Hamiltonian corresponding to (1.10) and giving directly the gaussian variational bound for the ground state energy. The idea consists in tuning the frequencies of the harmonic oscillators, whose eigenfunctions are used as a basis of the corresponding Fock space. We give an algorithm how to choose an optimal set of these frequencies such that the form of the Hamiltonian (1.10) expressed in the language of creation and annihilation operators is the simplest possible. This almost trivial observation (usually not spelled out explicitly in the literature though, see however [10]), based on the fact that the optimal Fock space frequencies in the quantum representation of (1.10) are not always the ω 0I 's, is the starting point of our study of the MMM (1.9), a system where there is no obvious choice of a basis consisting of harmonic oscillators since all the ω 0I 's are equal to zero. Thus one has to explore the quartic potential in more detail, which gives birth to a mass term (quadratic in x I ) in a properly chosen Fock space. In Section 3 we consider a toy model, the U (n)-invariant anharmonic oscillator (AO), and present Fock space-based techniques to get upper and lower bounds for the ground state energy in the planar limit, which agree with the exact answer [11,12,13,14,15,16]. In Section 4 we apply the method to the MMM. Finally, in Section 5 we show that the perturbative expansion suggested by the optimized Fock space decomposition serves as a very good approximation for the vacuum energy and for the spectral gap of the AO at least up to the third order, even in the strong-coupling regime g → ∞, restoring the correct scaling of the spectrum with the coupling constant ∝ g 1 3 . We also observe that the MMM contains a small hidden parameter (an effective coupling constant) and we perform the corresponding expansion, which turns out to be consistent with our lower and upper bounds for the vacuum energy, justifying the validity of the planar limit for this model. In particular, we find that the effective coupling constant is proportional to 1 d−1 , hence the perturbative expansion has better convergence properties in higher dimensions. Moreover, the perturbation series for the first SO(d) × SU (n) invariant excited state indicates that the spectral gap is finite at large n.
Although the purely bosonic model (4.14) is interesting on its own, more attention has been recently paid to its supersymmetric extensions, in particular for d = 9 dimensions (resp. for d = 9 matrices), see e.g. [17] and [6] for a more recent review of this topic. Remarkably, the spectrum of the supersymmetric version of (4.14) turns out to be continuous and equal to the interval [0, ∞), [18]. However, there have been also evidence of existence of discrete eigenvalues embedded in the continuous spectrum [19,20]. In particular, zero energy normalizable states are of high importance for the supermembrane as well as in the context of the BFSS conjecture of M-Theory [21]. Despite a number of profound results concerning zero energy eigenfunctions, e.g. an explicit construction in the pure fermionic sector for the n = 2 model [22], large x behaviour [23], or existence results on compact regions [24], the question posed in the full generality remains open. We believe that the approach discussed in this paper can shed new light on this problem since our results correspond to the purely bosonic sector of the supermembrane and our construction can be possibly extended to the fermionic sectors allowing to study the embedded part of the spectrum of the supermembrane. Moreover, as our method simplifies the Fock space representation of the Hamiltonian, it should also allow to optimize cut-off Fock space algorithms for studying various quantum mechanical systems related to (Super)Yang-Mills theories, introduced in [25].

Optimized Fock space
The canonically quantized Hamiltonian (1.10) becomes formally a Schrödinger operator acting on ⊗ N I=1 L 2 (R, dx I ) with the classical coordinates replaced by operators in the usual way, i.e. p I = −i∂ I and x I being the multiplication operator. Since we are interested in the large N limit, it is convenient to embed ⊗ N I=1 L 2 (R, dx I ) in the standard bosonic Fock space H ω , defined as the Hilbert space generated by states of the form We denote the norm and scalar product in H ω by ||.|| and ., . respectively. The vacuum state Ψ 0 (ω) has the form of an infinite product Ψ 0 (ω) : x I (ω I ) = 1 √ 2ω I (a I + a † I ). (2.4) Since ⊗ N I=1 L 2 (R, dx I ) is isomorphic to the subspace H ω,N ⊂ H ω generated by the first N creation operators a † I , I = 1, ..., N , the action of H N can be naturally extended to the whole Fock space by taking the tensor product of H N with the identity acting on H ⊥ ω,N . Therefore the operator induced on H ω by H N has the form IJKL are finite), but in general not for N = ∞. In order to assure that the domain of H N contains more than the zero vector, one has to subtract the divergent ground state energy by adding a multiple of the identity operator β N I to the Hamiltonian. We expect that for γ large enough (and c (N ) IJKL not too pathological) one should get at least a finite limit of ||(H N + β N I)ψ|| for a generic ψ ∈ H and some β N . This is unfortunately not always the case since the kinetic energy contains terms proportional to a I a I and a † I a † I , which are divergent at large N because e.g. || I a † I a † I ψ|| → ∞ for every non-zero ψ ∈ H ω . As a consequence, in order to simplify the large N behaviour of the theory one should cancel such terms by finding corresponding counter-terms in the potential. We will see that this is possible for a relatively large class of models including all models with a large symmetry group, e.g. SO(n) symmetric vector models or U (n)/SU (n) symmetric matrix models, where the interaction has the form of a trace. The idea is that the sequence of the Fock space frequencies {ω I } has to be adjusted to the Hamiltonian, i.e. the choice of the subspace H ω ⊂ ⊗ ∞ I=1 L 2 (R, dx I ) where the limit is taken, depends on the operator and in most cases the optimal choice is not the natural choice ω I = ω 0I (especially when all ω 0I = 0 like for the MMM).
Let us see how it works in detail. H N rewritten in terms of a I and a † I becomes (for the sake of transparency, we leave out the I H ⊥ ω,N part of (2.5)) We rewrite the quartic potential using the commutation relations (2.3) where we have defined A .., I π(k) } denotes the symmetrization and :: is the normal ordering with respect to Ψ 0 (ω). We get Let us now assume that there exists a sequence {ω I }, called the optimized Fock space frequencies (and Hω called the optimized Fock space), s.t. i.e. that the matrix A (N ) IJ is diagonal at large N (this is in fact the case for the mentioned above SO(n) symmetric vector models or U (n)/SU (n) symmetric matrix models 3 ). The smallest γ for which it is possible we call γ crit . From eq. (2.14) we get that lim N →∞ N −γ A (N ) . Moreover the sequence β N exhibits a very nice property given in the following lemma.
Proof. By noting that Ψ 0 (ω), H N Ψ 0 (ω) = −β N and using the variational principle we get that the optimized gaussian frequencies satisfying or equivalently

17)
3 One way to see it is to notice that for these models all double contractions of the tensor defining the quartic interaction produce Kronecker deltas, i.e. c (N ) converge to the solutions of (2.14). Moreover, the rest term R N originates from the matrix A (N −) IJ , which means that R N ≃ const. N I (a I a I + a † I a † I ) and therefore ||R N ψ||ω → 0 ∀ψ ∈ Hω.
Note that this result does not imply that the system described by the Hamiltonian (1.10) becomes necessarily a system of decoupled harmonic oscillators despite that according to Lemma 1 lim ∀φ, ψ ∈ Hω containing a finite number of elementary excitations (2.1). This means that, in order to avoid such a trivialisation, one has to employ states with infinitely many oscillatory modes and treat the optimized Fock space as the first step towards the correct quantum description of the model. This is the case for the two considered here matrix models, as we will see in the next sections, where the relevant space of interest is the space of SU (n) invariants.

U (n)-invariant anharmonic oscillator
Let us consider the U (n) symmetric matrix model where M is a hermitian n × n matrix and P its conjugate momentum. Since the exact value of the ground state energy of the model is known and even the U (n) symmetric sector has been solved in the large n limit one could expect that the Fock space approach should allow to rederive these results. Unfortunately, due to various technical difficulties, such as the fact that the Fock space representation of the Hamiltonian (3.1) does not annihilate the Fock space vacuum and does not preserve the single trace sector, there has been no exact solutions based on a Fock space formalism for this model so far. Nevertheless, one can still get some information about the spectrum using Fock space methods, which is of high importance for models where other methods fail. The Hamiltonian (3.1) is an excellent laboratory for testing our large N techniques before approaching the Membrane Matrix Models (1.5), since it exhibits all the properties which make the quantization of a system with a large number of degrees of freedom cumbersome. Let us start with the optimized Fock space decomposition for (3.1). By expanding M and P in the basis {T a } n 2 a=1 consisting of N := n 2 generators of U (n) with normalisation T r(T a T b ) = δ ab , satisfying the completeness relation The action of H N in the Fock space H ω , given in terms of the creation and annihilation operators, becomes (assuming that ω a = ω ∀a) As mentioned previously, the matrix A (N ) ab is diagonal 4 and The condition lim n→∞ A (N −) ab = 0 (or equivalently lim n→∞ ∂β N ∂ω = 0) implies the equation whose real solutionω is obviously different than the natural choice ω = 1 suggested by the original quadratic term. As we will see in Section 5,ω provides a crude approximation of the spectral gap for this model. According to Lemma 1, the Hamiltonian in the optimized Fock space Hω takes the following form where ||R N ψ|| → 0 ∀ψ ∈ Hω. Insertingω to (3.7) gives the gaussian variational upper bound for the ground state energy e (0) which is in an excellent agreement with the result of [11] (even for a large coupling g), where the authors obtained the exact value. Asymptotically, for large g, they have e 0 (g) ≃ 0.58993g is at most ≈ 9 wrong (see Table 1).

Spectral bounds
In order to produce a lower bound for the spectrum, one has to take into account the interaction term 1 n : V N :. This involves quite technical, but instructive estimates of matrix elements of the Hamiltonian between U (n) invariant wave functions, which we present in this subsection.
Proceeding along the lines of [26] we introduce a basis of the U (n) invariant subspace I (n) ω ⊂ Hω spanned by U (n)-invariant linear combinations of the first N = n 2 creation operators, called the partitions basis is a partition of a certain natural number k = |λ| := i iλ i . The partitions basis becomes orthonormal at N = ∞ for the properly chosen normalisation factors N λ ∝ n − |λ| 2 (generically), see [7]. As shown in [26], the matrix elements of the Hamiltonian (3.9) in the partitions basis contain three groups of divergent terms 1. the vacuum expectation value (Ψ 0 , H N Ψ 0 ) ∝ n 2 corresponding to our β N I, IJ ≃ O(1) and thus γ crit = 0. Note also that we perform the calculation for 2H N instead of H N in order to match it with the conventions from Section 2 and in the end we divide the results by 2 to compare them with the work of Brezin et al.
Renormalization of H N is based on a proper choice of basis (in particular the vacuum) such that the second and third group of divergent matrix elements would be "absorbed" into the first one as a constant shift of the whole spectrum. The divergent vacuum energy can be then easily subtracted. According to Lemma 1, the proper choice of the Fock space frequencies allows to absorb Group 2 into the ground state energy by eliminating the T r(a †2 + a 2 ) part from the description. Then, for the suitable β N , the only divergent matrix elements of H N + β N I belong to Group 3 but they are much more difficult to handle since the resulting vacuum is no longer a standard Fock vacuum (there are no counterterms which would cancel T r(a †4 + a 4 ) after an appropriate choice of the Fock space frequencies as it happened with T r(a †2 + a 2 ) ). In order to approach this problem let us point out that in the planar limit one can interpret T r(a †4 ) and T r(a 4 ) as composite creation-annihilation operators Intuition suggests that one should treat A and A † in a special way in the Fock space description. This can be realized as follows. Let us introduce a new basis, completely equivalent to the partitions basis, consisting of where N k,λ ∝ N λ n −2k . It turns out that the states ψ λ,k exhibit very useful properties, which we explore below.

Lemma 2. The action of the operator
on the states ψ λk , asymptotically 5 , at large n, becomes All operator inequalities are meant here in the usual sense, i.e.
. Also, terms of order O( 1 n ) are meant in the sense of norm in Hω and asymptotically equal terms differ by terms of order at most O( 1 n ). We call an operator T N to be of order O(n k ) iff ||T N ψ|| ≤ const.(ψ)n k , ∀ψ ∈ H ω . The key observation which allows to compute the leading terms at large n (i.e. of order O(1), O(n) and O(n 2 ) ) is the fact that they originate from Wick contractions of adjacent operators sitting in one U (n)/SU (n) trace, resp. corresponding to planar contractions in the diagramatic representation (see Section 5) and we refer to the large n limit taken by neglecting all non-planar contributions, resp. all subleading terms (i.e. of order O( 1 n ) and lower) as the planar limit. In Section 5 we give a perturbative justification of this limit up to the third order Indeed, a † a a a ψ λk = (4k + |λ|)ψ λk and One has to show that the first term in (3.19) converges to 0 in norm at large n ∀λ. According to Wick's theorem we have 4 cases • single contractions: we get n − 1 2 from the fact that the partition λ has been shortened by 1. Then we are left with three annihilation operators acting on (A † ) k ψ 0 , which prolongs λ by 1 and thus gives a factor of n 1 2 as well as n 2 (at most, when we perform a planar contraction, resp. when adjacent indices of U (N ) are contracted) from the contraction with A † . The total factor is then n −2 → 0; • double contractions: we get n 1 (at most) from the double contraction, n −1 from the fact that the partition λ has been shortened by 2. Then we are left with two annihilation operators acting on (A † ) k ψ 0 , which prolongs λ by 2 and thus gives a factor of n 1 as well as one more n 1 (at most) from the contraction with A † . The total factor is then n −2 → 0; • triple contractions: we get n 2 (at most) from the triple contraction, n − 3 2 from the fact that the partition λ has been shortened by 3. Then we are left with one annihilation operator acting on (A † ) k ψ 0 , which prolongs λ by 3 (and thus gives a factor of n 3 2 ). The total factor is n −2 → 0; • quadrupole contractions: we get n 3 (at most) from the quadrupole contraction, n −2 from the fact that the partition λ has been shortened by 4 and n 2 coming from the fact that k has increased by 1. The overall factor becomes then n −4 n 3 n −2 n 2 = n −1 → 0, which proves (3.17). Then we observe (using a similar justification as above) that at large n we get After introducing a new operator B = A + βn 3 α+βγ I, we obtain Now we come back to the full Hamiltonian, which can be rewritten as +ǫωa † a a a + g nω 2 (T r(a † a † a † a) + T r(a † aaa)) (3.23) is non-negative definite at large n if ω δ > 2.
0 +ẽ 0 n 2 I + O( 1 n ), we can easily find a lower bound for e 0 2 (and for the whole spectrum) in the planar limit e (lower) 0 This also shows the mechanism how the 1 n T r(a †4 + a 4 ) part of the Hamiltonian (so terms of order O(n)) shifts the whole spectrum by const.n 2 . We refer the reader to Table 1 for several numerical values.

Membrane Matrix Models
We will start with the optimized Fock space decomposition for the MMM (1.9) and obtain an upper bound for the ground state energy for arbitrary d. Later, we will produce a lower bound for the spectrum based on a direct generalization of Theorem 1 (and the preceding lemmas) to the multi-matrix case. We restrict the Hamiltonian to the constrained by (1.8) subspace of H ω i.e. to the SU (n) × SO(d) invariant sector and thus it is natural to assume that the optimized Fock space frequencies are all the sameω I ≡ω a I i I =ω ∀I, I = (a I , i I ), a I = 1, ..., n 2 −1, i I = 1, ..., d and consider only Fock spaces with all the frequencies equal. Using ) and the completeness relation (4.8) and therefore (4.11) Then the condition (2.14) is equivalent to which shows thatω = 3 4π 2 (d − 1). Therefore for the choice ω =ω,

the renormalized Hamiltonian becomes
where R N ∝ 1 , hence ||R N ψ|| → 0 ∀ψ ∈ Hω, and the ground state energy for the optimized Fock space approximation (i.e. the gaussian upper bound) is given by .

(4.15)
Let us point out that the scaling given by γ crit leads in fact to 't Hooft's scaling, since the quartic potential in the rescaled Hamiltonian (4.14) is multiplied by 1 n . Moreover, this is the only scaling for which the optimized Fock space frequency is non-trivial, i.e. 0 <ω < ∞ (cp. (3.8) for the AO, where 't Hooft's coupling is the only one leading to a non-trivial redefinition of the optimized mass/frequency). As we will see in Section 5, 4ω provides a crude approximation for the mass gap in the SU (∞) × SO(d) invariant sector.

Spectral bounds
In this section we will present estimates for matrix elements of various parts of the Hamiltonian (4.14) leading to a lower bound for the ground state energy in the planar limit, which is a direct generalisation of the procedure introduced in Section 3.1 for the AO. In order to obtain a lower bound for the spectrum, one has to take into account the quartic interaction term n −4 : V N : in the SO(d) × SU (n) invariant sector. As for the AO, one can show that the only divergent matrix elements of H N + β N I (4.14) in the appropriately modified partitions basis of the space of invariants are those coming from the operators T r(a †4 ), T r(a 4 ) due to the fact that the optimal choice of the Fock space frequencies (4.12) eliminates the T r(a †2 + a 2 ) part from the game. As previously, a special treatment of the T r(a †4 + a 4 ) part of H N is the main point of our construction.

Let us introduce two composite annihilation operators
A := T r(abcd)a ai a bi a cj a dj ≡ (iijj), (4.16)

17)
A, A † = 2dn 4 (d + 1) + O(n 2 ), (4.18) 20) and the following SU (N ) × SO(d) invariant states, keeping the dependence on A and B explicit, cp. (3.14), where Λ = {λ Λ , I Λ } is a partition of some k ∈ N equipped with the SO(d) invariant structure of the state ψ klΛ , i.e. λ Λ ⊢ k. I Λ is a sequence containing the information about SO(d) indices "compatible" with the SU (n) partition structure 6 , e.g. for λ = (2 2 ), i.e. λ 2 = 2, λ i = 0, i = 2, a possible I could be I = (ii, jj), but not I = (ij, ij). The corresponding states ψ klΛ would be then and the not allowed state We denote the subspace spanned by ψ klΛ by I (n) ω . Note that the basis (4.21) is not orthonormal and at large n we have the orthogonality relation where G(I Λ , I Λ ′ ) is the Gram matrix of (4.21) restricted to span(ψ klΛ ) with k, l, λ Λ fixed. The normal ordered quartic potential in (4.14) can be rewritten as and therefore it is useful to introduce the following operators asymptotically stabilizing the subspaces span(ψ klΛ , ψ k+1,l−1,Λ , ψ k−1,l+1,Λ ) λ Λ =λ (which means that they only mix up the SO(d) indices and not the SU (n) partition structure) which change also the SU (n)-partition structure (the λ Λ part). Now we come back to the full Hamiltonian, which can be rewritten in terms of the recently introduced operators for some 0 < ǫ < 1, cp. (3.22)-(3.24). Following the strategy established for the AO, we will express the action of the first part of the Hamiltonian, i.e. (4.36) in terms of A, B and the representation of S 1 + S 2 − 2S 3 in the finite dimensional blocks P λ,k,l Iω := span(ψ klΛ ) λ Λ =λ , then we will determine for which value of ǫ the middle part (4.37) is non-negative, which will allow to bound the full Hamiltonian from below by an operator of the form as in (4.36) modulo terms of order O( 1 n ) which do not affect the divergent vacuum energy in the planar limit. Let us perform this in several steps.

43)
n 2ẽ 0 = n 2ẽ 0 (ω, ǫ 1 , ǫ 2 , g) = Ω 1 α 2 + Ω 2 β 2 + 2Ω 3 αβ + 2 ǫ 1 α n + 2 ǫ 2 β n , (4.44) Proof. Repeating the argument from Lemma 1, as the SO(d) structure does not affect the order in n, one gets immediately that Using the relations (holding at large n) one can express the action of S 1 , S 2 , S 3 in terms of where G 1 (λ Λ ), G 3 (λ Λ ) are the finite dimensional representations of S 1 , S 3 in the subspaces P λ Λ ,k,l Iω corresponding to the partition Λ. Now, by combining (4.53)-(4.55), one can express the action of S 1 + S 2 − 2S 3 in terms of A † A, B † B, A † B + B † A and the finite dimensional representation G(λ Λ ) of 1 n (S 1 + S 2 − 2S 3 ) in P λ Λ ,k,l Iω, which determines Ω 1 , Ω 2 , Ω 3 . In order to get rid of the linear terms in A + A † and B + B † we introduce two new operators A = A + αI,B = B + βI, (4.56) and require that the coefficients in front of the A,B-linear terms are zero, which gives exactly (4.45) as well as the shift of the vacuum energyẽ 0 .
Note that in the planar limit, S 2 −S 3 ≥ 0, because S 2 ≃ (ǫij) † (ǫij) and S 3 ≃ (ǫji) † (ǫij), (ǫij) = (ǫab)a † ia a † jb , and thus By taking (α 2 + β 2 ) = 1 and 2αβ = −1 we get S 2 − S 3 ≥ 0. It turns out that the operator S 1 − S 3 is not positive definite, i.e. the eigenvalues of G(λ) can be negative. Since the operator in Lemma 4 will be used as a lower bound for the full Hamiltonian, one has to assure that (4.39) is non-negative definite. This is the case when ω ≥ g 3 because asymptotically . We also need further bounds.
Proof. It remains to prove that H − N ≥ 0 at large n. Note that H − N can be rewritten, after performing a Bogoliubov transformatioñ , as a sum of non-negative operators. Indeed, by demanding that the mixed termsÂ †B +B †Â vanish, one finds the value of x such that for someΩ 1 > 0,Ω 2 > 0. Moreover, gG(λ) + ω|λ| ≥ 0 ∀λ since ω > g.
Since H − N ≥ (e We refer the reader to Table 3 for several numerical values.

Perturbative expansion
We have seen that the Hamiltonian of an interacting model with a quartic interaction satisfying the assumption in Lemma 1 can be rewritten as a sum of a non-interacting HamiltonianĤ 0,N := H 0,N + N e k,N respectively, where the index k takes values in the appropriate index space J N . Using the standard technique of stationary perturbations in quantum mechanics (Rayleigh-Schrödinger perturbation theory), i.e. assuming that the actual eigenvalues and eigenstates of the full Hamiltonian, E k,N and ψ k,N , can be expanded in a power series in ǫ N := ǫN −γ , where ǫ is a book keeping parameter 8 , one can compute a first few terms of the perturbative expansion for finite N , take the limit N → ∞, and in the end put ǫ = 1. The corrections to the eigenvalue E k,N of H N up to the third order read where Q 0 is the orthogonal projection on span(ψ It is of course not clear whether the series converges, not even if separate terms do in the limit n → ∞ and one usually expects that such a series is only asymptotic. However we shall see below that the first three terms turn out to provide an astonishingly accurate approximation for the ground state energy and the spectral gap for the AO and to be consistent with the bounds obtained for the MMM. We will also show that the spectral gap for the MMM remains finite at large n at least up to the 2 nd order. Another important aspect of the considered here perturbative expansion is that it establishes a natural connection between the calculation presented in the previous sections and the planar limit in gauge field theories. In the next subsection we will give examples of planar and nonplanar contributions (resp. leading and subleading) to the vacuum energy using a diagramatic representation of certain terms popping up in the above perturbative series (5.3), which also justifies why one can neglect the operators of order O( 1 n ) in (3.24) and (4.38), at least in the vacuum energy calculations.

Vacuum energy corrections
Let us remind that the 0 th term gives the gaussian variational upper bound and point out that the 1 st order term is always zero. We will see that (at least up to 3 rd order) the leading terms of the perturbative contributions to the vacuum energy for the models considered in this note correspond to connected planar diagrams in the diagramatic representation and they are proportional to n 2 . Let us compute the second and third order corrections to the ground state energy of the AO (3.1). We put E 0,N = N e 0 and E where a ring with 4 outgoing branches represents the trace of a product of 4 matrices T a , a circle at the end represents an annihilation operator while a square stands for a creation operator carrying the indicated index. Thin lines between vertices represent Kronecker deltas resp. Wick contractions between annihilation and creation operators. It is clear that the maximal power of n is attained when adjacent indices in the two traces are contracted, which corresponds exactly to the drawn above planar contraction. If one contracts e.g. b withb and a withã instead then the contribution to (5.8) is subleading and the resulting diagram cannot be drawn in the plane without intersections of the contraction lines.
• 3 rd order correction for the AO If one replaces it with the operator : (a † aa † a) : T r(T e T f T g T h )a † e a † g a f a h then the resulting diagram is no longer planar and it leads to a subleading contribution to (5.10). These corrections decrease the error of determining the ground state energy at large g to 1.5 and 1.1 resp. (see Table 2).
where we have used the asymptotic formulas (4.47)-(4.52). A diagramatic representation of these contributions is qualitatively the same as for the AO, which leads to the conclusion that the operator : ((i † ij † j) + (i † jj † i) − 2(i † ji † j)) : in (4.38) is negligible in the planar limit.
The reader is referred to Table 3 for several numerical values, which show that the proposed perturbative expansion is consistent with our upper and lower bounds. One can see that the operators T r(a †4 ), T r(a 4 ), i.e. A, A † for the AO and (iijj), (ijij), (i † i † j † j † ), (i † j † i † j † ) for the MMM are the main sources of corrections to the vacuum energy for both models.

Spectral gap corrections
The spectrum of the singlet sector for the AO consists of a divergent vacuum energy and an infinite family of equally spaced excited states i.e. with a constant, finite energy gap ω(g) [14,15,16]: E λ = e 0 n 2 + ω(g) j jλ j . (5.14) The degeneracy of an energy level E k = n 2 e 0 + kω(g) is given by the number of partitions of k. This result is restored in our perturbative expansion, even in the 0 th order, so for U (n) invariant eigenstates ofĤ 0,N =ω(a † a) + e (0) 0 2 n 2 I. For later convenience we choose the second excited eigenstate ofĤ 0,N , i.e. ψ λ = N λ (a † a † )Ψ 0 (ω), λ i = 0, i = 2, λ 2 = 1, as a starting point for the perturbative expansion. The first order correction is of order O(1) The second and third order contributions E λ , E λ obtained in this way diverge like n 2 . However, the divergent parts are exactly equal to the corresponding corrections to the vacuum energy and thus the renormalized energies are finite. Therefore the spectral gap ω(g) up to the third order becomes (5.20) Table 4 shows several numerical values. The accuracy of our approximation is very good, also in the strong coupling limit g → ∞. The reason why it works so well is the following. In the optimized Fock space formulation the Hamiltonian takes the form, cp. (3.22)- (3.24), hence the effective coupling constant is not g, butg := g 4ω 3 . By using eq.(3.8), one gets max gg = 1 16 , attained in the limit g → ∞, which shows that the system in the optimized Fock space is coupled weakly even if the coupling g in the original formulation is strong. Let us also point out that each order ing gives the correct scaling of the spectrum as (5.22) Table 4: Comparison of the exact value of the spectral gap ω(g) [13] with the perturbative expansion in the optimized Fock space ω (0) =ω, ω (1) , ω (2) , ω (3) . Let us repeat the calculation for the MMM. As the starting point for the perturbation expansion we choose an analogue of the previously considered for the AO ψ λ , i.e. the first SO(d) × SU (n) invariant excited state ofĤ 0,N = 2ω(i † i) + e (0) 0 n 2 I, namely ψ Λ = N Λ (i † i † )Ψ 0 (ω) and thus E  23) and the second order is again divergent with the singular part equal to the corresponding correction to the vacuum energy e Λ − e is finite at latge n.
hence the effective coupling constant isg := π 2 ω 3 = 1 4(d−1) . We have seen in Section 2 that the special choice of the Fock vacuum (resp. Fock space frequencies,ω I ) eliminates one of the divergent parts of the Hamiltonian, i.e. T r(a 2 + a †2 ) ∝ O(n) and minimizes the vacuum expectation value Ψ 0 (ω), H N Ψ 0 (ω) = e (0) 0 n 2 providing a rigorous upper bound for the ground state energy at large n in a concise way for a rather general family of Hamiltonians with a quartic potential (1.10). A further study of the remaining divergent operators, i.e. 1 n T r(a †4 + a 4 ) allows to produce astonishingly good lower bounds for the ground state energy for the AO as well as for the MMM (Theorems 1 and 2) in the planar limit. A similar mechanism shall lead to the true vacuum in Fock space, however the task is rather challenging. Our results suggest that one should consider bases consisting of non-linear coherent states instead of polynomials in simple U (n)/SU (n) invariants (3.11), (4.21), which would fully capture the divergent behaviour of 1 n T r(a †4 + a 4 ). For instance, a possible vacuum state φ 0 for the operator 1 n 4 B † B from Lemma 2 has the following form and C can be found recursively as an infinite sum of single-trace operators with increasing length, C = k=0 C k , C 0 = 1 4n 4 A. Nevertheless, the two considered here matrix models admit a perturbative formulation, which provides evidence for the validity of the planar limit and it seems to be efficient in approximating the spectrum, including the strong-coupling regime for the AO, allowing to rederive the wellknwon result of Brezin et al [11] for the ground state energy with high accuracy. In particular, the Hamiltonian for the Membrane Matrix Models (1.7), originally not containing a quadratic term, gains an effective mass in the optimized Fock space and can be treated as a perturbation of a non-interacting Hamiltonian by the original normal ordered quartic potential with the effective coupling constantg = 1 4(d−1) , which means that the model contains a small hidden parameter. This is perhaps not surprising in the context of the established validity of 1/D expansions (where D = d + 1 is the dimension of space-time) in various Yang-Mills theories, see [27,28]. Our numerical results, Table 3 and 5, show that the perturbative expansion is stable, especially for higher dimensions of the embedding space, the results are more accurate for larger values of d, and that it leads to a finite energy gap at least up to the second order. One expects that cancellations of the vacuum contributions (corresponding to vacuum diagrams), like in formulas (5.16), (5.19) and (5.24), should occur in any order, depending on the validity of the linked cluster theorem for this model. Therefore one can conjecture that the spectrum of the properly rescaled family of Membrane Matrix Models (4.14) in the large n limit remains purely discrete with a finite spectral gap and a divergent vacuum energy, i.e. a generic energy level E Λ can be written as E Λ = e 0 n 2 + e Λ , 0 < e Λ < ∞. One can perform a perturbative expansion according to (5.3), treating : V := 1 2 ǫ abc ǫ ab ′ c ′ : x ib x ib ′ x jc x jc ′ : as a perturbation. Table 6 contains a comparison of the exact ground state energies for d = 2, [29], and d = 3, [30], with our computation.  (ω + 4αβ)b † a b a + (ωα 2 + 2αβγ)N I, (B.6) provided αβ = − γ ω , resp. x = − 1 2 arcsin 2γ ω . Therefore the rescaled ground state energy is given explicitly (as the whole spectrum of H N ) and it satisfies the following e 0 := ωα 2 + 2αβγ = − 2γ 2 ω + ω[sin( This essentially shows what happens if one uses a 'wrong' basis for a given Hamiltonian with infinitely many degrees of freedom. The original divergent terms a a a a + a † a a † a ∝ O(n) can be eliminated after switching to the correct Fock space generated by b † a at the cost of producing a divergent multiple of the identity operator as in (B.6).
Another method of getting a lower bound for e 0 is based on our considerations from Section 4.1 and 3. Using the same argument as in Lemma 2, one can show that at large n a † a a a ≃ Let us introduce a new operator B = A − ǫI , which allows to complete the first two terms of (B.11) to a full square |λ|P λ +ẽ 0 N I, (B.12) withẽ 0 = 2γǫ + ω N ǫ 2 , provided ǫ = − N γ ω . Now it follows that H N ≥ẽ 0 N I + O( 1 n ) and the true rescaled ground state energy is bounded from below byẽ 0 e 0 ≥ẽ 0 = − γ 2 ω , (B.13) which agrees with the exact computation (B.7). This shows again that a special treatment of divergent matrix elements of the Hamiltonian (||A † ψ λ || ∝ O(n)) allows to produce a reasonable lower bound for the divergent part of the ground state energy in the planar limit, i.e. after ignoring terms of order O( 1 n ).