Solving anharmonic oscillator with null states: Hamiltonian bootstrap and Dyson-Schwinger equations

As basic quantum mechanical models, anharmonic oscillators are recently revisited by bootstrap methods. An effective approach is to make use of the positivity constraints in Hermitian theories. There exists an alternative avenue based on the null state condition, which applies to both Hermitian and non-Hermitian theories. In this work, we carry out an analytic bootstrap study of the quartic oscillator based on the small coupling expansion. In the Hamiltonian formalism, we obtain the anharmonic generalization of Dirac's ladder operators. Furthermore, the Schrodinger equation can be interpreted as a null state condition generated by an anharmonic ladder operator. This provides an explicit example in which dynamics is incorporated into the principle of nullness. In the Lagrangian formalism, we show that the existence of null states can effectively eliminate the indeterminacy of the Dyson-Schwinger equations and systematically determine $n$-point Green's functions.


Introduction
Two main goals of the bootstrap methods are to achieve a deeper understanding of the strong coupling physics and to provide concrete computational schemes for extracting precise predictions of strongly coupled theories. Before delving into the intricate quantum field theories in physical dimensions, a useful strategy is to first study their low dimensional counterparts, such as zero-dimensional and one-dimensional models, hoping that certain insights may be independent of the spacetime dimension. Analogously, the perturbative expansion in a small coupling constant may also elucidate some strong coupling physics if certain general structure is independent of the coupling constant. With these motivations in mind, we study the quantum mechanical bootstrap of the quartic oscillator analytically based on the weak coupling expansion in this work. 1 Recently, matrix models and quantum mechanical models have been investigated by bootstrap methods . They are usually implemented with positivity constraints associated with the physical assumption of unitarity. 2 However, the violation of reflection positivity frequently occurs in statistical physics models. The relevant theory can be related to non-unitary quantum systems, where the positivity principle does not apply. The bootstrap study of such models necessitates alternative principles. One of the potential candidates is the principle of nullness, i.e. the existence of many null states [25]. In the 1 The meaning of the quantum mechanical bootstrap is that the observables, such as energy spectra and matrix elements, are studied using consistency relations, without referring to explicit wave functions. This approach can be traced back to Heisenberg's original perspective that led to the establishment of quantum mechanics. 2 See however [17] for the use of positivity constraints in non-Hermitian models.
context of 2d conformal field theory [26,27], the existence of null states is closely related to the quantization conditions on the scaling dimensions and the central charges of the minimal models, which imply that these physical parameters can only take certain discrete values. Only a small subset of the minimal models further obey the unitarity assumption. A prominent example of the non-unitary case is the M(5, 2) minimal model [28,29], which describes the critical behaviour of the Yang-Lee edge singularity [30][31][32][33]. 3 The principle of nullness postulates that many states are orthogonal to all states. From the algebraic perspective, the null states are related to the left ideals in the operator algebra since the action of any operator on a null state also gives a null state. For the standard quantum mechanics with single position operator, the operator algebra is generated by the position operator x and the momentum operator p. They satisfy the canonical commutation relation (1.1) Below, we will set ℏ to one. A representation of the abstract operator algebra can be induced by a state which is a linear functional mapping the elements of the operator algebra to complex numbers. Then one may construct the space of states as a representation of A on H π : A → End(H) , (1.3) and show the existence of a vector ψ ρ ∈ H with ρ(A) = ⟨ψ ρ |A|ψ ρ ⟩ := ⟨ψ ρ , π(A) ψ ρ ⟩ , (1.4) for all A ∈ A. Typically, H is a quotient vector space H := A/N , (1.5) where N is a left ideal in A, corresponding to the subspace of null states. The null subspace plays a crucial role in the null bootstrap program, which aims to classify physical solutions and extracts concrete predictions from the null state condition [25]. From the algebraic viewpoint, this can be viewed as a classification program based on the ideals in operator algebra. Under some conditions, the rigorous construction of a Hilbert space H with a cyclic vector ψ ρ is known as the Gelfand-Naimark-Segal construction [38,39]. For physicists, the dynamics of a concrete quantum mechanical model is specified by a Hamiltonian, whose eigenstates are labelled by the energy E. 4 The measurable information includes the energy spectrum and the matrix elements. The choice of a Hamiltonian 5 and then an energy eigenstate leads to a concrete representation of the operator algebra. The mapping (1.2) is realized by the expectation values of different operators in the chosen state. One can also reconstruct the space of states. Matrix elements can be obtained from ladder operators that connect different eigenstates. For example, the non-diagonal matrix elements in the energy representation are given by ⟨E|A|E ′ ⟩ = ⟨E|A L E ′ E |E⟩.
For concrete applications of the null bootstrap, let us consider some basic quantum mechanical models. In the textbook example of the harmonic oscillator the eigenstates satisfying H|n⟩ = E n |n⟩ are connected by Dirac's ladder operators where the lowering and raising operators are The energy levels are labeled by n. It is well known that the Hamiltonian H = a † a + 1 2 is linear in the number operator N = a † a . (1.9) Together with Dirac's ladder operators, they form a closed algebra A direct consequence of the commutators is that the energy spectrum has a constant spacing, i.e. E n+1 −E n = 1. Therefore, Dirac's ladder operators furnish a natural set of building blocks for the operator algebra of the harmonic oscillator. For a spectrum that is bounded from below, the ground state with the lowest energy should be annihilated by the lowering operator a|0⟩ = 0 . (1.11) This annihilation equation provides an example of the null state condition generated by the lowering operator. We can also construct the null states from excited states, such as a k |n⟩ = 0 with k = n + 1. The stationary Schrödinger equation also gives rise to null states which is called trivial in [25] because it is satisfied by definition and does not lead to any constraint on E.
As the harmonic oscillator is well understood, it is more interesting to study the anharmonic oscillator, which is usually not exactly solvable at finite coupling. 6 Since quantum 6 Some special potentials can also lead to exact solutions, such as the Morse potential and the Pöschl-Teller potential. We refer to [40] for a review of the factorization method and the ladder operators associated with the underlying Lie algebra. We would like to emphasize that the existence of ladder operators does not rely on dynamical symmetries. mechanics can be viewed as a (0 + 1)-dimensional quantum field theory, the anharmonic oscillators provide a testing ground for novel field theory methods. For example, a potential with a quartic term x 4 can be viewed as a ϕ 4 theory in (0+1) dimension. A curious question is whether there exists a natural set of building blocks for the anharmonic operator algebra as well. We will focus on the quartic case: 13) which is related to the Ising universality class in higher dimensions. It is known that the corresponding energy spectrum does not have a constant spacing, which in fact depends on the occupation number nonlinearly. Energy eigenstates are still expected to be connected by certain ladder operators. We would like to know if these ladder operators have a simple algebraic structure. If not, we may need completely different operators to connect different pairs of eigenstates.
Recently, the nonperturbative null bootstrap results of the quartic and cubic anharmonic oscillators suggest the existence of some underlying algebraic structure in the anharmonic ladder operators [25]. However, these properties are only studied numerically and approximately due to the nonperturbative truncation scheme. In order to obtain analytical and exact results, we assume g is small and make use of perturbation theory in this work. The null state in the anharmonic oscillators receives perturbative corrections (1.14) The trivial null state again takes the form (H AH − E 0,AH ) |0⟩ AH = 0. We will use the null state condition to formulate the bootstrap constraints for the observables, to determine the energy spectrum and to derive the analytic expressions of the ladder operators. More ambitiously, the bootstrap program aims to classify and solve the dynamical information by basic principles and consistency constraints. We have a curious question • How is the dynamics encoded in the null bootstrap?
To address this question to some extent, we will show that dynamical constraints from the Schrödinger equation are related to certain null states generated by ladder operators.
After investigating the quantum mechanical bootstrap in the Hamiltonian formalism, it is natural to consider the Lagrangian formalism. Therefore, we also apply the null state condition to solving Dyson-Schwinger (DS) equations [41][42][43], the self-consistency equations for the n-point 7 Green's functions. Since the DS equations can serve as an alternative to operator theory, we expect to recover the same results in the Hamiltonian formalism. When solving the DS equations, one obstacle is that they form an underdetermined system, as higher DS equations involve higher-point Green's functions. In the traditional approach, the DS equations can be solved order by order in perturbation around the free theory, due to the additional constraints from the existence of the weak-coupling expansion. However, for the strong-coupling expansion, one needs to impose extra constraints to resolve the indeterminacy. 8 In a simple scheme, one can close the system by setting high-point connected Green's functions to zero, but this produces results that do not converge to the exact values, as emphasized recently in [45]. A more sophisticated approach is to replace high-point connected Green's functions by their large-n asymptotic behavior [45], which gives numerically accurate results. This approach has been carried out at d = 0 and seems more challenging at higher dimensions. A novel avenue proposed recently in [46] is to resolve the DS indeterminacy by the null state condition. It was shown that the approximate, numerical results converge rapidly to the exact values for both d = 0 and d = 1.
To obtain analytic and exact results, we will investigate the null state approach in perturbation theory. To be more explicit, we want to solve the following set of DS equations: The Green's functions are the correlation functions of the Heisenberg picture operators where T is the time-ordering operator and |0⟩ denotes the ground state.
In the traditional approach, the Green's functions can be determined order by order starting from the zeroth-order free theory solutions, together with the asymptotic behavior at infinity. In the null state approach, the derivation of the physical solutions does not rely on the asymptotic behavior at infinity or free theory solutions. Instead, it is crucial that certain linear combinations of x(t) and d dt x(t) acting on the ground state amounts to higher-order terms in the perturbative expansion. In the harmonic limit g = 0, we have 17) which is precisely the annihilation equation (1.11). According to the Heisenberg equation of motion, we have d dt , H] = p(t) and this relation remains exact after turning on the quartic perturbation in x(t). Since the DS equations are formulated in the Lagrangian formalism, we will use the time derivative d dt x(t) instead of the momentum p(t). As before, the annihilation equation receives perturbative corrections in g An important consequence is that its inner products with certain states give rise to a set of relations for the Green's functions. We can solve for the Green's functions order by order using the perturbed ladder operators. The rest of the paper is organized as follows. In section 2, we use the null bootstrap to investigate the quartic anharmonic oscillator perturbatively. We present two different procedures in 2.1 and 2.2, and discuss the algebraic properties of the operator algebra in 2.3. We then consider the DS equations in section 3, where we use the null state condition to resolve the indeterminacy and solve for the n-point Green's functions. In section 4, we summarize the results and discuss future directions. For comparison, the results from the traditional perturbation method are summarized in appendix A.

The null bootstrap
In the null bootstrap, physical solutions are derived from the null state condition. By definition, an exact null state ψ null is orthogonal to arbitrary test states, so we have where ψ test can be any state. By considering more general types of test states, one can deduce stronger constraints, then the properties of the physical states annihilated by the ladder operators are determined more precisely. At finite coupling, this can be performed numerically and approximate results of high precision are obtained by truncating the search spaces of the null and test states. In perturbation theory, we can carry out the null bootstrap analytically and derive the exact perturbative series using truncated search spaces of finite dimensions. We will focus on the quartic anharmonic oscillator with H = 1 2 p 2 + 1 2 x 2 + gx 4 . To simplify the notation, we will not write "AH" explicitly. In the small g expansion, the null state condition (a k + O(g))|ψ⟩ = 0 determines the k low-lying states, where a is Dirac's lowering operator. We will focus on the expectation values associated with an energy eigenstate labelled by E ⟨E|O|E⟩ = ⟨O⟩ E . (2. 2) The perturbative null bootstrap will be carried out based on three assumptions: 1. The Hamiltonian is Hermitian and the eigenvalue E is real. We have the Schrödingerlike equations which can be derived from the Schrödinger equation H|E⟩ = E|E⟩. It turns out that the expectation value ⟨x m 1 p m 2 ⟩ E can be expressed in terms of the coupling constant g, the energy E, the expectation values ⟨x 2 ⟩ E and ⟨1⟩ E . We choose the normalization convention ⟨1⟩ E = 1.
2. The independent parameters E and ⟨x 2 ⟩ E can be written as perturbative series which are formal power series in the coupling constant g.
3. The expectation value ⟨x m 1 p m 2 ⟩ E is regular in the g → 0 limit, where m 1 , m 2 are non-negative integers. This implies that ⟨x 2 ⟩ E can be expressed in terms of g and E so ⟨x m 1 p m 2 ⟩ E is a function of the coupling constant g and the energy E.
The third assumption is particularly interesting. It is not immediately clear why we have (2.5). In fact, this is similar to the additional constraints from the existence of a weakcoupling expansion when solving the DS equations. The crucial point is that the consistency relations, whether they are the Schrödinger-like equations (2.3) or the DS equations (1.15), allow singular behavior of the expectation values or the Green's functions in the limit g → 0.
The assumption that the g → 0 limit is regular or the weak-coupling expansion exists implies the absence of singularities and leads to additional constraints on the free parameters. 9 The basic idea behind this type of constraints is that the physical data of a weakly interacting theory should allow continuous deformation into the free theory limit.
To be more explicit, let us examine some concrete expectation values. Under the first assumption, ⟨x m 1 p m 2 ⟩ E is expressed as where P i (E, ⟨x 2 ⟩ E ) are polynomials in E and ⟨x 2 ⟩ E , and q is the highest order of 1/g. We assume that the parity symmetry of the quartic anharmoinc oscillator is not broken, so the expectation value vanishes if m 1 + m 2 is odd. The important point is that the expression (2.6) contains terms with negative powers in g, which are singular in the g → 0 limit. To eliminate these singularities, we impose additional constraints on the small g expansion of E and ⟨x 2 ⟩ E . This will lead to a set of relations among ⟨x 2 ⟩ (j) E and E (j) . For example, the case of m 1 = m 2 = 2 reads whose g → 0 limit is singular if E and ⟨x 2 ⟩ E are completely independent. To avoid the singular behavour, the perturbative series of E and ⟨x 2 ⟩ E should satisfy the constraint ⟨x 2 ⟩ (0) E = E (0) . At higher powers, the expectation value ⟨x 4 p 2 ⟩ E reads Note that this is consistent with (2.8) because the 1/g 2 singularity is removed by ⟨x 2 ⟩ (0) E = E (0) . In addition, the 1/g singularity is eliminated by (2.10) By considering higher powers in x and p, we obtain more stringent constraints and determine higher order terms in the perturbative series, such as The regularity of E . If a high power expectation value has a regular limit, then the expectation values with low powers are automatically regular. The regularity constraints of different expectation values are consistent with each other. In fact, it suffices to impose that ⟨x m ⟩ is regular in the g → 0 limit. Furthermore, we can repackage these relations by expressing ⟨x 2 ⟩ E in terms of E, which is shown in (2.5).
In principle, the variable ⟨x 2 ⟩ E is completely determined by E to all orders in perturbation theory. In practice, we only need to know ⟨x 2 ⟩ E in terms of E to certain order. This is because we will deal with a finite set of expectation values. Suppose that the strongest singularity among them is 1/g q 0 and we are interested in perturbative corrections up to order g i . We should examine an expectation value with a singularity 1/g q 0 +i+1 , which is not in this set of expectation values. The regularity assumption then determine ⟨x 2 ⟩ E in terms of E to order g q 0 +i . As a result, all the expectation values under consideration are regular and expressed only in terms of E to order g i .
We are now in the position to carry out the complete procedure of the null bootstrap, which consists of two steps: 1. Determine the full spectrum and exact ladder operators by Schröding-like equations.
2. Impose the null state condition on the unboundedly low energy states and solve the remaining parameters.
In general, it is challenging to derive the complete, non-perturbatively spectrum in the first step. 10 This obstacle is circumvented by the reduced procedure, in which we directly solve for the low-lying data using the null state condition. In perturbation theory, we can show that the low-lying energy spectrum and matrix elements obtained in the reduced procedure match with those from the complete procedure. Therefore, the reduced procedure is as strong as the complete procedure concerning the low-lying information.

Complete procedure
In the complete procedure, we first determine the energy spectrum by the Schrödinger-like null state condition which corresponds to the stationary Schrödinger equation The two eigenstates |ψ E ⟩ and |ψ E ′ ⟩ is connected by the ladder operator L E ′ E . We require that (2.13) is valid for arbitrary O test in the form of polynomials in x and p. We first focus on the level-1 ladder operators and solve the null state condition to order g 3 . Then we derive the level-k ladder operators using the level-1 ladder operators. In the end, we use these ladder operators to compute matrix elements. Before presenting the details, it is useful to note some general features of the solutions to (2.13). In the g → 0 limit, the reference energy E is labeled by the energy level n. The solutions to the Schrödinger-like equation with energy E ′ are labeled by k, which denotes the number of energy levels shifted by the ladder operator L E ′ E . We adopt the following notations for convenience: (2.14) which L ±k are the level-k ladder operators. The discrete energy spectrum and ladder operators can be continuously deformed to the anharmonic case and we assume the existence of the following perturbative series In general, there exists a trivial family of solutions for L ±k due to the Schrödinger equation 10 We notice that the solutions for the full energy spectrum and exact (level-1) ladder operators are always derived at the same time. We are led to the question whether they are equivalent information. It is clear that we can deduce the full spectrum from the exact (level-1) ladder operators. On the other hand, the equation where L ±k trivial |E n ⟩ = 0 automatically. As Dirac's ladder operators for the harmonic oscillator, a nontrivial solution for L ±k should connect two energy eigenstates where we have used "n.t." to indicate the nontrivial part. In addition, the Schrödingerlike equation (2.13) leads to recursion relations for E n and one can deduce the complete energy spectrum from one energy level. For a stable system, the energy spectrum should be bounded from below. The states lower than the ground state should satisfy the null state condition, such as In this way, the ground-state energy E 0 is determined order by order in g. Using this boundary condition, we can further deduce the energy spectrum from the energy recursion relations. The null state condition (2.20) can be generalized to the expectation values of the excited states, which reads ⟨O test L −k ⟩ En = 0 with n = k − 1.
The order g 0 First, we consider (2.13) at order g 0 . The explicit expressions of L ±k are formulated in terms of polynomials in x and p. We truncate the search space of null states by the ansatz where K denotes the degree of the polynomial and the coefficients are assumed to be real.
To remind the reader, we use k to denote the changes in the energy level number. At the lowest truncation order K = 1, we find two sets of solutions labeled by ±1. They correspond to the raising and lowering operator respectively ±1 are free real parameters. The trivial terms are absent because they are at least quadratic in x and p. For higher K, 11 the Schrödinger-like equation (2.13) determines A (0) m 1 m 2 up to some free real parameters, which can depend on E (0) n . The general solutions for the level-1 ladder operators take the form 12 Since the sum of a nontrivial solution and a trivial solution is also nontrivial, there are some ambiguities in the explicit expressions of the nontrivial part. Below, we will remove these ambiguities by fixing the normalization and requiring the nontrivial part is E n -independent. 11 The k > 1 solutions appear when K > 1, but we focus on the k = 1 solutions at the moment. 12 We assume that n is sufficiently large such that |n⟩ cannot be annihilated by ladder operators in any truncation K.
Since the expectation values are expressed as functions of E n , it is useful to find the explicit expression of E n first. At order g 0 , the Schrödinger-like equation (2.13) gives the energy recursion relation 13 The null state condition (2.20) yields E We will choose a specific normalization for L ±1 |n⟩ based on this result.
To determine the free parameters, we first impose the following condition: • Normalization condition: |L +1 |n⟩| 2 = n + 1 and |L −1 |n⟩| 2 = n. 14 However, this does not constrain the trivial terms since they do not contribute to the normalization. To fix the expressions of the ladder operators completely, we further impose • Gauge-fixing condition: the nontrivial part is independent of E n .
In this way, the trivial part is quotient out. However, the phase ambiguity has not yet been removed. Since the parameters are real, we have two sets of results for the ladder operators with an overall sign difference. We can choose the phase such that the nontrivial parts are identically the same as Dirac's ladder operators where "n.t." means the nontrivial part. The general solution (2.23) can be recovered by adding trivial solutions and overall normalization factors. Furthermore, we fix the relative phases of the energy eigenstates by The conditions we have imposed to determine the explicit expressions of the ladder operators are not limited to the zeroth-order calculations; higher-order results also follow from the same conditions.
The order g 1 We extend the analysis to order g 1 . Since the zeroth-order ladder operators have been solved, we set the zeroth order part to (2.26)

28)
13 There is also an equivalent relation E n + 1. 14 The difference in the norms is consistent with the commutation relation of L+1 and L−1. and use K to denote the truncation order of the g 1 ansatz As the free parameters start at first order, the trivial part at order g 1 takes a simple form 30) and the anharmonic corrections to H and E n are of higher order.
To solve the Schrödinger-like equation (2.13) for arbitrary test operators, it turns out that the truncation parameter in (2.29) should satisfy K ⩾ 3, which corresponds to the minimal shift in the energy level, i.e. k = 1. We gradually increase the value of K and extract the general solution The remaining freedom resides in the choice of the normalization and the trivial terms.
Here the phase factor is determined because we require that the parameters are real and the normalization does not depend on g.
As above, we first solve for E n before choosing a normalization. At order g 1 , the energy recursion relation reads (2.32) The null state condition (2.20) leads to the boundary condition E (1) 0 = 3 4 , so the first-order energy corrections are given by Then we impose the normalization and the gauge-fixing conditions and obtain a unique, explicit expression for the nontrivial part One can check that the lowering and raising operators are Hermitian conjugate to each other In Appendix A, we summarize the results from the traditional perturbation method and they agree exactly with the above.

Higher orders
At order g 2 , we use the known expressions of the nontrivial parts at order g 0 and g 1 ±k + . . . , (2.36) and the unknown g 2 order terms are When solving the null condition (2.13), the lowest truncation parameter is K min = 5, which corresponds to the case k = 1. By increasing K, we obtain the general solution The recursion relation from (2.13) 39) and the boundary condition E Imposing the normalization and the gauge-fixing conditions, we have a unique expression for the nontrivial part It is straightforward to repeat the procedure at order g 3 and the results are and All these results agree with those from the traditional method in Appendix A. In principle, we can perform the complete procedure of the null bootstrap and determine the ladder operators and the energy spectrum to arbitrary order in g.
Ladder operators with higher level k > 1 So far we have not considered the k > 1 solutions. In fact, we can construct nontrivial part of the level k > 1 ladder operators from multiple level-1 ladder operators They can be obtained from the normalization and gauge-fixing conditions: • The norms are set by |L +k |n⟩| 2 = (n + 1) k and |L −k The second condition is the same as the harmonic oscillator, while the first one can be understood as the consequences of the repeated action of L ±1 . Although the general solutions are much more involved, we verify that they can be written in terms of L ±1 n.t.
Therefore, there is no new independent solutions at k > 1. The bounded-from-below energy spectrum corresponds to the case with L −1 . All the nontrivial algebraic information is encoded in the level-1 ladder operators L ±1 n.t. . Below are some general comments from the algebraic perspective. In contrast to the ideals in commutative algebra, the non-commutative nature of operator algebra leads to nontrivial constraints on the eigenenergies E k and the fundamental ladder operators L ±1 that are compatible with the Hamiltonian H. The null bootstrap is a program about the systematic classification of the set of consistent data 15 We can generalize the Hamiltonian and energy eigenvalues to a set of mutually commuting operators {H (i) }, i.e. conserved quantities, and the corresponding "good" quantum numbers {E (i) }. For example, we could include the Z 2 parity operator that commutes with the Hamiltonian of the quartic anharmonic oscillator. The meaning of "consistent data" is that the dynamical constraints associated with the Hamiltonian is encoded in the null state conditions associated with the ladder operators. This will be discussed in more detail in the section 2.3.

Matrix elements
Besides the energy spectrum, there are other physical observables, such as the matrix elements of an operator O. The diagonal elements are the expectation values discussed above, which are expressed in terms of the energy E n and the coupling constant g. The off-diagonal elements can be computed using the ladder operators where we have written ⟨. . .⟩ n ≡ ⟨. . .⟩ En . Let us consider O = x as a simple example. Note that the position operator x can be written as (2.47) Due to the orthogonality of different energy eigenstates, the non-vanishing matrix elements, to order g 1 , are ⟨n|x|n ± 1⟩ and ⟨n|x|n ± 3⟩ The rest can be obtained by a change of variable ⟨n|x|n − j⟩ = ⟨n|x|n + j⟩| n→n−j , where j = 1, 3.

Reduced procedure
The reduced procedure is motivated by the difficulties in obtaining the complete energy spectrum and the exact ladder operators in a non-perturbative scheme. Nevertheless, we can study the low-lying state directly and obtain their energies and matrix elements using the reduced procedure. Below, we will show that the perturbative results are in exact agreement with those from the complete procedure. To obtain the low-lying eigenenergies, we consider the null state condition 16 ⟨O testL±k ⟩ n = 0 , (k > n) . (2.50) We have put a tilde on L ±k to emphasize that the operatorL ±k is not a ladder operator. Instead, it could be a combination of all the ladder operators whose levels are larger than n. In practice, we need to consider a truncation scheme labelled by the number k. We will see how the truncation depends on k below. The small g expansion reads (2.51) We will consider the null state condition (2.50) order by order by assuming thatL ±k is a polynomials in x, p. The k = 1 truncation is imposed with the ansatz L ±1 = degree-1 polynomial + g (degree-3 polynomial) (2.52) The reason for the choices of the degrees is that they are the minimal ones, i.e. the null condition is not satisfied for arbitrary test operators if the degrees are lower. In fact, these minimal degrees correspond to the minimal level-1 ladder operators, which is sufficient for the null state condition associated with the ground state n = 0. For the excited states n > 0, we should consider the level-k generalizatioñ L ±k = degree-k polynomial + g (degree-(k + 2) polynomial) which have greater minimal degrees. Using this ansatz, the null state condition (2.50) determines the first k eigenenergies and yields some constraints onL ±k . In practice, we consider some specific values of k and solve the null state condition (2.50) order by order in g. For the solutions corresponding to lowering operatorsL −k , we obtain the low-lying energies from the k = 1, 2, . . . , 5 constraints We can extract the general expression for E n by assuming that each coefficient E n is a degree-(i + 1) polynomial in n. The results agree with those from the complete procedure. Some examples for the low-order solutions ofL −k arẽ The ellipses represent arbitrary terms that are allowed by the ansatz. We use L −k to denote the level-k lowering operators, which should be understood as of appropriate degrees such that they do not contradict the relevant ansatz. For a specific n, The operatorL −k is a combination of L −(n+1) , L −(n+2) , . . . L −k , making it is difficult to obtain the ladder operators using this result. Below we will consider a different type of null conditions to determine the ladder operators in order to compute the matrix elements. Now we consider the matrix elements. Suppose that we have obtained the eigenenergies of two states |n⟩ and |n ′ ⟩. We can also compute the matrix element using a slightly modified version of (2.46) where L n ′ ,n is the operator that connects the two states |n⟩ and |n ′ ⟩. Note that we have specified the two energy eigenstates connected by the ladder operator; the reason will be explained shortly. The ladder operator in (2.61) is obtained by first considering where we use the ansatz (2.63) The energies E n ′ and E n are known from the above discussion, instead of variables to be determined as in the complete procedure. For some low-lying energy eigenstates, there are differences between ladder operators obtained in this way and those from the complete procedure. For example, the ladder operator L 1,0 is where the last term amounts to a null state when acting on |0⟩. Despite the differences in the ladder operators, we obtain the same results for the matrix elements because the differences are related to null states and orthogonal to other states. For example, after choosing the normalization, although L 1,0 is not exactly L +1 , the results for the matrix elements are identical up to a phase factor ⟨0|O|1⟩ = (phase factor)⟨0|OL 1,0 |0⟩ = ⟨0|OL +1 |0⟩ . (2.65) To produce the correct factors in (2.61), we impose the normalization condition: • For n ′ ̸ = n, we have |L n ′ ,n |n⟩| 2 = (min(n ′ , n) + 1) |n ′ −n| .
In general, the matrix elements obtained in the reduced procedure are the same as those in the complete procedure up to a phase factor. Therefore, the reduced-procedure results for the low-lying matrix elements are as complete as those from the complete procedure.

Anharmonic operator algebra
In the complete procedure, we have derived the explicit expressions of the anharmonic ladder operators. We can further study their algebraic properties. It is natural to construct the anharmonic number operator from the ladder operators. which form a closed algebra as that in the harmonic case. However, the Hamiltonian H is a nonlinear function of the number operator and the commutators involving H are more complicated. As discussed earlier, the level-1 raising operator L +1 is precisely the Hermitian conjugate of the level-1 lowering operator L −1 (2.66) Here and below, we omit writing "n.  It is precisely the nontrivial dependence on H that leads to the nonlinear energy spacing in the occupation number n. Furthermore, G + and G − are not independent as they are closely related to the energy differences. If the operator H in G ± is replaced by a number E, we have which can be equivalently written as where n = 0, 1, 2, . . .. This perturbative series gives a precise approximation of the energy levels for sufficiently small g at low n. 18 Accordingly, the Hamiltonian can be expressed in terms of the anharmonic number operator which is consistent with replacing the occupation number n in (2.80) with N = L +1 L −1 . From the operator algebraic perspective, the left ideal generated by (H − E 0 ) is a subset of that generated by L −1 . Here {H, E 0 , L −1 } provides a concrete example of a set of consistent data. For higher states, we can use the generalization of (2.81): where m denotes the number of lowering operators and S is given by (

2.83)
For m = n, the stationary Schrödinger equation (H − E n )|n⟩ = 0 is encoded in the leveln null state condition (L −1 ) n+1 |n⟩ = 0, together with the assumption (L −1 ) n |n⟩ ̸ = 0. Although the statements in this subsection are examined to order g 3 , we believe that some are valid to arbitrarily high orders and even non-perturbatively.

Dyson-Schwinger equations
In section 2, we have solved the anharmonic oscillator in the Hamiltonian formalism. Before presenting a parallel discussion in the Lagrangian formalism, let us first give a brief overview of the Dyson-Schwinger(DS) equations.
In the discussion of the DS equations, we consider the Heisenberg picture. To be consistent with the results in the Hamiltonian approach, the Lagrangian in the generating functional

1)
18 Since the anharmonic corrections are of higher orders in n, the g series for the energy levels will cease to be a good approximation at high energies. At large n, it is more reasonable to use the WKB method. It would be interesting to figure out the algebraic counterpart of the WKB method from the null bootstrap perspective.
where J is the classical source. An infinitesimal change of the integration variable ϕ(t) gives 19 We can derive the DS equations by taking its functional derivatives with respect to J and then setting J = 0 where we have introduced the full Green's function .
In the Heisenberg picture, we should write the t dependence of the operators explicitly. Then the matrix element associated with two energy eigenstates |m 1 ⟩ and |m 2 ⟩ reads which can be written in a differential form 20 To derive the DS equations, we first assume the time order t 1 > t 2 > . . . > t j−2 > t j−1 > t > t j > t j+1 > . . . > t n−2 > t n−1 . Then we multiply (3.8) by the matrix elements ⟨0|x(t 1 ) . . . x(t j−1 )|m 1 ⟩ and ⟨m 2 |x(t j ) . . . x(t n−1 )|0⟩ and sum over m 1 , m 2

9)
19 This corresponds to the classical equation of motion Then the closure relation m |m⟩⟨m| = 1 implies (3.10) Similarly, for the slightly different time order t 1 > . . . > t j > t > t j+1 . . . > t n−1 , we have where the dependence on t is removed by the canonical commutation relation [x(t), p(t)] = i. The normalization is chosen to be ⟨0|0⟩ = 1. In terms of the Green's function the equation (3.12) implies that the first-order t-derivative of ⟨x(t)x(t 1 )x(t 2 ) . . .⟩ 0 is discontinuous at the coincident limit t j → t. This is realized by introducing the contact term with δ(t − t j ) to the second-order differential equation. Consequently, the complete differential equation reads which is precisely the DS equation (3.4). One can also verify explicitly that the DS equations are solved by the energy spectrum and the matrix elements in section 2. As an example, we consider where Θ(x) is the Heaviside step function defined by We have used the half-maximum convention Θ(0) = 1 2 to make sure that the formula produces the correct result when t 1 = t 2 . 21 Furthermore, the function ⟨x 3 (t 1 )x(t 2 )⟩ 0 is As expected, they satisfy the DS equation (3.4) with n = 2 Below, we will now investigate the anharmonic oscillator in the DS approach, without reference to the Hamiltonian. We first make the comparison to the Hamiltonian approach by considering one-point functions of composite operators. The Dyson-Schwinger equations in the one-point limit give the constraint where the ellipses represent arbitrary operators at time t. The time-translation invariance implies which is equivalent to the one from ⟨[H, xp]⟩ 0 = 0. However, the constraint (3.20) is less stringent than (2.3). To extract more information, we consider higher-point functions G n (t 1 , t 2 . . .). To solve the DS equations for these functions, we will impose the null state condition. In parallel to section 2, we can use the exact expressions of the ladder operators to construct the null state condition, which corresponds to the complete procedure in the Lagrangian formalism. Alternatively, we can also solve the DS equations by only assuming the existence of some null states, without knowing the exact expressions of the ladder operators. This can be seen as the reduced procedure for solving the DS equations.

Complete procedure
In the complete procedure of section 2, we obtain the exact expressions of the ladder operators. We will use the corresponding null state condition to solve the DS equations.
To derive the constraints for G n , we start with the null state condition in the Heisenberg picture which can be rearranged into the form where the right hand side correspond to the perturbative corrections to the lowering operator. To relate the states in (3.23) to G n , we need to specify a time order. Below, we will always assume t 1 > t 2 > . . . > t n and sometimes omit the time dependence of G n on (t 1 , t 2 , . . . , t n ) to simplify the notation. Note that G n is symmetric in its arguments, so it suffices to solve for G n in this specific time order. Using ⟨0|x(t 1 )x(t 2 ) . . . x(t n−1 ) as the a state, the null state condition (3.23) implies the null differential equation are associated with the perturbative corrections to the lowering operator. The explicit expressions of U (j) n are composed of Green's functions of n-point or more. After deducing the t n dependence from (3.24), we want to determine the dependence on other time variables, which are constrained by differential equations containing the time derivative ∂ t k with k < n. It seems that the differential equation (3.24) is not useful as it only involves the t n derivative. Can we take advantage of the null state condition again? We notice that certain limits of G n + i∂ t k G n are also constrained by the null state condition. In more detail, we consider the coinciding limit of several time variables, i.e. (G n + i∂ t k G n ) t k+1 ,t k+2 ,...,tn→t k . Then the derivative with respect to t k is not distinguishable from ∂ tn and we can use (3.24) to constrain the t k dependence as well. For example, the limit of two coinciding time variables is where we have used the DS equation in the t n → t n−1 limit itself admits a small g expansion.
and (3.24). From the perspective of operator theory, we move the momentum operator p(t k ) to the right by the commutation relation, so we can take advantage of the null state condition. More generally, we obtain the following differential equation for Green's functions (3.28) One main difference between the k = n and k < n differential equations is the coefficient of G n−2 . The factor (n − k) is related to the use of (n − k) different DS equations with coincident time variables, which is equivalent to the number of commutation relations used in the operator theory perspective . The null differential equations (3.28) are the main ingredients to solve the DS equations in the complete procedure. Now, the question is whether we can determine G n completely using these constraints, since one may find it worrisome that the null differential equations only constrain G n in certain limits. But the short answer is yes. Let us assume that G n admits the small g expansion To address the question in more detail, it is simpler to consider the zeroth order. We will show by induction that we can in principle determine all G n−2 is known. First, the k = n null differential equation determines the t n dependence in G (0) n . Next, we consider the k = n − 1 null differential equation. The crucial point is that the t n dependence is known, so taking the limit t n → t n−1 does not lose any information about the unknown dependence on t n−1 of G (0) n . We proceed as this from k = n to k = 1 one by one, and the functional form of G (0) n is determined up to a free parameter, which can be fixed by time-translation invariance. 23 Therefore, if G (0) n−2 is known, then we can determine the complete time dependence of G (0) n . Since by definition G (0) 0 = 1, we can determine G (0) n one by one. At higher orders in g, we need to account for the contributions from U n , but they can be computed using lower order results for G n . Therefore, the argument extends to higher orders. In conclusion, with time-translation invariance, we can determine all G n completely order by order using the null differential equation (3.28).
Examples: G 2 , G 4 and G 6 As an example of the general discussion above, we consider some low-point Green's functions G 2 , G 4 and G 6 . At zeroth order, the null differential equation (3.28) with n = 2, k = 2 reads 30) and the solution reads n . To simplify the notation, the unknown function will always be denoted by Q, instead of introducing a new symbol each time we solve a differential equation. Using (3.31), we see that the null differential equation (3.28) with n = 2, k = 1 becomes where by definition G (0) 0 = 1. The solution to this differential equation is On the right hand side, the first term is the particular solution to the differential equation, and the second term is the general solution that violates time-translation invariance. Therefore, the time-translation invariant solution of G 2 . First we have the n = 4, k = 4 null differential equation 35) and the solution is G 4 (t 1 , t 2 , t 3 ). Next, the k = 3 null differential equation reads Instead of the simple G 2 case, the right hand side here takes a more complex form, which leads to the solution Then we solve the k = 2, 1 null differential equations and obtain where the free parameter is again fixed by time translation invariance. Repeating the procedure above for G At first order in g, we need to take into account the contribution from U (1) n . According to the definition, it is related to G n , G n+2 and their derivatives in a certain limit. For example, the contribution from the term xp 2 (t n ) in L (1) −1 (t n ) can be computed by (3.40) Using the exact expressions of the nontrivial lowering operator and G where U 2 | g 0 denotes the zeroth-order terms in U 2 . At first order, the n = 2, k = 2 null differential equation reads which leads to the solution Then the n = 2, k = 1 null differential equation reads where by definition G Together with time-translation invariance, we find that The above procedure gives (3.48) We will not consider G The zeroth-order solutions for G At order g 2 , the n = 2, k = 2 null differential equation is which has the solution In the end, the n = 2, k = 1 null differential equation reads As before, the free parameter is fixed by time-translation invariance and we obtain (3.55) We will not consider G 4 because that will require the knowledge of U (1) 4 at order g 1 and U (2) 4 at order g 0 , that is, the calculation involves G

A simplified approach
For the reader's convenience, we recall the definition of U n When calculating U (j) n in the above discussion, we write it in terms of higher-point functions, and then take the coincident time limit, which involves more time variables and thus more complicated functions that seem irrelevant. In a nonperturbative setting, the complexity of the Green's function can grow much faster in the number of time variables than the perturbative case. To avoid introducing more variables in the intermediate steps, we want to express U (j) n directly in terms of n-point functions and their derivatives. In this approach, we can obtain G n without introducing more time variables.
To illustrate this approach, let us recall the example (3.40). In fact, it can also be computed from where we have used the DS equations with coincident points (or equivalently the canonical commutation relation). In short, the main difference is that we exchange the order of the coincident time limit and the time derivatives in the computation of U (j) n . As we can see, we need to consider the Green's functions with coincident times. They can be obtained by which is similar to (3.28), but we take the limit first and then take the time derivative.
As an example, let us solve for G 2 to order g 2 using only two time variables. At zeroth order, the G (0) 2 calculation is the same as before. To obtain G  The n = 4, k = 1 null differential equation (3.58) reads The solution is Q which is precisely (3.41) and this allows us to determine G 2 . To obtain G (2) 2 , we need to consider U (2) 2 | g 0 and U (1) 2 | g 1 , which are related to the following Green's functions: 6 (t 1 , t 2 , t 2 , t 2 , t 2 , t 2 ) , where G (0) 6 (t 1 , t 2 , t 2 , t 2 , t 2 , t 2 ) and G (1) 4 (t 1 , t 2 , t 2 , t 2 ) are unknown. At zeroth order, the null differential equation (3.58) with n = 6, k = 2 yields which is precisely (3.50). Next we compute G 4 (t 1 , t 2 , t 2 , t 2 ). The n = 4, k = 2 null differential equation (3.58) reads (3.69) The second term on the right hand side can be computed using the zeroth-order solutions for G 4 (t 1 , t 2 , t 2 , t 2 ) and G (0) 6 (t 1 , t 2 , t 2 , t 2 , t 2 , t 2 ) In the end, using the n = 4, k = 1 differential equation (3.58), we obtain and the result for U which is precisely (3.49). Therefore, we can obtain G 2 to order g 2 using only two time variables. We have also verified explicitly that G

Reduced procedure
We have shown that the DS equations can be solved in the complete procedure, i.e. by using the null state condition from the exact ladder operators. In the reduced procedure, the ladder operators are considered unknown, so the exact form of the null state condition seems unclear. However, we want to emphasize that the null state condition should be consistent with the DS equations, and cannot be arbitrary. In fact, the DS equations allow only two types of null state conditions, both of which can determine G n completely order by order in g.
Although we do not know the exact expressions of the ladder operators, we have some general idea about the form of the null state condition. We assume that where L ±1 (t) is of the form L ±1 (t) = degree-1 polynomial + g (degree-3 polynomial) (3.74) As we have mentioned in the reduced procedure in section 2, these are the minimal degrees for constructing the level-1 ladder operators. Since we consider the Heisenberg picture, the polynomials are in x(t) and p(t). We will see later that there are two choices ±1 corresponding to the two types of null state conditions that the DS equations allow. As indicated in the complete procedure, to apply the null state condition, we rearrange it into a more convenient form which leads to the null differential equation for G n G n + ic ± ∂ tn G n = gU (1) n,± + g 2 U n,± + O(g 3 ) . (3.76) We have defined (3.77) The null differential equation with t k derivative reads n,± + g 2 U (2) n,± t k+1 ,t k+2 ,...,tn→t k For consistency, the solutions to the DS equations should also solve the null differential equations. We will plug the solutions to the DS equations into the null differential equation (3.76), which yields strong constraints on c ± and U (j) ± . There are two types of choices satisfying these constraints, corresponding to the null state condition for the raising and the lowering operators respectively. Explicitly, the DS equations at zeroth order yield n (t 1 , t 2 , . . . , t n ) = e itn Q (0) n (t 1 , t 2 , . . . , t n−1 ) + e −itn R (0) n (t 1 , t 2 , . . . , t n−1 ) . (1 − c ± )e itn Q (0) n (t 1 , t 2 , . . . , t n−1 ) + (1 + c ± )e −itn R (0) n (t 1 , t 2 , . . . , t n−1 ) = 0 . (3.80) The nontrivial solutions are 24 In this way, we determine the zeroth-order null state condition. We repeat the analysis for the first order term, and it turns out that each choice leads to a concrete k = n null differential equation. In other words, the value of U n,± is completely fixed by the consistency requirement, which will be explained more explicitly below. Therefore, the null differential equations for all k are determined, and we can solve them to obtain G (1) n . The argument extends to higher orders, and consequently we can solve for G n order by order in g.
Examples: G 2 , G 4 and G 6 As concrete examples, we consider G 2 , G 4 and G 6 . Let us choose the more physical case c + = +1. The zeroth-order calculation is the same as that in the complete procedure. At first order, the n = 2, k = 2 null differential equation (3.78) reads On the other hand, we have the DS equation which gives The consistency with (3.82) determines R 4 and their derivatives in the limit t 3 , t 4 → t 2 , which cannot give the factor e −it 2 . Therefore, we have Then the n = 2, k = 1 null differential equation (3.78) is also fixed, so we can determine G 4 , the null differential equation (3.78) and the DS equation give respectively 4,+ g 0 , (3.86) . 6 . Therefore, we have So the n = 4 null differential equation (3.78) are determined to order g 1 , and consequently we can solve for G 4 . At second order, the n = 2, k = 2 null differential equation (3.78) reads (3.89) The DS equation gives Based on the first-order and the second-order solutions, one can check that U 2,+ | g 1 +U 2,+ | g 0 cannot have the factor e −it 2 . So we have Consequently, we can obtain G 2 using the null differential equation (3.78).

Remark on L ±1
One may wonder if we could determine L ±1 by considering more Green's functions and determining more U (j) n . The answer is that we can determine L ±1 up to some free parameters, but it is not exactly the level-1 ladder operators in the complete procedure. The reason is that we only impose that L ±1 annihilates |0⟩, instead of being a raising or lowering operator for all energy eigenstates. In fact, the constraints on L ±1 here are equivalent to the null condition in the reduced procedure in section 2 where we use the ansatz (2.52).

The simplified approach
In addition, in parallel to the complete procedure, we present the calculation that involves only two time variables. We restrict the discussion to the case c + = +1. Since the zerothorder calculation is the same as that in the complete procedure and G 2 is already discussed in the example above, the only new ingredient here is to obtain G   the Green's functions can be determined order by order from the null differential equation (3.28). In the reduced procedure, the exact expression for the lowering operator is not needed. Using the null differential equation (3.58) and the DS equations, the n-point Green's functions can be determined order by order as well. In the explicit examples, the numbers of points in the Green's functions are n = 2, 4, 6. We also presented simplified methods that do not introduce additional points in the intermediate steps, which significantly reduce the complexity of the functional forms in the computation.
We would like to extend these perturbative results to genuine quantum field theory with at least two spacetime dimensions. It would be interesting to examine if the standard issues of divergences and the renormalization procedure are simplified in the bootstrap approach. Some insights from our perturbative analysis should also extend to the nonperturbative bootstrap approach.
Another interesting direction is to revisit the multiplet recombination method for conformal field theory [47], which is closely related to the Dyson-Schwinger equations [48]. The null state condition may be crucial to the derivation of higher order corrections in the ϵ expansion. 25 In addition, the CFT classification program at higher dimensions should share some features with the two-dimensional minimal models, in which the null condition plays a central role. This may also shed light on the more ambitious goal of classifying more generic QFTs by the principle of nullness. |n⟩ = |n (0) ⟩ + g|n (1) ⟩ + g 2 |n (2) ⟩ + g 3 |n (3) ⟩ + . . . . We have written the terms involving the occupation number n in terms of Dirac's ladder operators, which is more natural for the operator algebra perspective and more convenient for the discussion of the anharmonic ladder operators below. For the second-order corrections, we consider (A.5). Following the same procedure of projection, we obtain E (0) m ⟨m (0) |n (2) ⟩ + ⟨m (0) |H ′ |n (1) ⟩ = E (0) n ⟨m (0) |n (2) ⟩ + E (1) n ⟨m (0) |n (1) ⟩ + E (2) n ⟨m (0) |n (0) ⟩ . where the part parallel to |n (0) ⟩ is fixed by the condition that ⟨n|n⟩ is independent of g. We can also study the corrections to Dirac's ladder operators. We define the raising and the lowering operator respectively by their actions on a generic energy eigenstate L ±1 |n⟩ = C ± |n ± 1⟩ , (A. 19) where C ± are c-numbers. The ladder operators L ±1 and C ± have the perturbative expansion L ±1 = L