Quantization of the ModMax Oscillator

We quantize the ModMax oscillator, which is the dimensional reduction of the Modified Maxwell theory to one spacetime dimension. We show that the propagator of the ModMax oscillator satisfies a differential equation related to the Laplace equation in cylindrical coordinates, and we obtain expressions for the classical and quantum partition functions of the theory. To do this, we develop general results for deformations of quantum mechanical theories by functions of conserved charges. We show that canonical quantization and path integral quantization of such deformed theories are equivalent only if one uses the phase space path integral; this gives a precise quantum analogue of the statement that classical deformations of the Lagrangian are equivalent to those of the Hamiltonian.


Introduction
Historically, quantum field theories first arose via the quantization of classical field theories. Following the modern usage of the term [1], we understand a quantum field theory (QFT) to mean any model that is compatible with certain physical principles including quantum mechanics, locality, and Lorentz invariance on a fixed (d + 1)-dimensional spacetime manifold. When d = 0, the Lorentz structure becomes essentially trivial and one has an ordinary theory of quantum mechanics. However, we still lack a systematic understanding of the process of quantization for several reasons. One reason is that it is not known how to uniquely quantize a general classical theory, except in the case of theories which can be brought into a conventional form with a quadratic kinetic term. A famous example is the Nambu-Goto action of string theory; rather than attempting to quantize this theory directly, one first rewrites it in the classically equivalent form of the Polyakov action, which can then be quantized because the theory is quadratic in derivatives. A second reason is that not all quantum field theories admit classical limits. This means that certain QFTs cannot ever be understood by quantization of a classical theory, defined for instance by a Lagrangian or Hamiltonian (indeed, many QFTs are non-Lagrangian and thus do not even admit such a description).
Because of these observations, it is sometimes said that "quantization is not a functor." In order to better understand quantization and the space of QFTs, it seems that one must develop new tools. One such tool is to describe new quantum field theories using controlled deformations of old ones. An example of such a deformation, which has generated considerable interest in the past several years, is the T T deformation of two-dimensional QFTs. The T T operator refers to the coincident point limit which was shown in [2] to define a local operator in any translation-invariant 2d QFT.
Using any such 2d QFT as a seed theory, one can define a family of theories, labeled by a flow parameter λ, which arise from deforming the seed theory by T T . At the classical level, we think of this parameterized family of actions as solving the flow equation where T (λ) µν is the stress tensor computed from S λ , However, the interpretation of the differential equation (1.2) for the classical Lagrangian can be somewhat subtle. Because O T T exists in the spectrum of local operators in the seed theory, deforming by this operator should lead to a well-defined quantum theory.
One could ask whether this quantum theory corresponds to the quantization, in some appropriate sense, of the classical action S λ .
To address this question, one can use an alternative characterization of the quantum theory obtained by a T T deformation. For instance, it is known [3,4] that the S-matrix for scattering in a T T -deformed QFT is obtained by dressing the S-matrix of the undeformed theory with a momentum-dependent phase known as a CDD factor [5]. This gives an independent description of scattering in the quantum theory which can be compared to predictions from quantization of the solution to (1.2). At one-loop level, one must add specific counter-terms when renormalizing the classical Lagrangian in order to reproduce the expected behavior of the T T -deformed S-matrix [6][7][8]. 1  Lagrangian can be ambiguous. Indeed, as we have stressed above, quantization is not a functor: except in simple cases such as free theories, we do not understand a unique and systematic prescription for turning classical theories into quantum ones. Rather, what we mean by the quantum theory of a T T -deformed seed is determined by other characterizations such as the S-matrix or torus partition function [11][12][13]. These independent pieces of data should be viewed as picking out the correct prescription for performing the quantization of the T T -deformed Lagrangian. This is in analogy with the viewpoint that the proper quantization prescription for the Nambu-Goto string is the one which proceeds 1 Because the T T operator is irrelevant in the sense of the renormalization group, this behavior is partly expected. Adding a generic irrelevant operator will typically activate infinitely many counterterms; the surprise is that the irrelevant T T deformation does not lead to a loss of analytic control.
by first rewriting the theory in Polyakov form and then quantizing using the path integral.
It is natural to ask whether adopting this perspective offers us insights into the quantization of other models. Recently, a family of related theories which exhibit non-analytic square-root structures in their Lagrangians have been introduced, all of which satisfy some classical flow equation similar to (1.2). We will now take a detour to describe some purely classical aspects of this collection of theories before returning to issues of quantization.
The first member of this class to be introduced was a four-dimensional gauge theory known as the Modified Maxwell or ModMax model [14], which is described by the action where F µν is the field strength of the Abelian gauge field A µ and F µν = 1 2 ε µνρτ F ρτ is its Hodge dual. When γ = 0, the action (1.4) reduces to that of the usual Maxwell theory.
As a classical theory, the ModMax model (1.4) exhibits several intriguing properties.
It is the unique conformally invariant and electric-magnetic duality-invariant extension of the Maxwell theory. It also satisfies a flow equation driven by a function of the energymomentum tensor [15,16], namely where T (γ) µν is the stress tensor of the ModMax theory (1.4) at parameter γ. Unlike the flow (1.2) for the Lagrangian of a T T -deformed 2d QFT, the operator on the right side of (1.5) is classically marginal. Note that, since the ModMax theory is conformally invariant and thus its stress tensor has vanishing trace, the operator driving the flow (1.5) need not have any dependence on T (γ)µ µ . However, it is convenient to define another combination which does involve the trace and which reduces to (1.5) in the conformal limit. For a theory in D spacetime dimensions with energy-momentum tensor T µν , let (1. 6) In terms of the traceless part of the stress tensor, which we write as T µν = T µν − 1 D g µν T ρ ρ , this operator is simply Including this dependence on the trace allows us to extend certain flow equations to non-conformal theories. For instance, there is a two-parameter family of ModMax-Born-Infeld theories labeled by couplings (λ, γ), which reduces to (1.4) when λ = 0 and to the Born-Infeld theory when γ = 0. This family satisfies two commuting classical flow equations, one driven by a four-dimensional version of the T T operator and one driven by the operator R (4) [17][18][19]. The operator R (3) also appears in the flow equation which deforms the 3d Maxwell Lagrangian into the Born-Infeld theory in three dimensions [20].
When D = 2, the combination R (2) is the root-T T operator introduced in [21]. 2 Applying this deformation to the seed theory which describes N massless free scalar fields φ i produces a second example of a non-analytic classical Lagrangian. The resulting deformed theory can also be obtained from the 4d Modified Maxwell theory by dimensional reduction [16], so we will sometimes refer to this model as the Modified Scalar theory.
This model is described by the action where the index i = 1, . . ., N labels the N scalars. This action satisfies the flow equation as shown in [21,26]. Like the 4d ModMax theory, the Modified Scalar theory is classically conformally invariant and thus the trace of its stress tensor vanishes, so only the term T µν T µν appearing under the square root in R (2) gives a non-zero contribution.
The two-dimensional root-T T deformation which gives rise to the theory (1.8) appears to share some of the interesting properties of the T T deformation, such as preserving classical integrability in several examples [27]. However, unlike the case of T T , it is not yet known how to define the root-T T deformation at the quantum level and obtain flow equations for quantities like the S-matrix; proposed flow equations for the finite-volume spectrum and torus partition function of a root-T T deformed CFT were given in [28] and supported using evidence from holography, but there is no general proof of these results.
This presents an obstruction to carrying out the procedure that we have described above -namely, identifying the correct prescription for the quantization of these models using some additional input -for root-T T deformed models such as (1.4) and (1.8). 2 We refer the reader to [22][23][24][25] for other work related to the root-T T operator.
In this work, we take up the task of studying the quantization of such non-analytic models in a simplified setting where one can carry out this program explicitly, namely in the arena of (0 + 1)-dimensional theories. By performing a particular dimensional reduction described in [29], either the ModMax theory or its Modified Scalar analogue can be reduced to a 1d model which describes a harmonic oscillator with a non-analytic interaction term. We refer to this system, which was first studied in [30], as the ModMax oscillator. The simplest version of this theory features two position variables x(t), y(t), and is described by the Lagrangian When γ = 0, this theory reduces to a two-dimensional isotropic harmonic oscillator with unit mass and frequency. This gives the third example of a non-analytic theory. 3 At first glance, it is not obvious that the full Lagrangian (1.10) at finite γ will be amenable to exact quantization because of the velocity-dependent square root interaction.
However, one might become more optimistic about the prospects of quantization after observing that this Lagrangian obeys a root-T T -like flow equation, where E γ and J γ are the energy and angular momentum, respectively, of the theory (1.10) at parameter γ. This is the dimensional reduction of the flow equations driven by R (4) and R (2) that are obeyed by the ModMax and Modified Scalar models, respectively.
Because the ModMax oscillator can be described as a deformation of the harmonic oscillator by conserved charges, this suggests that one might perform canonical quantization of this theory using the prescription described in [35,36] for quantum mechanical deformations by functions of the Hamiltonian. That is, we first choose a basis of simultaneous eigenfunctions of the Hamiltonian and total angular momentum operators in the undeformed harmonic oscillator theory. We then declare that the eigenfunctions of the ModMax oscillator are the same as those of the harmonic oscillator, but with energy eigenvalues that have been shifted by the square root combination appearing in (1.11).
This prescription gives a simple and elegant way to define a quantum theory of the ModMax oscillator. However, as we have emphasized, quantization is not a functor: it is not clear that this is the only prescription, or even the correct one. For instance, because this quantization scheme is so simple, one might expect that it is also possible to quantize (1.10) using the path integral formulation and get equivalent results. However, it is generally very difficult to perform the path integral for any theory which is not quadratic in derivatives. One of our goals in this work is to perform a detailed comparison of quantization prescriptions for the ModMax oscillator and check that they agree.
We will show that any deformation of a 1d theory by conserved charges induces a flow

Classical Deformations
In this section, we will consider deformations of (0+1)-dimensional theories at the classical level. We focus on theories that describe the dynamics of a collection of real bosons x i (t), i = 1, . . . , N. The generalization to theories with fermions ψ i (t) or more general degrees of freedom is straightforward, although we will not consider such cases here. 5 We view such a theory as being defined by either a Lagrangian or a Hamiltonian, where p i is the momentum which is canonically conjugate to the variable x i , Throughout this work, we assume that the indices i, j, etc. that label positions and momenta are raised or lowered with the trivial Euclidean metric δ ij . We therefore do not distinguish between upstairs and downstairs indices.
We will often write expressions like (2.1) in which the argument of a function of positions, velocities, or momenta carries an index like i. In such expressions, the index i is not meant to be a free index, but is merely shorthand to indicate dependence on all of the corresponding variables as i runs from 1 to N. Explicitly,

Flow Equations for Lagrangians and Hamiltonians
Our first goal is to study deformations, or flows, in the space of classical theories, which are defined as follows. Let O(x i ,ẋ i ; λ) be some function of the coordinates x i and their time derivatives, which may also depend on a real variable λ. The differential equation

4)
5 A convenient way to incorporate fermions is to define flow equations in superspace. These manifestly supersymmetric flows have been extensively studied; see [38] and references therein for a review of such deformations in field theory, or [29,39] for the corresponding flows in 1d theories.
defines a one-parameter family of Lagrangians L(x i ,ẋ i ; λ). We refer to equation (2.4) as a flow equation and we say that the function O is the operator which drives the flow. 6 A theory may equivalently be described in the Hamiltonian formulation by the function which is the Legendre transform of L(x i ,ẋ i ). In equation (2.5), one must view all instances of the velocitiesẋ i =ẋ i (p j ) as being implicitly defined in terms of the conjugate momenta.
In analogy with the Lagrangian flow (2.4), we may consider a differential equation  one-parameter family of Lagrangians L λ which satisfy the differential equations with the initial conditions L λ → L 0 as λ → 0. Then the Hamiltonian associated with L λ , satisfies the flow equation where the function O is defined by

11)
and whereẋ i (p j ; λ) represents the functional dependence between the velocities and conjugate momentum in the theory L λ .
Conversely, given a function O (x i , p i ; λ), consider the family of Hamiltonians H λ which obey

14)
and where p i (ẋ j ; λ) represents the functional dependence between the conjugate momenta and velocities in the theory H λ .
The interpretation of this theorem is that the diagram (2.7) commutes, so long as the Lagrangian deformation O and the Hamiltonian deformation O are correctly related using the constraint between the velocities and conjugate momenta in the theory at finite λ. Note that this is not the same as using the relationship between the p i andẋ i in the undeformed theories L 0 and H 0 . Unsurprisingly, if one uses the relation betweenẋ i and p i which is valid in the seed theories, the corresponding flows commute only to leading order in the deformation parameter λ. This was proven for field theories describing a single field φ and conjugate momentum π = ∂L ∂φ in appendix A of [40]. To make the present work self-contained, we include the analogue of their leading-order proof for (0 + 1)-dimensional theories of N positions x i (t) in our appendix A.
Proof. We first prove the forward direction, beginning with the flow equation (2.8) for the Lagrangian. We view the velocitiesẋ i as independent variables, while at each value of λ, the conjugate momenta p i (ẋ i ; λ) are determined via the relation The Legendre transform which defines the Hamiltonian, written in a way which emphasizes the λ dependence, is We differentiate both sides of (2.16) with respect to λ to find By the Hamilton equations of motion, we have The object on the right side of (2.19) is precisely the operator O(x i , p i ; λ). This completes the first half of the proof. Now we show the reverse direction. Suppose that the Hamiltonian obeys the flow equation (2.12). We now view the p i as independent variables while theẋ i are fixed aṡ Again making all functional dependence explicit, the Legendre transform which defines the Lagrangian is Differentiating with respect to λ gives After using the relation we see that the terms p j ∂ẋ j ∂λ cancel on either side of equation (2.22), and we are left with which establishes the converse.
Note that we have stated this theorem and its proof for (0 + 1)-dimensional theories.
However, one can repeat this argument almost verbatim, making the replacements to obtain the corresponding theorem and its proof in any d-dimensional quantum field theory for a collection of fields φ i and their conjugate momenta π i = ∂L ∂φ i .

Examples of Classical Lagrangian and Hamiltonian Deformations
It is instructive to see how Lagrangian and Hamiltonian flows are related in several examples, both when the assumptions of Theorem 1 hold and when they do not.
First let us consider a non-example of this theorem. We begin from an undeformed theory which has only a free kinetic term: The relationship between the undeformed velocity and momenta is simply p = mẋ. Now consider the pair of flows with solutions (2.28) The Lagrangian L λ and Hamiltonian H λ are not related by a Legendre transformation.
The conjugate momentum evaluated using the Lagrangian L λ is (2.29) and thus the Legendre transform of L λ , which we call H λ , is In this simple example, the difference between the Legendre transform H λ and the Hamiltonian H λ is only a constant term, but nonetheless the two quantities do not agree. Next let us consider a modification of the above flow: This deformation agrees with (2.27) at leading order in λ, but beyond first order, it has been altered in order to satisfy the assumptions of Theorem 1. We will solve these differential equations with initial conditions that are arbitrary functions of velocities or momenta, L 0 (ẋ) and H 0 (p). One finds The Lagrangian L λ and Hamiltonian H λ of (2.32) are indeed related by a Legendre transform 7 to all orders in λ, as guaranteed by Theorem 1.
Let us consider one more, slightly less trivial, example. Again beginning from an arbitrary velocity-dependent seed Lagrangian L 0 (ẋ) and Hamiltonian H 0 (p), consider the flow equations This deformation was first considered in appendix A of [35]. The Hamiltonian flow equation can be solved for any initial condition H 0 (x, p): However, the Lagrangian flow equation is more complicated. If the seed theory is L 0 =ẋ 2 , the solution is given in terms of a hypergeometric function: This same hypergeometric function has appeared in several contexts related to classical deformations by conserved charges, including the T T deformation of the 2d Maxwell theory [17] and Yang-Mills [41]. Despite the complicated form of L λ , one can check that it is indeed related to the simpler function H λ of (2.34) by a Legendre transform when H 0 = 1 4 p 2 . This is again required by the general argument of Theorem 1.

Deformations by Conserved Charges
The two examples (  More precisely, what we mean by a deformation by conserved charges is the following. Suppose that a seed theory L 0 has a collection of symmetries generated by variations δ a and which are associated with conserved charges Q a , for a = 1, . . . , M, according to equation (2.37). We will always use early Latin indices like a, b, c to label charges Q a and middle Latin indices like i, j, k to refer to coordinates x i . We would like to define a flow equation of the form a is the Noether charge associated with the symmetry δ a in the theory L λ . The corresponding Hamiltonian flow is In this equation, the Hamiltonian charges Q a (x i ,ẋ i ) in terms of conjugate momenta, using the relation between p i andẋ i in the theory L λ .
We must make an additional assumption in order to define such a flow: the variations δ a must generate symmetries of the entire family of theories L λ , rather than only for the seed theory L 0 . That is, we must assume that this deformation does not break any of the symmetries. This will be true if all of the charges Q a are Poisson-commuting, This assumption is sufficient because, if Q a is the Noether charge associated with a symmetry variation δ a , then the Poisson bracket with Q a generates the transformation of any function g as {Q a , g} = δ a g. Thus the condition (2.41) implies that all of the charges Q a are invariant under all of the symmetries generating these charges, and thus any deformation of a Lagrangian by a function of these charges will still enjoy the same symmetries.
For instance, we could consider a theory with two coordinates x 1 = x and x 2 = y and which has two conserved momenta p x , p y along with a conserved angular momentum J and a Hamiltonian H. In this case, In this case, we cannot define a flow equation (2.39) if, for instance, the function f depends on J and p x but not p y . Such a deformation breaks the rotational symmetry between x and y, and thus J is no longer a conserved quantity in the deformed theory. However, we are free to construct a flow equation using the three quantities and thus these three charges Q a Poisson-commute.
The first examples of deformations by conserved charges, which we will also call f (Q a ) flows, are those driven only by a function of the Hamiltonian H. This is the class of f (H) deformations considered in [35,36] and which includes the dimensional reduction of the T T deformation, in the special case where the 2d seed theory is conformally invariant. This dimensional reduction leads to the flow equation which has the solution See [42] for the extension of these f (H) flows to the case of non-Hermitian Hamiltonian deformations. Another class of examples are those which involve the product of the conserved energy E and a second charge Q, which were studied in [43] and which are similar to the JT deformations of 2d field theories.
However, in the present work we will be primarily interested in the class of f (E, J 2 ) deformations studied in [29], which was motivated by the flow equation obeyed by the ModMax oscillator of [30]. These deformations can be applied to theories which enjoy an additional SO(N) symmetry which rotates the N coordinates x i as The conserved currents associated with each of the rotation generators is a component of angular momentum, We define the total angular momentum by A general deformation by a function of both the energy and angular momentum, written in the Lagrangian formulation, is then The main operator of interest in the present work is harmonic oscillator, one always has the bound |E| 2 ≥ |J| 2 so that the argument of the square root in (2.52) is non-negative; we will assume this to be true in what follows. We will always use the symbol γ for the flow parameter when deforming by the operator R, in contrast with λ, which we use as the parameter for a general deformation.
We refer to the combination (2.52), in either formulation, as the 1d root-T T operator.
The reason for using this term is that, as shown in [29], this object arises from a certain dimensional reduction of the flow equation which defines the root-T T deformation of (1 + 1)-dimensional field theories [21]. More precisely, given a field theory describing the dynamics of a collection of scalar fields φ i (x, t) on a spatial circle x ∼ x + 2πR, one can Fourier-expand each scalar field as Truncating the theory to the dynamics of a single non-zero mode c i m (t), m > 0, and integrating over the circle then yields a dimensionally reduced theory for the functions c i m (t). Performing this reduction for the family of theories that arises from deforming a collection of free scalars by the root-T T operator (2.53) then yields a family of 1d theories which satisfy a flow equation driven by the combination (2.52). In particular, applying this 1d root-T T deformation to a seed theory of N bosons x i (t) subject to a harmonic oscillator potential yields the theory which we refer to as the ModMax oscillator. This will be the subject of section 4.
To conclude this subsection, we point out that -although deformations by conserved charges are quite general -not all models of interest satisfy flow equations of this form.
Interesting non-examples include the Born oscillator, and generalized Born oscillator, which were recently studied in [44,45]. A version of the Born oscillator for N coordinates x i can be described by the Hamiltonian Despite its very symmetrical form, this Hamiltonian has the property that and thus it cannot obey any flow equation of the form ∂ λ H λ = O (H λ , Q a ).

Flow of the Classical Partition Function
We are ultimately interested in quantum observables, such as the propagator, for theories deformed by functions of conserved charges, which will be studied in section 3. The periodic Euclidean-time propagator reproduces the quantum thermal partition function, and the classical limit of this object is the ordinary classical partition function. It will therefore be useful to study the classical partition function in order to have a check against which to compare the results of section 3, since these quantities should agree in the limit → 0. We will see that the flow equations satisfied by the classical and quantum partition functions under a general deformation by conserved charges are identical. which would give M + 1 charges in total. We define the grand canonical partition function (2.57) Here β = 1 T is the inverse temperature and µ a is the chemical potential, or fugacity, for the conserved charge Q a . Note that the sum over charges in (2.57) begins at a = 1 so that the Hamiltonian is not included, although we could treat the Hamiltonian symmetrically by defining µ 0 = −β and beginning the sum at a = 0. From now on, we will set = 1.
It is convenient to define the expectation value of a function f in this ensemble as (2.58) The derivative of the partition function with respect to λ is and the derivatives with respect to the inverse temperature and chemical potentials are The partition function therefore obeys the flow equation The right side of (2.61) is defined by expanding the operator O as a power series in each of its variables H λ , Q a , and then replacing each variable with the appropriate derivative.
This is only possible if the deforming operator is an analytic function of its arguments, but we will see shortly how to extend this argument to some non-analytic deformations.
For example, let us consider the flow equation for the one-dimensional T T deformation, We may schematically write the corresponding flow equation for the partition function as where the rational function of derivatives is again defined by the Taylor series expansion whose first few terms are shown in (2.62). We can invert this infinite series of derivatives to write the equivalent flow equation which can be expressed as This is the flow equation for the classical partition function of a theory deformed by the 1d T T flow. It also turns out that the quantum partition function obeys the same flow equation. 8 We will see in section 3 that this is true more generally: the flow equations for the classical and quantum partition functions are always identical for deformations by any function of conserved charges.
An interesting feature of the differential equation (2.65) is that its solution can be written as an integral transform of the undeformed partition function Z 0 (β) at λ = 0.
This can be understood as an analogue of the solution to the heat equation, written as a convolution against the heat kernel, since (2.65) takes the form of a diffusion-type equation where the parameter λ plays the role of time. This integral kernel solution is which was obtained and studied in [36]. It is easy to check that the integral expression (2.66) automatically solves the differential equation (2.65) We now turn our attention to deformations which take the form where f is an analytic function. In the case where f (H λ = 0, Q a = 0) = 0, this deforming operator does not admit a Taylor series expansion because the square root function is not analytic around 0. For such deformations, we cannot define the differential operator appearing on the right side of equation (2.61) via a series.
We can attempt to circumvent this difficulty in one of two ways. The first way is to attempt to define a fractional derivative by diagonalizing the differential operator. That is, if we can identify a complete basis of eigenfunctions ψ n with the property for some non-negative eigenvalues ν n , then we simply define the fractional differential operator to act as We may then expand the partition function Z in this basis as and obtain a flow equation Although it should be possible, in principle, to carry out this procedure of defining the fractional derivative -at least in some examples -we will not pursue it further here.
Instead, we will attempt to remedy the non-analyticity by taking a second derivative.
In certain cases, this can convert a first-order flow equation driven by a square-root operator to a second-order flow equation driven by an analytic operator. For a general deformation by an operator O = O(H λ , Q a ), the second derivative of the partition function with respect to λ is and for an operator O = f (H λ , Q a ) of the form in equation (2.67), this is The second term now depends only on f but not its square root, so this term can be expressed in terms of a power series in derivatives of Z with respect to β and the µ a as before. The first term, however, still depends on √ f and is not manifestly analytic for a deformation involving a generic function f . However, we will revisit this expression in section 4 for the special case f (H λ , Q a ) = H 2 λ − J 2 λ and the seed theory of a harmonic oscillator. In this case, we will see that the first term also becomes analytic, and the flow equation for Z collapses to a conventional second-order differential equation in three variables. Although this PDE does not admit an integral kernel solution like (2.66), its general solution can be written in terms of exponentials and Bessel functions.

Quantum Deformations
In this section we turn to the deformation of quantum-mechanical theories by conserved charges in (0 + 1) spacetime dimensions. As we mentioned in the introduction, there is no universal method for quantizing a general classical theory; see, for instance, [47] for a survey of quantization methods from a mathematical perspective.
Here we will focus on canonical quantization and path integral quantization. When discussing operators in the canonical formalism, we will use hats to distinguish them from the corresponding classical variables; for instance, we writê (3.1) We will use vector symbols x, p to represent the collection of all components x i and p i , respectively, for i = 1, . . . , N. The component indices i, j, etc. are not to be confused with the subscripts A, B, which we will introduce shortly and which refer to the initial and final configurations that determine the boundary conditions of the path integral.
One of the results of section 2 is that deformations by conserved charges in the Hamiltonian and Lagrangian formulations are equivalent, if one uses the correct relationship between variables in the deforming operators. Because the quantum theory features either the Hamiltonian, in canonical quantization, or the Lagrangian, in conventional path integral quantization, a natural first question to ask is whether deformations by conserved charges in these two formalisms are again equivalent quantum-mechanically.
For example, one might ask whether computing the propagator using the unitary time evolution operator U(t B , t A ) associated with a given Hamiltonian H(t), agrees with a definition using the Feynman path integral, after deforming H 0 by an operator O( Q a ) in (3.2) and deforming L 0 by the corresponding However, as we will review, this is not the right question to ask. The expression (3.3) for the propagator, in terms of a path integral over the position coordinates x with the standard measure, is only valid for Lagrangians which are quadratic in derivatives. After deforming a seed theory by a function of conserved charges, the resulting deformed Lagrangian will often have more general dependence on the derivativesẋ i . In such situations, the conventional path integral (3.3) does not correctly compute the deformed propagator, and instead one must use a phase space path integral: The quantity appearing in the exponential of (3.4) is not the classical action, because the functions p i (t) in the first term are not the canonical momenta, but rather dummy functions which are path-integrated over. Thus, for a general non-quadratic action, the classical Lagrangian plays no role in the path integral quantization of the theory, and only the phase space path integral defined in (3.4) is important.
Given this observation, we should not ask whether deformations of the Hamiltonian and Lagrangian are equivalent in the quantum theory. Instead we should ask whether deforming the Hamiltonian operator in the expression (3.2) is equivalent to deforming the classical function of phase space variables H 0 ( p, x) in (3.4). This will be the topic of section 3.2. First we will take a detour to review the phase space path integral.

Phase Space Path Integral
In this section, we will review the path integral quantization of a general Hamiltonian which need not be quadratic in the momentap i that are conjugate to the position operatorŝ x i . Only in the case of this quadratic dependence on momenta does the general phase space path integral reduce to the ordinary Feynman path integral. 9 To begin, we will assume that the Hamiltonian is an analytic function of the variableŝ x i andp i . Later we will be interested in deformations of such Hamiltonians by non-analytic functions of charges, but for now we will require H to admit an expansion One can always bring a general analytic Hamiltonian into this form by using the canonical commutation relation to move all position operatorsx i to the right of momentum operatorsp i . The form (3.6) of the Hamiltonian is said to be normal-ordered. Taking the Hermitian conjugate of this expression reverses the order of the operators, and is thus anti-normal-ordered. We also assume that the Hamiltonian is Hermitian, so that H = H † . Our conventions for the position and momentum eigenstates are and we take = 1.
We are interested in computing the propagator where the unitary time evolution operator U is related to the Hamiltonian according to (3.2). Following the usual time-slicing procedure, we subdivide the time interval T = t B − t A into a large number M + 1 of smaller intervals of length ǫ, where we have defined The time evolution operator decomposes into a product of operators U (t j+1 , t j ) over each of the smaller time intervals, where the integration measure d x is shorthand for N i=1 dx i , to insert several resolutions of the identity and write (3.14) Here we have defined x 0 = x A and x M = x B . Note that the subscripts j, k, etc. on the position variables do not refer to the components x i of the vector x but rather to labels which index the different integration variables.
We can now focus on the propagator over one of the smaller time intervals of length ǫ. Over such an interval from t to t + ǫ, even if the Hamiltonian H has explicit time dependence, we can approximate the unitary time evolution operator to O(ǫ) as Inserting a complete set of momentum eigenstates using the completeness relation we can then write a single factor in the integrand of (3.14) as It is convenient that we have two representations of the Hamiltonian (3.6) and (3.7), one with position operators to the right and one with momentum operators to the right. We evaluate the second matrix element in (3.17) to order ǫ using the normal-ordered form, where we use the symbol h( x j , p) with no indices to refer to the normal-ordered Hamiltonian (3.6) with all operators replaced with classical variables.
Similarly, we use hermiticity of H along with the anti-normal-ordered form (3.7) for the Hamiltonian to evaluate the first matrix element appearing in (3.17), where we similarly write h * (t; x j+1 , p) for the anti-normal-ordered Hamiltonian (3.6) with operators replaced by classical variables.
Using these results (3.18) and (3.19) for the matrix elements, we find (3.20) This discretization suggests that we should define the classical Hamiltonian To leading order at small ǫ, the propagator over a small time interval ǫ is therefore (3.22) The full time-sliced propagator (3.14) is obtained from the product of the individual factors (3.22) in the limit as ǫ → 0 and M → ∞, (3.23) In the limit of small ǫ, the sum in the argument of the exponential becomes an integral: (3.24) Here we have passed from discrete collections of x j , p j to continuous trajectories x i (t), p i (t). We conclude that the propagator for a general analytic Hamiltonian takes the form where the path integral measures D x and D p are defined as the limits of the products in (3.23). To respect causality, we set the propagator equal to (3.25) when t B > t A and Similarly, by performing this phase space path integral in Euclidean time with periodic boundary conditions, one can obtain a phase space integral expression for the finite-temperature partition function.
Again, it is important to emphasize that the integral (3.25) runs over all phase space paths ( x(t), p(t)). For a generic path, there is no relationship between the coordinates and momenta; in particular, it is not the case that p i is constrained to be equal to the canonical momentum which is conjugate to x i . In the special case where the Hamiltonian is quadratic in the momenta, for instance if

Deformations in Canonical Quantization and Path Integral Quantization
We now wish to study how observables in quantum mechanics are modified when the theory is deformed by conserved charges. Our goal is to show that the propagator, and hence the finite-temperature partition function, of a general theory satisfies a flow equation which is identical to the one derived in section (2.3) for the classical partition function.
As a consistency check, we will also see that this flow equation for the propagator can be equivalently derived using either canonical methods or path integral methods.

Canonical Analysis
First let us see how to understand this flow equation in the canonical formalism. For the moment, we will specialize to the case of Hamiltonians which do not depend on time explicitly. In this case, the unitary time evolution operator can be written as 27) and the propagator is Suppose that the one-parameter family of Hamiltonian operators H λ satisfies a flow equation driven by a combination of conserved charge operators Q a , If the operator O depends on charges besides the Hamiltonian, it will not be possible to derive a closed flow equation for the usual propagator (3.28). Instead, we must consider a more general propagator which includes sources for the various conserved charges Q a .
We therefore define the quantity where µ a are a collection of couplings that serve the same purpose as the chemical potentials in the analysis of the classical partition function in section 2.3.
For simplicity, we define T = t B − t A , and we will suppress the arguments of the propagator in what follows. One has the relations Summation is implied in the expression µ a Q a = a µ a Q a . Because the deforming operator is itself a function of the operators H and Q a , we arrive at a differential equation We may also think of this flow equation in terms of a generalization of the prescription of [35,36] for quantizing theories which are deformed by functions of the Hamiltonian.
To do this, let us first recall how to derive the kernel representation of the propagator.
Suppose that we can identify a complete basis of simultaneous eigenstates of the Hamiltonian H λ and each of the charge operators Q a . We write these simultaneous eigenstates as |φ n , which satisfy H λ |φ n = E n (λ) |φ n , Q a |φ n = q a,n |φ n . we can evaluate the propagator as Here we have used the orthogonality relation φ n | φ m = δ m,n , along with the definition of the position space wavefunction x | φ n = φ n ( x) . . All that has changed is that each ket |φ n now has a deformed energy eigenvalue E n (λ) which obeys the differential equation where on the right side of equation (3.38), O is now a classical variable which is evaluated on the energy eigenvalue E n and charge eigenvalues q n,a of the state |φ n .
For instance, in the case of the the deformation by O = H 2 λ 1 2 −2λH λ whose classical solution was discussed around (2.46), each energy eigenstate |φ n with undeformed energy E n (0) remains an energy eigenstate in the deformed theory, but with a new energy There is a sharp difference in the behavior of the deformed spectrum depending on the sign of λ. If λ < 0, then the argument of the square root remains positive for arbitrary large positive undeformed energies. This choice is called the "good sign" of the deformation parameter. However, if λ > 0, then for sufficiently large undeformed energies E n (0), the deformed energy levels become complex. This is the "bad sign" of the deformation. The same qualitative behavior occurs for the T T deformation of a 2d CFT. 10

Path Integral Analysis
We will now see how the above flow equations can be derived using the path integral. For the same reasons as we mentioned above, if the deforming operator O depends on charges Q a besides the Hamiltonian, we cannot obtain a flow equation for the propagator using the unflavored path integral (3.4). Instead we must introduce a flavored version which includes sources for the various charge operators, (3.40) To ease notation, we will again suppress the arguments of the propagator, omit the upper and lower bounds of path integration (which are always understood to take the values in (3.40)), and write µ a Q a for a µ a Q a .
Let us emphasize two points about the phase space path integral expression for the propagator. First, as is typical of path integals, all dynamical quantities appearing inside the integrand are simply classical variables x i (t) and p i (t) rather than quantum operatorŝ x i andp i , so there are no ordering ambiguities. Second, and more importantly, it is critical that the momentum variables p i (t) inside the path integral are not the conjugate momenta to x i (t). The path integral runs over all phase space trajectories with the specified endpoints, and there is no constraint between the functions x(t) and p(t) along these trajectories. This is important because it implies that Let us contrast this situation with that of the proof of Theorem 1. In that context, the momenta and velocities were related by equation (2.18), the Hamilton equation of motion: Therefore, if we choose to treat p i as an independent variable which does not depend on λ, it follows that only λ dependence appears in the Hamiltonian itself, so one finds (3.44) We now use the fact that O is only a function of conserved charges, which are independent of time. This means that the path integral expectation value of O is itself also independent of time, so we may interchange the time integral with the path integral to conclude where we have defined the path integral expectation value Here we have used that the quantity p(t)ẋ(t) is odd in both p i (t) and x i (t), and a path integral of an odd quantity over all paths vanishes by symmetry. Therefore the first term in the path integral expectation value of the first line of ( Similarly, the derivatives of the propagator with respect to the chemical potentials µ a generate expectation values of charges: We conclude that the expressions for ∂ λ K, ∂ T K, and ∂ µa K computed using the path integral formulation are identical to those computed using the canonical analysis. The phase space path integral representation for the propagator therefore obeys the same differential equation, where the right side is defined for any analytic function O of conserved charges.

Comments on Non-Analytic Deformations
Our derivation of the phase space path integral in section 3.1 assumed that the Hamiltonian operator H admits a power series expansion in the operatorsx i andp i . We also in almost exactly the same way. At the risk of repeating ourselves, let us quickly check that this is true. We will consider a Hamiltonian with a form that is slightly more general than a first-order deformation, namely where H 0 is analytic and O is a (possibly non-analytic) function of charges. When f 1 = 1 and f 2 = λ, this is the leading-order correction to H 0 generated by a flow driven by O.
As before, using the kernel representation of the propagator one would similarly argue that a deformation of H by a non-analytic function O of conserved charges has the effect of leaving all of the eigenfunctions φ n unchanged, and only shifts the energy eigenvalues as E n (λ) = f 1 (λ)E n (0) + f 2 (λ)O. One can still differentiate, although it may no longer be possible to express (3.52) in terms of derivatives of K with respect to T and the µ a when O is non-analytic.
A similar analysis is possible using the path integral. By the Baker-Campbell-Hausdorff formula, one has since all commutator terms in the BCH expansion vanish by virtue of the fact that O is a function only of conserved quantities. Suppose that we repeat the time-slicing prescription of section 3.1 for this Hamiltonian. When evaluating (3.14), one has where we have inserted two complete sets of eigenstates |φ n of both the undeformed Hamiltonian H 0 and the charge operators Q a . We can then evaluate the middle factor by replacing O with its classical value O, giving and then we may evaluate the remaining matrix element for the analytic part f 1 (λ) H 0 of the Hamiltonian using the same steps as before. This would lead us to the same result, (3.56) Taking f 1 = 1 and f 2 = λ allows us to conclude, using either formalism, that a deformation of an analytic Hamiltonian by a non-analytic function of charges -to leading order in the deformation parameter -is described by the same differential equation for the propagator. We emphasize that the only difference in the case of a non-analytic deforming operator is that the expectation value O may not be expressible in terms of derivatives of the propagator with respect to T and the µ a .
One might then ask how one can extend this analysis to higher orders in the deformation parameter. For instance, suppose that we use the analysis above to define the By construction, this is equivalent to using the corresponding solution for the deformed energy levels E n (λ) and inserting them into the kernel representation (3.51) of the propagator. In this way, we obtain a consistent prescription for the quantization of a non-analytic theory which agrees with the above analysis to leading order around an analytic seed Hamiltonian. In this case, the analysis which we carried out for the leading-order deformation is sufficient to derive differential equations that hold to all orders along the flow.

Flow of Quantum Partition Function and Comparison to 2d
In the preceding subsection, we have developed general differential equations obeyed by and thus In fact, we could have derived the flow equation for the classical partition function by first arguing that the quantum partition function satisfies the differential equation (3.60) and then taking the limit → 0.
We will discuss some examples of such flow equations only briefly, because they take the same form as the corresponding flow equations for the classical partition function described in section 2.3. However, since these differential equations now hold in the quantum theory, we will comment on the relationship to the analogous flow equations for the quantum mechanical partition functions of two-dimensional field theories which are deformed by the T T operator.
For instance, under the flow (2.33) which is the 1d version of the T T deformation, the propagator K and thermal partition function Z obey the differential equations This flow equation for Z was also considered in [46]. It can be understood as follows.
Suppose that we begin with a two-dimensional conformal field theory whose torus partition function is Z 0 (τ, τ ), where τ is the modular parameter of the torus. One can then deform this theory by the T T operator to obtain a one-parameter family of deformed torus partition functions Z λ which obey the differential equation This differential equation can be obtained by performing a Hubbard-Stratonovich transformation which replaces the T T deformation with a random metric [11]. Modular properties of the deformed partition function were studied in [12,13]; although the T T -deformed theory no longer enjoys conformal symmetry, the partition function Z λ is still invariant with respect to a modular transformation under which the parameter λ also transforms.
Let us specialize to a torus with purely imaginary modular parameter τ = iβ 8 . Using  It is interesting to consider a similar dimensional reduction of the two-dimensional root-T T flow. The analogous flow equation for the torus partition function of a root-T T deformed 2d CFT was conjectured in [28] to be based on a proposal for the flow of the cylinder spectrum which was justified by holographic considerations. If we take a torus with modular parameter then the flow equation (3.65) reduces to The factor of 1 4 multiplying the third term in (3.67) is related to the normalization of the root-T T operator, and can be rescaled to 1 by an appropriate redefinition. Up to this choice of scaling, this is the same differential equation which is satisfied by the partition function of the ModMax oscillator, which we will present in equation (4.24).

Application to ModMax Oscillator
In this section, we will apply the general results of the preceding sections to the main example of interest in the present work, which is the ModMax oscillator. This theory was first introduced in [30] and is a particular deformation of an isotropic harmonic oscillator.
Given a collection of position variables x i , for i = 1, . . . , N, we begin by defining the undeformed theory with the Lagrangian where we have set the mass m and frequency ω of the harmonic oscillator to 1 for convenience. This theory has a conserved energy which is the Noether current associated with time translation symmetry, along with a collection of conserved angular momenta which are the conserved currents associated with rotations x i → R i j x j , R ∈ SO(N). The total angular momentum is For any γ, we can now define the Lagrangian for the ModMax oscillator as Note that there is a choice of the relative sign between the two terms in (4.5) which is correlated with the choice of sign for the root-T T operator that drives the flow equation (4.7) below. One can also view this sign choice as a convention for the sign of the parameter γ, since sending γ → −γ reverses the relative sign.
Even though this Lagrangian L γ is written in terms of the conserved quantities E 0 and J 2 0 in the undeformed theory, it satisfies a differential equation which involves the conserved currents in the deformed theory at finite γ, namely One can show that L γ obeys which we refer to as the 1d root-T T flow equation. The corresponding Hamiltonian, satisfies a flow equation with the opposite sign, as required by Theorem 1. In equations (4.8) and (4.9), the quantities E 0 , E γ , J 2 0 , and J 2 γ are defined by beginning with the appropriate Noether currents computed in the Lagrangian formulation, and then expressing these quantities in terms of conjugate momenta p i rather than velocitiesẋ i .
Although the ModMax oscillator can be defined for any number N of position variables x i , i = 1, . . . , N, we will focus on the case N = 2 for simplicity. First let us review some features of the classical dynamics of the ModMax oscillator.

Classical Aspects
We now specialize to the case of N = 2 coordinates x i , and we will use the notation x 1 = x, x 2 = y. In terms of these variables, the general Lagrangian (4.5) for the ModMax oscillator can be written as (4.10) The conserved angular momentum, which is the Noether charge associated with rotations in the (x, y) plane, is (4.11) It is interesting to express J γ in terms of the conjugate momenta p x and p y , as appropriate for formulating flows for the Hamiltonian. The conjugate momenta computed from the Lagrangian (4.10) are After expressing the angular momentum J γ in terms of p x and p y , one finds That is, when written in Hamiltonian variables, the deformed angular momentum J γ takes the same functional form as the undeformed angular momentum J 0 . This is a special case of the observation, which we first made in the text below equation (3.36), that f (H, Q a ) deformations modify the Hamiltonian but not the other charges such as J.
Similarly, for N = 2 one can write the Hamiltonian for the ModMax oscillator as (4.14) Let us consider the symmetries of this theory in somewhat more detail. It is well-known that the undeformed theory, which is the 2d isotropic harmonic oscillator, enjoys an SU (2) symmetry. To see this, it is convenient to define complex variables 15) so that the harmonic oscillator Hamiltonian is  is also invariant under the U(1) transformations (4.19).
We have commented before that the ModMax oscillator is a particular dimensional reduction of the four-dimensional ModMax theory, which enjoys electric-magnetic duality invariance. In fact, the U(1) invariance (4.19) of the ModMax oscillator descends directly from this electric-magnetic duality symmetry, which can be written as where F µν is the field strength of the 4d electrodynamics theory and F µν is its Hodge dual.
It is straightforward to see that any deformation of the isotropic harmonic oscillator which is constructed from the Hamiltonian and the conserved angular momentum will preserve invariance under the U(1) duality transformation (4.19). This is the 1d version of the statement that any deformation of a theory of self-dual electrodynamics in four spacetime dimensions, where the deforming operator is a function of the energymomentum tensor of the theory, will preserve electric-magnetic duality invariance. See [52] for futher discussion and examples of such duality-preserving stress tensor flows.

Flow Equation for Partition Functions
As we have seen in sections 2 and 3, both the classical and quantum partition functions for a theory deformed by a function of conserved charges satisfy the same differential equation.
We will now study this differential equation in the case of the ModMax oscillator, which obeys a flow driven by the operator R or R introduced in equation (2.52). This falls into the class of non-analytic deformations which we briefly considered around equation (2.67). In this case, the differential equation (2.73) simplifies considerably.
The reason for this simplification is the following. We have seen that the solution to the flow equation in this case is given by the Lagrangian (4.10) or Hamiltonian (4.14), which satisfy the equations Because ∂ γ L γ = ±R (γ) and ∂ γ H γ = ∓R (γ) , this means The flow equation (2.73) then becomes where in the last step we have expressed quantities in terms of derivatives of the partition function. Note that, if we had instead chosen a different normalization for the 1d root-T T operator, so that the flow equation for the Lagrangian were ∂ γ L γ = c 0 R (γ) for some constant c 0 , the flow equation would have been For the choice c 0 = 1 2 , this matches the dimensional reduction of the conjectured flow equation for the torus partition function of a root-T T deformed CFT given in (3.67).

Equation (4.24) is very nearly of a familiar form. To see this, it is convenient to
Wick-rotate µ → iµ and γ → iγ, which reverses the signs on two terms. The resulting differential equation can be written as This is identical to the Laplace equation for a function f : R 3 → R written in cylindrical coordinates (r, θ, z), namely where the roles of the coordinates (r, θ, z) are played by (β, γ, µ), respectively.
One may therefore solve the flow equation for Z(β, γ, µ) by separation of variables, in a manner analogous to that which is done when studying electrodynamics in cylindrical coordinates. The original differential equation We note that formal analytic continuation of flow equations to imaginary values of the parameters, like that which relates (4.24) and (4.26), has sometimes been useful in previous work. For instance, in [19] such a continuation of a T T -like parameter λ was useful in relating the flow equations which produce the 4d Born-Infeld and reverse-Born-Infeld theories, which are two of the four solutions to the zero-birefringence condition for 4d nonlinear electrodynamics [53].

Direct Computation of Classical Partition Function
As a warm-up for our study of the quantum partition function, and in order to illustrate an example of the Laplace equation which the deformed flavored partition functions satisfy, we will now perform a direct computation of the classical partition function for the 2d ModMax oscillator. The resulting formulas will turn out to be tidier if we evaluate this partition function with an imaginary chemical potential for the angular momentum, which merely reverses a sign in the corresponding flow equation. We will therefore compute Z(β, γ, µ) = 1 (2π) 2 dx dy dp x dp y exp (−βH γ + iµJ γ ) , (4.29) where all integrals run from −∞ to ∞. Writing this integrand explicitly in terms of the positions and momenta, and making the sign choice for which the two terms in the Hamiltonian H γ are both manifestly positive, the integral we wish to evaluate is Z(β, γ, µ) = 1 (2π) 2 dx dy dp x dp y exp − β It is convenient to perform the change of variables so that the integral becomes (4.32) Note that this change of variables has decoupled the square root interaction into two factors. We can now go to polar coordinates in the (u 1 , u 2 ) and (v 1 , v 2 ) planes as Then our partition function is (4.34) The angular integrals give factors of 2π, whereas the resulting radial integrals can be evaluated in closed form, and we find can also check by explicit computation that it obeys the partial differential equation When the chemical potential is set to zero, the deformed partition function (4.35) is simply a γ-dependent rescaling of the undeformed partition function. However, when µ and γ are both finite, the temperature dependence is modified in a more interesting way.
We will see shortly that, in the quantum theory, even for µ = 0 the deformed partition function is not simply a rescaling of the undeformed partition function.

Quantum Aspects
We now turn to the main subject of this work, which is the quantum mechanics of the ModMax oscillator. At first glance, it is not so clear that one should be able to quantize this theory at all. Naïvely, one would like to begin with the classical Hamiltonian (4.14) and promote all position and momentum variables to operators. This requires one to make sense of the operator square root in the second term, which takes the form It is not immediately obvious what this operator should mean. First note that we would not expect that it is possible to define an operator square root for a generic combination of position and momentum operators, at least without additional assumptions like positivity.
For instance, the expression √x does not give a conventional Hermitian operator; even if one attempts to define it by diagonalizing the position operator and declaring √x |x = √ x |x , this operator will have imaginary eigenvalues for negative positions x.
In our case, we are aided in interpreting the operator (4.37) by the fact that it is positive definite -which allows us to define it by diagonalization and taking square roots -and because it is a function of conserved charges in the undeformed theory, which allows us to write flow equations for quantities in the deformed theory using the results of section 3. We will begin by attempting to understand the operator (4.37) directly using raising and lowering operators in the theory of the undeformed harmonic oscillator.

Ladder Operator Representation
We can develop one useful perspective on the ModMax oscillator by rewriting the Hamiltonian in terms of creation and annihilation operators. As usual, when studying the undeformed theory of an isotropic 2d harmonic oscillator with Hamiltonian 38) it is natural to define the annihilation operatorŝ whose Hermitian conjugates are the creation operators, In terms of these operators the Hamiltonian takes the standard form where we have defined the number operators N i =â † iâ i . However, the angular momentum operator can be made more transparent by a change of basis. We can instead define the "circularly-polarized" linear combinationŝ The corresponding number operators, N L =â † Lâ L and N R =â † Râ R , count the numbers of left-moving and right-moving circular quanta. This is a useful way to leverage the rotational symmetry of the problem, since the angular momentum operator is now simply 43) and the Hamiltonian is We now turn to the 1d root-T T deformation which generates the interaction term in the ModMax oscillator Lagrangian. This operator is The argument of the square root factorizes into left-moving and right-moving pieces.
Because the left-moving and right-moving operators commute, we are free to split the square root into separate factors: Each of the operators 1 + 2 N L and 1 + 2 N R have strictly positive eigenvalues, and it is therefore possible to define an operator square root. This is equivalent to defining the square root operators by the Taylor series expansions Either of these infinite sums is convergent and well-defined when acting on any state in the Hilbert space. We can therefore write the Hamiltonian of the ModMax oscillator as where we have again chosen the positive sign for concreteness, although one can obtain the other sign choice by taking γ to be negative.
One could study the quantum mechanics of this theory in essentially the same way that we have described above for the canonical quantization prescription. That is, one considers a complete basis of eigenstates |N L , N R of the undeformed Hamiltonian and angular momentum, and then shifts each of their energy eigenvalues according to (4.48).
However, this number operator representation also allows us to see a simple way to generate the ModMax oscillator in one step from the undeformed isotropic harmonic oscillator. Let us introduce operators 49) which are defined by the convergent Taylor series expansions (4.47). Then the undeformed Hamiltonian can be written as Suppose that one now performs the transformation Using this basis, the flavored partition function with an imaginary chemical potential for the angular momentum J = N R − N L can be written as Here we have used the fact that J γ = J 0 = N R − N L , following the comments around (4.13), and taken the positive sign choice in the Hamiltonian as usual.
Unlike in the case of the classical partition function, it does not seem to be possible to obtain a simple closed-form expression for the sum (4.53). However, it is straightforward to evaluate the trace perturbatively in the flow parameter γ. For instance, to leading order one finds where Φ(z, s, a) is a special function known as the Lerch transcendent and defined by (4.55) Even when µ = 0, the expression (4.54) for the partition function to order γ is not simply a rescaling of the undeformed partition function by a γ-dependent factor. This is unlike the classical partition function (4.35), to which the expression (4.53) reduces in the limit → 0. 11 This gives one way to see that the quantum theory of the ModMax oscillator is richer than its classical counterpart, since even without a chemical potential µ there is a non-trivial interplay between the inverse temperature β and flow parameter γ.
One can check that the infinite sum (4.53) satisfies the flow equation (4.24) for a theory deformed by the 1d root-T T operator, after specializing to an imaginary value of µ. As we mentioned, this differential equation can solved by separation of variables, which gives a general solution that is a sum of factorized terms involving exponentials and Bessel functions. It is instructive to see how (4.53) can be brought into this form, since as written this sum is not obviously related to Bessel functions. This can be accomplished using a variant of the Jacobi-Anger expansion, which expresses a plane wave as a superposition of cylindrical waves. For any z, θ ∈ C, these identities take the form See, for instance, section 10.35 of [54]. To apply these identities to the partition function Although rather unwieldy, this expansion in Bessel functions makes the connection between the trace form (4.53) of the partition function, and the Laplace-type equation which it satisfies, more explicit.

Description of Deformed Propagator
To conclude, we will comment on the characterization of the propagator for the ModMax oscillator. Here we will be brief, since this is a direct application of the general results of section 3 and leads to flow equations which are essentially identical to those for the classical and quantum partition function discussed above.
Because the Hamiltonian for the ModMax oscillator is of the form (3.50), there is no ambiguity in defining the propagator by a path integral representation, even at all orders in γ. Therefore, the flavored propagator is defined by the general phase space path integral given in equation (3.40). This propagator, with real chemical potential, satisfies which is the same as the differential equation for the flavored partition function after making the replacement β = iT . However, we emphasize that the result is more general, since this holds for the propagator K( x B , t B ; x A , t A ; λ; µ) with any initial position x A and final position x B . In contrast, the thermal partition function is obtained from the Euclidean time propagator with periodic boundary conditions, which means that the initial and final positions are equal.
The differential equation (4.58) fully determines the propagator for the ModMax oscillator, given the initial condition K(γ = 0) and the first derivative ∂ γ K| γ=0 , which is related to the expectation value of the 1d root-T T operator in the undeformed theory.
The initial condition K(γ = 0) is essentially the propagator for a 2d harmonic oscillator in a background magnetic field, which plays the role of the chemical potential for the angular momentum. This quantity can be computed either by canonical methods or path integral methods; for the path integral computation, see the pedagogical review [55].
We also note that the propagator can be written using the kernel representation and the basis of states |N L , N R , which gives where the φ N L ,N R are harmonic oscillator wavefunctions, which are known in closed form.
Our characterization of the propagator K, including that it satisfies the Laplace-like differential equation (4.58), is one of the main results of this work. Because the propagator for the ModMax oscillator is completely determined by the above considerations, this essentially constitutes a full solution of the model. Any physical question involving time evolution of states can in principle be extracted from the function K. This therefore completes our study of the quantum mechanical theory of the ModMax oscillator.

Conclusion
In this work, we have studied general deformations of 1d theories by conserved charges, both at the classical and quantum level. This has allowed us to obtain flow equations for quantities in the theory of the ModMax oscillator, which is the dimensional reduction of the 4d ModMax theory. In particular, we have found that the thermal partition function in the quantum theory of the ModMax oscillator -or, relatedly, the real-time propagator -satisfies a certain partial differential equation which is related by Wick-rotation to the Laplace equation in 3d cylindrical coordinates.
One way of summarizing our analysis is to say that any deformation of a quantum mechanical theory by conserved charges is essentially "solvable" in the sense that one can write differential equations which relate quantities in the deformed theory, such as the propagator or partition function, to those in the undeformed theory. The results on the ModMax oscillator are a special case of this fact when the deformation is driven by the 1d root-T T operator. Furthermore, the quantization of such charge-deformed models is unambiguous, since one obtains equivalent flow equations using either canonical quantization or path integral quantization. We therefore conclude that the fairly broad class of theories obtained through deformations by conserved charges should be included among other examples of solvable deformations of quantum mechanics, such as the one in which the quadratic kinetic term is replaced by one involving a hyperbolic cosine [56].
There are several directions for future investigation, some of which we outline below.

One-Loop Calculation
In this manuscript, we have focused on finding exact flow equations for observables in the quantum theory of the ModMax oscillator, such as the propagator. We have also studied certain quantities in the classical theory, such as the classical partition function, for which it is possible to obtain a closed-form expression.
However, it would be also interesting to study semi-classical expressions for quantum observables by performing an expansion in . In the limit as γ → 0, the ModMax oscillator reduces to the ordinary harmonic oscillator, which is one-loop exact. It seems very unlikely that the deformed theory is also one-loop exact due to the complicated nature of the interaction term. One could attempt to compute the one-loop correction around the classical solution to the equations of motion for the ModMax oscillator and examine how closely this reproduces the full quantum results.
To do this, one would need to expand the phase space path integral which defines the propagator for the ModMax oscillator and retain terms up to quadratic order in fluctuations around the classical path. Fortunately, it is straightforward to write down the general classical solution to the equations of motion for the ModMax oscillator following [30]. Given a set of initial conditions (x 0 , y 0 ) and (p x,0 , p y,0 ), one can evaluate the conserved energy H 0 and angular momentum J 0 corresponding to this initial condition, and then define The general solution to the deformed equations of motion is given by x(t) = sin(At) (p x,0 cos(Bt) + p y,0 sin(Bt)) + cos(At) (x 0 cos(Bt) + y 0 sin(Bt)) , y(t) = sin(At) (p y,0 cos(Bt) − p x,0 sin(Bt)) + cos(At) (y 0 cos(Bt) − x 0 sin(Bt)) , p x (t) = cos(At) (p x,0 cos(Bt) + p y,0 sin(Bt)) − sin(At) (x 0 cos(Bt) + y 0 sin(Bt)) , p y (t) = cos(At) (p y,0 cos(Bt) − p x,0 sin(Bt)) − sin(At) (y 0 cos(Bt) − x 0 sin(Bt)) . (5.3) One could then perform a one-loop computation by defining where x i cl (t) and p i cl (t) are a solution to the equations of motion (5.3), and then performing the phase space path integral over δp i and δx i . This is slightly more involved than a semiclassical computation in the ordinary Feynman path integral, which gives a one-loop determinant. In this case, one would first expand the Hamiltonian action to write H = f 1 (δx) δp 2 + f 2 (δx)δp + f 3 (δx) , where we suppress indices on the fluctuations. It is still possible to carry out the D(δp) path integral over momentum fluctuations for such a Hamiltonian, as described in section 1.4 of [50]. After performing the momentum path integral, one is left with a path integral over D(δx) with a modified Lagrangian. It should then be possible to complete the semi-classical expansion by computing the one-loop determinant associated with the fluctuations δx around the classical solution in this Feynman path integral.

Laplace Equation for Flavored Partition Function and Narain Moduli Space
One of the main results of this manuscript is that the propagator for a root-T T deformed theory satisfies a partial differential equation which -up to signs which can be eliminated where Θ(m, τ ) is a Siegel-Narain theta function, which obeys the differential equation where ∆ M D is the natural Laplacian on the Narain moduli space that parameterizes the T D . For instance, in the case of a single compact boson, the target space is a circle of radius R and the Laplacian is (5.8) The structure of equation (3.65) is almost identical to (5.7), except with the Laplacian on moduli space replaced with the second derivative with respect to γ. Also note that the root-T T flow equation involves the partition function itself, while (5.7) holds for the function Θ appearing in the numerator of the partition function, not for the full combination including the eta function in the denominator.
It would be interesting to understand whether there is a deeper relationship between these flow equations for root-T T deformed partition functions, and their dimensional reductions to 1d, and properties of Narain moduli space.

Coupling to Worldline Gravity
As we have emphasized, there may be several inequivalent quantization schemes -or "UV completions" -for a particular classical theory. In particular, this is true for theories obtained from the (0 + 1)-dimensional version of the T T deformation. One way to see the difference between two such choices of quantization scheme is by examining the resulting thermal partition functions. As we reviewed in section 2. However, there is a second UV completion of this deformation which is defined by coupling the seed theory to worldline gravity [35]. In this prescription, the deformed partition function is defined by the path integral Z λ (β) = De DX Dσ Vol exp (−S 0 (e; X) − S(λ; e, σ)) , (5.9) where X represents the fields of the undeformed theory, which are now minimally coupled to an einbein e, and σ is an auxiliary scalar field with an action S(λ; e, σ) = 1 2λ After evaluating the path integral, one finds that this quantization scheme produces the deformed partition function which is a different partition function than the result (2.66), although they agree in the unit winding sector (m = 1) up to normalization for the flow parameter. 12 It would be very interesting to find an analogue of this worldline gravity prescription for the 1d root-T T -like deformation. It seems likely that one would need to couple the undeformed theory to both an einbein e(τ ) and a gauge field for the SO(N) symmetry, which plays the role of a time-dependent magnetic field. This would be in accord with the fact that, in order to obtain a flow equation for the partition function using our simpler quantization prescription for the root-T T deformed quantum mechanics, we were forced to turn on a chemical potential µ for the angular momentum. Finding a path integral expression which represents this alternative, worldline gravity quantization prescription would allow us to better understand the available choices of UV completions for ModMaxlike theories of quantum mechanics.

A First-Order Analysis of Lagrangian and Hamiltonian Flows
In this appendix, we will consider the analogue of Theorem 1 where one uses the relationship between the conjugate momenta p i and velocitiesẋ i in the undeformed theory (defined by L 0 and H 0 ) rather than in the deformed theory (defined by L λ and H λ ). If one does not correct the definition of the conjugate momenta, the resulting deformations of the Lagrangian and Hamiltonian will be equivalent only to leading order in the deformation parameter. The logic of this proof was originally presented in appendix A of [40], which focused on T T deformations of two-dimensional quantum field theories describing a field φ and conjugate momentum π. Here we will instead consider 1d theories which describe the dynamics of a collection of positions x i and velocitiesẋ i , since this is the primary focus of the present work, although the reasoning is almost identical. 12 In particular, one can redefine λ → −4λ in (2.66) to match the conventions of (5.11).
Let us represent the deformation of the Hamiltonian to first order as follows: H(x i , p i ) = H 0 (x i , p i ) + ǫO(x i , p i ) . (A.1) We claim that this is equivalent to a deformation of the Lagrangian to first order as Here we define f i 0 to be the function that relatesẋ i and p i when ǫ = 0. More precisely, Because we are defining the deformation in terms of the Hamiltonian, we view the momenta p i as independent variables which do not depend on the deformation parameter λ; this corresponds to the forward direction of Theorem 1, rather than the converse. Consequently the definition of the velocitiesẋ i changes due to the deformation.
We use the Legendre transform to write the deformed Lagrangian in terms of the Hamiltonian, Next, we solve the equations of motion to write p i as a function of x i andẋ i . We assume that this solution can be written as an infinite series as follows: p i = f i (x j ,ẋ j ) = f i 0 (x j ,ẋ j ) + ǫf i 1 (x j ,ẋ j ) + O(ǫ 2 ). (A.5) By Hamilton's equations of motion,ẋ i = ∂H ∂p i . We can use the equation for p i to first order to obtain,ẋ We expand the first term perturbatively to obtain the following expression forẋ i : Matching the terms of order ǫ 0 on either side of (A.7) giveṡ This relation describes how f i 1 is implicitly determined in terms of H 0 and O. Since the Legendre transform is an involution, we can express the deformed Lagrangian in terms of the deformed Hamiltonian, (A.10) Using the equation for p i , we obtain By equation (A.8), the term proportional to ǫ must vanish, leading to This is the claim we sought to prove since deforming the Hamiltonian will give rise to the same results as deforming the Lagrangian according to (A.2) to first order in ǫ.
Since we are only looking at the first order, it does not matter that the relation betweeṅ x i and p i changes. However, as we saw in Theorem 1, one can extend this argument to all orders in the deformation parameter by using the corrected relationship between velocities and conjugate momenta in the deformed theory.