Polar decomposition of the Wiener measure: Schwarzian theory versus conformal quantum mechanics

TL;DR

This paper establishes a mathematical connection between conformal quantum mechanics and Schwarzian theory by deriving the polar decomposition of the Wiener measure, enabling evaluation of complex functional integrals and solving the Schrödinger equation in imaginary time.

Contribution

It provides an explicit polar decomposition of the Wiener measure and links functional integrals in conformal quantum mechanics to those in Schwarzian theory, offering new computational tools.

Findings

Derived the explicit polar decomposition of the Wiener measure.

Connected functional integrals in conformal quantum mechanics to Schwarzian theory.

Obtained the fundamental solution of the Schrödinger equation in imaginary time.

Abstract

We derive the explicit form of the polar decomposition of the Wiener measure, and obtain the equation connecting functional integrals in conformal quantum mechanics to those in the Schwarzian theory. Using this connection we evaluate some nontrivial functional integrals in the Schwarzian theory and also find the fundamental solution of the Schroedinger equation in imaginary time in the model of conformal quantum mechanics.

Equations

$$I=-\frac{1}{{\sigma }^2}\underset{S^1}{\int }\left[{\mathcal{S}}_{\phi }(t)+2{\pi }^2{\left({\phi }^{\prime }(t)\right)}^2\right]dt,$$

$${\mathcal{S}}_{\phi }(t)={\left(\frac{{\phi }^{\prime \prime }(t)}{{\phi }^{\prime }(t)}\right)}^{\prime }-\frac{1}{2}{\left(\frac{{\phi }^{\prime \prime }(t)}{{\phi }^{\prime }(t)}\right)}^2$$

$${\mu }_{\sigma }(d\phi )=\exp \left\{\frac{1}{{\sigma }^2}\underset{0}{\overset{1}{\int }}{\mathcal{S}}_{\phi }(t)dt\right\}d\phi$$

$${\mu }_{\sigma }(d\phi )=\frac{1}{\sqrt{{\phi }^{\prime }(0){\phi }^{\prime }(1)}}\exp \left\{\frac{1}{{\sigma }^2}\underset{0}{\overset{1}{\int }}{\mathcal{S}}_{\phi }(\tau )d\tau +\frac{1}{{\sigma }^2}\left[\frac{{\phi }^{\prime \prime }(0)}{{\phi }^{\prime }(0)}-\frac{{\phi }^{\prime \prime }(1)}{{\phi }^{\prime }(1)}\right]\right\}d\phi$$

$$\phi (t)=\frac{\underset{0}{\overset{t}{\int }}{e}^{\xi (\tau )}d\tau }{\underset{0}{\overset{1}{\int }}{e}^{\xi (\eta )}d\eta },\xi (t)=\log {\phi }^{\prime }(t)-\log {\phi }^{\prime }(0),$$

$$w_{\sigma }(dx)=\exp \left\{-\frac{{\sigma }^2}{8{\rho }^2}\right\}{\left({\phi }^{\prime }(0){\phi }^{\prime }(1)\right)}^{\frac{3}{4}}{\mu }_{\frac{2\sigma }{\rho }}(d\phi )d\rho ,$$

$$x\in C_{+}([0,1]),\phi \in Dif{f}_{+}^1([0,1]),0<\rho <+\infty ,$$

$$w_{\sigma }(dx)=\exp \left\{-\frac{{\sigma }^2}{8{\rho }^2}\right\}{\mu }_{\frac{2\sigma }{\rho }}(d\phi )d\rho ,$$

$$x\in C_{+}(S^1),\phi \in Dif{f}_{+}^1(S^1),0<\rho <+\infty .$$

$$\underset{C_{+}(S^1)/SL(2,\text{R})}{\int }\Phi (x)\exp \left\{-\frac{1}{2{\sigma }^2}\underset{S^1}{\int }\left[{\left(x^{\prime }(t)\right)}^2-2{\pi }^2{x}^2(t)+\frac{2g}{x^2(t)}\right]dt\right\}dx$$

$$=\underset{0}{\overset{+\infty }{\int }}d\rho \exp \left\{-\left(\frac{{\sigma }^2}{8}+\frac{g}{{\sigma }^2}\right)\frac{1}{{\rho }^2}\right\}$$

$$\times \underset{Dif{f}_{+}^1(S^1)/SL(2,\text{R})}{\int }\Phi (x(\rho ,\phi ))\exp \left\{\frac{{\rho }^2}{4{\sigma }^2}\underset{S^1}{\int }\left[{\mathcal{S}}_{\phi }(t)+2{\pi }^2{\left({\phi }^{\prime }(t)\right)}^2\right]dt\right\}d\phi .$$

$$w_{\sigma }(dx)=\exp \left\{-\frac{1}{2{\sigma }^2}\underset{0}{\overset{1}{\int }}{\left(x^{\prime }(t)\right)}^{2}dt\right\}dx.$$

$$x\mapsto fx,(fx)(t)=x\left(f^{-1}(t)\right)\frac{1}{\sqrt{{\left(f^{-1}(t)\right)}^{\prime }}},$$

$$x\in C_{+}([0,1]),f\in Dif{f}_{+}^3([0,1]).$$

$$\frac{1}{{\rho }^2}=\underset{0}{\overset{1}{\int }}\frac{1}{x^2(t)}dt.$$

$${\phi }^{-1}(t)={\rho }^2\underset{0}{\overset{t}{\int }}\frac{1}{x^2(\tau )}d\tau .$$

$$x(t)=\rho \frac{1}{\sqrt{{\left({\phi }^{-1}(t)\right)}^{\prime }}}.$$

$$\underset{0}{\overset{1}{\int }}{x}^2(t)dt={\rho }^2\underset{0}{\overset{1}{\int }}{\left({\phi }^{\prime }(\tau )\right)}^{2}d\tau .$$

$$-\frac{1}{2{\sigma }^2}\underset{0}{\overset{1}{\int }}{\left(x^{\prime }(t)\right)}^{2}dt=\frac{{\rho }^2}{4{\sigma }^2}\left\{\underset{0}{\overset{1}{\int }}{\mathcal{S}}_{\phi }(\tau )d\tau +\left[\frac{{\phi }^{\prime \prime }(0)}{{\phi }^{\prime }(0)}-\frac{{\phi }^{\prime \prime }(1)}{{\phi }^{\prime }(1)}\right]\right\}.$$

$$w_{\sigma }(dx)={\mathcal{P}}_{\sigma }(\rho ){\left({\phi }^{\prime }(0){\phi }^{\prime }(1)\right)}^{\frac{3}{4}}{\mu }_{\frac{2\sigma }{\rho }}(d\phi )d\rho .$$

$$\underset{C_{+}([0,1])}{\int }\delta \left(\tilde{\rho }-{\left(\underset{0}{\overset{1}{\int }}\frac{1}{x^2(t)}dt\right)}^{-\frac{1}{2}}\right)\delta \left(x(0)-x(1)\right)w_{\sigma }(dx)=\underset{0}{\overset{+\infty }{\int }}\delta (\tilde{\rho }-\rho ){\mathcal{P}}_{\sigma }(\rho )$$

$$\times \underset{Dif{f}_{+}^1([0,1])}{\int }\delta \left(\frac{\rho }{\sqrt{{\left({\phi }^{-1}(0)\right)}^{\prime }}}-\frac{\rho }{\sqrt{{\left({\phi }^{-1}(1)\right)}^{\prime }}}\right){\left({\phi }^{\prime }(0){\phi }^{\prime }(1)\right)}^{\frac{3}{4}}{\mu }_{\frac{2\sigma }{\rho }}(d\phi )d\rho$$

$${\mathcal{P}}_{\sigma }(\tilde{\rho })\frac{2}{\tilde{\rho }}\underset{Dif{f}_{+}^1([0,1])}{\int }\delta \left(\frac{{\phi }^{\prime }(1)}{{\phi }^{\prime }(0)}-1\right){\phi }^{\prime }(0){\mu }_{\frac{2\sigma }{\tilde{\rho }}}(d\phi ).$$

$${\mathcal{P}}_{\sigma }(\tilde{\rho })\frac{2}{\tilde{\rho }}\underset{C_0([0,1])}{\int }\delta \left(\xi (1)\right)\frac{1}{\underset{0}{\overset{1}{\int }}{e}^{\xi (\tau )}d\tau }{w}_{\frac{2\sigma }{\rho }}(d\xi ).$$

$$\underset{C_0([0,1])}{\int }\delta \left(\xi (1)\right)\exp \left\{\frac{-2{\beta }^2}{{\kappa }^2(\beta +1)}\frac{1}{\underset{0}{\overset{1}{\int }}{e}^{\xi (\tau )}d\tau }\right\}{w}_{\kappa }(d\xi )$$

$$=\frac{1}{\sqrt{2\pi }\kappa }\exp \left\{-\frac{1}{2{\kappa }^2}{\left(2\log (\beta +1)\right)}^2\right\}$$

$$\delta \left(\tilde{\rho }-{\left(\underset{0}{\overset{1}{\int }}\frac{dt}{x^2(t)}\right)}^{-\frac{1}{2}}\right)=\frac{2}{{\tilde{\rho }}^3}\delta \left(\frac{1}{{\tilde{\rho }}^2}-\underset{0}{\overset{1}{\int }}\frac{dt}{x^2(t)}\right).$$

$$\frac{1}{\sqrt{2\pi }\sigma }{\mathcal{P}}_{\sigma }(\tilde{\rho })=\frac{2}{{\tilde{\rho }}^3}\exp \left\{-\frac{{\sigma }^2}{8{\tilde{\rho }}^2}\right\}\underset{C_{+}([0,1])}{\int }\delta \left(\frac{1}{{\tilde{\rho }}^2}-\underset{0}{\overset{1}{\int }}\frac{dt}{x^2(t)}\right)\delta \left(x(0)-x(1)\right)w_{\sigma }(dx).$$

$$w_{\sigma }(dx)=\exp \left\{-\frac{1}{2{\sigma }^2}\underset{0}{\overset{1}{\int }}{\left(x^{\prime }(t)\right)}^{2}dt\right\}dx,$$

Full text

I Introduction

In recent years, the Schwarzian theory has become extremely popular. It is given by the action

[TABLE]

where

[TABLE]

is the Schwarzian derivative, and $\varphi(t)$ is an orientation preserving $(\varphi^{\prime}(t)>0)$ diffeomorphism of the unit circle $(\varphi\in Diff^{1}_{+}(S^{1}))\,.$

The Schwarzian action appears to be the effective action in the quantum mechanical model of Majorana fermions with a random interaction (Sachdev-Ye-Kitaev model), in the holographic description of the Jackiw-Teitelboim dilaton gravity, in open string theory and in some other models (Kit) - (WBCK) .

An extraordinary universality of the Schwarzian action is a consequence of its $SL(2,\textbf{R})$ invariance. At the same time, the Schwarzian theory is not the only $SL(2,\textbf{R})$ invariant action. This symmetry manifests itself in conformal quantum mechanics (Alfaro) , Liouville quantum mechanics (PP) -(T) and some other models that are used to describe the near-horizon geometry of Reissner-Nordström black hole and $AdS_{2}/CFT_{1}$ duality, gravity near space-like singularity (see, e.g., (Kallosh) -(Pioline) ), and other physical problems where a universal regime is reached. Therefore, the attempts (MSY) , (BAK) - (SW) to connect in some way the Schwarzian theory to other conformal invariant quantum mechanical models are quite natural.

In this case, a temptation to substitute the simpler conformal action for the Schwarzian action (1) may emerge. However, as we will see below, the relationship between these theories is more difficult.

Here, to connect theories with each other we propose to study the corresponding functional integration measures.

In (Shavgulidze1978) -(Shavgulidze2000) , the measures

[TABLE]

on the group $Diff^{1}_{+}(S^{1})$, and

[TABLE]

on the group $Diff^{1}_{+}([0,\,1])$ were constructed.

The measure (4) can be generated from the Wiener measure under some special substitution of variables (Shavgulidze1978) -(Shavgulidze2000) . Namely, let us consider the function $\xi\in C_{0}([0,\,1])\,$ (that is, $\xi(t)$ is a continuous function on the interval satisfying the boundary condition $\xi(0)=0\,$). Then, under the substitution

[TABLE]

the measure $\mu_{\sigma}(d\varphi)$ on the group $Diff^{1}_{+}([0,\,1])$ turns into the Wiener measure $w_{\sigma}(d\xi)$ on $C_{0}([0,\,1])\,.$

The measures (3) and (4) are quasi-invariant under the action of the subgroup consisting of three times differentiable transformations $Diff^{3}_{+}(S^{1})$ $(\,Diff^{3}_{+}([0,\,1])\,)\,.$

Quasi-invariance means that under the action of the subgroup $G=Diff^{3}_{+}(S^{1})$ ($\,G=Diff^{3}_{+}([0,\,1])\,)$, the measure $\mu_{\sigma}(d\varphi)$ transforms to itself multiplied by a function $\mathcal{R}_{g}(h)$ parametrized by the elements of the subgroup $g\in G\,:$ $\mu\left(\,d(g\circ h)\,\right)=\mathcal{R}^{\mu}_{g}(h)\,\mu(\,dh\,)\,.$)

Using the quasi-invariance of the measure $\mu_{\sigma}(d\varphi)$, we have evaluated functional integrals for the partition function and the correlation functions in the Schwarzian theory explicitly, and derive the general rules of functional integration in the theories of the Schwarzian type (BShExact) -(BShRules) .

The Wiener measure on the infinite-dimensional space of continuous positive functions $C_{+}([0,\,1])$ is also known to be quasi-invariant under the action of $Diff^{3}_{+}([0,\,1])$ (Shepp) , (Shavgulidze2000) (see also (BSh) ).

In this paper (sections II, III), we find that the two measures are connected with each other by the equation

[TABLE]

[TABLE]

which we name ”Polar decomposition of the Wiener measure”.

The corresponding polar decomposition of the measure on the circle has the form

[TABLE]

[TABLE]

In section IV, for $SL(2,\textbf{R})$ invariant functionals $\Phi$ we obtain the equation

[TABLE]

[TABLE]

[TABLE]

It connects functional integrals in conformal quantum mechanics to corresponding functional integrals in the Schwarzian theory, and makes it possible to improve the technique of functional integration significantly to evaluate nontrivial functional integrals in both theories.

In section V, we demonstrate an example of the application of above equation in conformal quantum mechanics. Using already evaluated functional integrals in the Schwarzian theory, we find the fundamental solution of the Schrödinger equation in imaginary time in the quantum mechanical model with Calogero-type potential.

In section VI, we give the concluding remarks.

II Stratification of the space $C_{+}([0,\,1])$ and polar decomposition of the Wiener measure

Consider the Wiener measure with the variance $\sigma$ on the space of continuous positive functions $x(t)$ on the interval $[0,\,1]\,,$ with arbitrary values $x(0)$ and $x(1)$ independent of each other. Formally it is written as

[TABLE]

The measure (6) is quasi-invariant under the following action of the group of diffeomorphisms $Diff^{3}_{+}([0,\,1])$ on $C_{+}([0,\,1])$ (Shepp) , (Shavgulidze2000) :

[TABLE]

[TABLE]

There is the invariant under the action (7) of the group $Diff^{3}_{+}([0,\,1])$. It is given by the integral

[TABLE]

Define $\varphi\in Diff^{1}_{+}([0,\,1])$ by the equation

[TABLE]

Then $x(t)$ is expressed in terms of $\rho$ and $\varphi(t)$

[TABLE]

In this case, we have

[TABLE]

Therefore, there is a one-to-one correspondence $(\rho\,,\ \varphi)\leftrightarrow x\,,$ and the space $C_{+}([0,\,1])$ is stratified into the orbits with different values of the invariant $\rho\,.$

For smooth $x(t)$ $(x\in C^{1}_{+}([0,\,1]))$ and three time differentiable $\varphi(t)$ $(\varphi\in Diff^{3}_{+}([0,\,1]))\,,$ we also obtain

[TABLE]

Thus, for the Wiener measure on the space $C_{+}([0,\,1])$ the following polar decomposition is valid:

[TABLE]

The normalizing factor $\mathcal{P}(\rho)$ (to be evaluated below) determines the relative weight of the input to the measure from the path $x(t)$ with the definite value $\frac{1}{\rho^{2}}$ of the invariant (8).

III Normalizing factor in the polar decomposition of the Wiener measure

To find the explicit form of the density $\mathcal{P}(\rho)$, we multiply (13) by the two $\delta-$functions and consider the following equality of functional integrals:

[TABLE]

[TABLE]

First, we evaluate the functional integral in the right-hand side of (14). It can be rewritten as

[TABLE]

After the substitution (5), the equation (15) takes the form

[TABLE]

To evaluate the integral (16), we differentiate the equation

[TABLE]

[TABLE]

proved in (BShCorrel) . Also we use the equality

[TABLE]

As a result, (14) is represented in the form

[TABLE]

To perform functional integration in the above equation, we take the Fourier transform of the first $\delta-$function in the integrand of (18), represent the Wiener measure as

[TABLE]

and substitute the variable $y(t)=\frac{x(t)}{\sigma}\,.$ As the result, the functional integral in the right-hand side of (18) has the form:

[TABLE]

[TABLE]

Re-scaling also the parameters $\tilde{\lambda}=\sigma^{2}\lambda\,,\ \tilde{\rho}^{2}=\sigma^{2}\eta^{2}\,,$ we rewrite (19) as

[TABLE]

The integral over $C_{+}([0,\,1])$ in (20) is nothing less than $Tr\,\exp\{-A_{0}\}\,,$ that is, the functional integral for the partition function in the Calogero model given by the Euclidean action

[TABLE]

although with an imaginary coupling constant $g=i\lambda\,.$

Note that the appearance of the action of conformal quantum mechanics is tightly connected to the reduction of the Wiener measure to the orbit (8).

Since the action (21) has the continuous spectrum, we first consider the regularized action

[TABLE]

calculate the Fourier transform

[TABLE]

and then tend the parameter $\omega$ to zero.

The solution of the quantum problem for the action of the quantum oscillator with the square-inverse potential (22) is well known. The wave eigenfunctions $\psi_{n}(x)\ (\psi_{n}(0)=0)$ form a basis in the Hilbert space of square integrable on the semiaxis $0<x<+\infty$ functions (see, e.g., (Perelomov) ) with the energy levels

[TABLE]

Therefore the partition function for the action (22) equals to

[TABLE]

[TABLE]

The integral over $\lambda$ in (20) has the form

[TABLE]

To find the integral (25), we first substitute $\zeta=2i\lambda+\frac{1}{4}\,,$ divide the integral

[TABLE]

and make the substitutions $\zeta=i\sqrt{z}\,,\ \zeta=-i\sqrt{z}$ in the first and in the second integrals correspondingly.

After taking the limit $\omega\rightarrow 0$ in the above equations, from (18) we obtain

[TABLE]

Thus we prove the polar decomposition of the Wiener measure

[TABLE]

[TABLE]

The polar decomposition is also valid for the Wiener measure on the space $C_{+}(S^{1})$. However in this case, to evaluate the factor $\mathcal{P}(\rho)$ we are to normalize the measure in some way. For example, we can parametrize the circle of unit length $S^{1}$ by the interval $[0,\,1]\,,$ and consider the measure on $C_{+}(S^{1})$ as the measure on $C_{+}([0,\,1])$ with the ends of the interval ”glued”: $x(0)=x(1)\,.$ In this case, it has the form

[TABLE]

[TABLE]

IV Connection between functional integrals in the Schwarzian theory and in conformal quantum mechanics

From the polar decomposition of the Wiener measure (28 ), we have the equality for the functional integrals

[TABLE]

[TABLE]

[TABLE]

In particular, for

[TABLE]

with $\Phi(x)$ good enough, the integrals are well defined. Therefore we have a transparent connection between conformal quantum mechanics and the Schwarzian theory.

If the functional $\Phi$ is $SL(2,\textbf{R})$ invariant then it can be reduced to the orbits. In this case, the integrals can be factorized with the result

[TABLE]

[TABLE]

[TABLE]

Evaluating functional integral in conformal quantum mechanics, we thus find the corresponding functional integral in the Schwarzian theory, and vice versa, the technique of functional integration over the group of diffeomorphisms makes it possible to evaluate nontrivial functional integrals in conformal quantum mechanics.

Consider a simple example. Let us put $g=0,\ \sigma=1$ and take

[TABLE]

The right-hand side defines the partition function in the Schwarzian theory

[TABLE]

The left-hand side of (30) leads to the result (BShUnusual)

[TABLE]

obtained early (BShExact) by direct functional integration in the Schwarzian theory.

In the same way, one can use functional integrals in conformal quantum mechanics to evaluate other Schwarzian functional integrals.

V An example of application of the polar decomposition in conformal quantum mechanics

In (BShUnusual) , we used the exact solutions of the quantum oscillator model with the Calogero potential to evaluate the functional integral assigning the partition function in the Schwarzian theory.

The polar decomposition of the Wiener measure (27) can be also used in the opposite direction. Namely, we can use already evaluated functional integrals of the Schwarzian theory to perform nontrivial functional integration in conformal quantum mechanics. In this way, we find the fundamental solution of the Schrödinger equation in imaginary time in the quantum mechanical model with Calogero-type potential.

Consider now the heat transfer equation of the form

[TABLE]

It is the Schrödinger equation in imaginary time with the potential $V(q)=\frac{g}{q^{2}}\,.$

To exclude the paths with negative values of $x(\tau)$ in the subsequent functional integrals, we put the boundary condition $\psi_{g}(q=0,\,\tilde{\tau})=0\,,$ and integrate over the space $C_{+}([0,\,t])\,.$

Let the initial condition be $\psi_{g}(q,\,0)=\psi_{0}(q)=\delta(q(0)-q_{0})\,,\ \ \ \ \ q_{0}>0\,.$

The fundamental solution of the Cauchy problem is given by the functional integral

[TABLE]

After the substitution

[TABLE]

it turns to

[TABLE]

Now we use the polar decomposition of the Wiener measure (27) and get

[TABLE]

[TABLE]

In terms of the special functional integral

[TABLE]

it is written as

[TABLE]

[TABLE]

[TABLE]

In (BShCorrel) , we performed the functional integration in (35) and represented the function $\mathcal{E}_{\sigma}\left(u,\,v\right)$ in the form of the ordinary integral:

[TABLE]

[TABLE]

With the explicit form of $\mathcal{E}_{\sigma}(u,\,v)$ given by (37), the equation (36) is transformed into

[TABLE]

[TABLE]

Substituting $z=\theta-i\pi$ or $z=\theta+i\pi$, we rewrite the integral over $\theta$ as

[TABLE]

[TABLE]

where

[TABLE]

Thus we have

[TABLE]

[TABLE]

Now we change the order of the integrals and integrate over $\rho\,.$ Then after the substitution $\zeta=\cosh(R+i\tau)\,,$ we take the limit $R\rightarrow+\infty\,.$ As the result, (40) is transformed into

[TABLE]

where

[TABLE]

$\Gamma_{2}$ and $\Gamma_{3}$ are the lower and the upper edges of the cut along the real axis in the complex plane of $\zeta$ from $-\infty$ to $1\,.$

Note that for $-1<x<1$

[TABLE]

and for $-\infty<x<-1$

[TABLE]

After corresponding reorganization of the integrals in (41) and obvious substitutions, we obtain the fundamental solution of the Cauchy problem in the final form

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

If $g=0$, the integrals can be evaluated explicitly. In this case,

[TABLE]

Representing

[TABLE]

and making substitutions

[TABLE]

in the integrals (41), we get the expected result

[TABLE]

VI Concluding remarks

In this paper, we demonstrate that, due to the existence of the $Diff_{+}^{3}$-invariant (8), there is the one-to-one correspondence

[TABLE]

and the Wiener measure can be written as (27). The factor (26) determines the relative weight of the paths $x(t)$ with a definite value of the invariant (8).

Having in mind the analogy with the form of the Riemann-Lebesgue measure on two-dimensional plane in polar co-ordinates, we give the name ”polar decomposition of the Wiener measure” to the representation (27). Elements of the group $\varphi\in Diff^{1}_{+}$ play the role of angles, and values of the invariant $\rho$ correspond to lengths of radius vectors.

The equation (30) connecting Schwarzian functional integrals with those in conformal quantum mechanics gives us the possibility to choose the most successful strategy of functional integration.

After the substitution (5) we can rewrite the polar decomposition of the Wiener measure (27 ) as

[TABLE]

In (BShRules) , we have already noticed the apparent violation of Markov property by the function $\varphi(t)\,.$ Although $x(t)$ and $\xi(\tau)$ are both Wiener processes , the Markov behaviour of $x(t)$ with respect to the time $t$ of ”its own world” obviously does not imply its Markov behaviour with respect to the time $\tau$ of the ”shadow world”, and vice versa.