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

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.

I.

INTRODUCTION
In recent years, the Schwarzian theory has become extremely popular. It is given by the action where is the Schwarzian derivative, and ϕ(t) is an orientation preserving (ϕ ′ (t) > 0) diffeomorphism of the unit circle (ϕ ∈ Dif f 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 [1] - [35].
An extraordinary universality of the Schwarzian action is a consequence of its SL(2, R) invariance. At the same time, the Schwarzian theory is not the only SL(2, R) invariant action.
This symmetry manifests itself in conformal quantum mechanics [36], Liouville quantum mechanics [37]- [39] and some other models that are used to describe the near-horizon geometry of Reissner-Nordström black hole and AdS 2 /CF T 1 duality, gravity near space-like singularity (see, e.g., [40]- [45]), and other physical problems where a universal regime is reached.
Therefore, the attempts [10], [19] - [25] 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 [46]- [48], the measures on the group Dif f 1 + (S 1 ), and on the group Dif f 1 + ([0, 1]) were constructed. The measure (4) can be generated from the Wiener measure under some special substitution of variables [46]- [48]. Namely, let us consider the function is a continuous function on the interval satisfying the boundary condition ξ(0) = 0 ). Then, under the substitution the measure µ σ (dϕ) on the group Dif f 1 The measures (3) and (4) are quasi-invariant under the action of the subgroup consisting of three times differentiable transformations Dif f 3 + (S 1 ) ( Dif f 3 + ([0, 1]) ) . Quasi-invariance means that under the action of the subgroup G = Dif f 3 + (S 1 ) ( G = Dif f 3 + ([0, 1]) ), the measure µ σ (dϕ) transforms to itself multiplied by a function R g (h) parametrized by the elements of the subgroup g ∈ G : µ ( d(g • h) ) = R µ g (h) µ( dh ) .) Using the quasi-invariance of the measure µ σ (dϕ), 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 [49]- [51].
In this paper (sections II, III), we find that the two measures are connected with each other by the equation which we name "Polar decomposition of the Wiener measure".
The corresponding polar decomposition of the measure on the circle has the form In section IV, for SL(2, R) invariant functionals Φ we obtain the equation 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.
The measure (6) is quasi-invariant under the following action of the group of diffeomor- [52], [48]: x There is the invariant under the action (7) of the group Dif f 3 + ([0, 1]). It is given by the integral Define Then x(t) is expressed in terms of ρ and ϕ(t) In this case, we have Therefore, there is a one-to-one correspondence (ρ , ϕ) ↔ x , and the space C + ([0, 1]) is stratified into the orbits with different values of the invariant ρ .
For smooth Thus, for the Wiener measure on the space C + ([0, 1]) the following polar decomposition is valid: The normalizing factor P(ρ) (to be evaluated below) determines the relative weight of the input to the measure from the path x(t) with the definite value 1 ρ 2 of the invariant (8).

III. NORMALIZING FACTOR IN THE POLAR DECOMPOSITION OF THE WIENER MEASURE
To find the explicit form of the density P(ρ), we multiply (13) by the two δ−functions and consider the following equality of functional integrals: First, we evaluate the functional integral in the right-hand side of (14). It can be rewritten After the substitution (5), the equation (15) takes the form To evaluate the integral (16), we differentiate the equation proved in [50], and use the equality As a result, (14) is represented in the form To perform functional integration in the above equation, we take the Fourier transform of the first δ−function in the integrand of (18), represent the Wiener measure as and substitute the variable y(t) = x(t) σ . As the result, the functional integral in the right-hand side of (18) has the form: Re-scaling also the parametersλ = σ 2 λ ,ρ 2 = σ 2 η 2 , we rewrite (19) as The integral over C + ([0, 1]) in (20) is nothing less than T r exp{−A 0 } , that is, the functional integral for the partition function in the Calogero model given by the Euclidean action although with an imaginary coupling constant g = iλ .
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 calculate the Fourier transform and then tend the parameter ω to zero.

Therefore the partition function for the action (22) equals to
The integral over λ in (20) has the form To find the integral (25), we first substitute ζ = 2iλ + 1 4 , divide the integral , and make the substitutions ζ = i √ z , ζ = −i √ z in the first and in the second integrals correspondingly.
The polar decomposition is also valid for the Wiener measure on the space C + (S 1 ).
However in this case, to evaluate the factor P(ρ) 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 x ∈ C + (S 1 ) , ϕ ∈ Dif f 1 + (S 1 ) , 0 < ρ < +∞ .

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 In particular, for with Φ(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 Φ is SL(2, R) invariant then it can be reduced to the orbits. In this case, the integrals can be factorized with the result 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, σ = 1 and take The right-hand side defines the partition function in the Schwarzian theory The left-hand side of (30) leads to the result [54] Z Schw (κ) = 1 √ 2πκ 3 exp 2π 2 κ 2 obtained early [49] 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. Consider now the heat transfer equation of the form ∂ ∂τ ψ g (q,τ ) = 1 2 It is the Schrödinger equation in imaginary time with the potential V (q) = g q 2 . To exclude the paths with negative values of x(τ ) in the subsequent functional integrals, we put the boundary condition ψ g (q = 0,τ ) = 0 , and integrate over the space C + ([0, t]) .
The fundamental solution of the Cauchy problem is given by the functional integral After the substitutionτ Now we use the polar decomposition of the Wiener measure (27) and get In terms of the special functional integral it is written as In [50], we performed the functional integration in (35) and represented the function E σ (u, v) in the form of the ordinary integral: With the explicit form of E σ (u, v) given by (37), the equation (36) transforms into Substituting z = θ − iπ or z = θ + iπ, we rewrite the integral over θ as where Thus we have Now we change the order of the integrals and integrate over ρ . Then after the substitution ζ = cosh(R + iτ ) , we take the limit R → +∞ . As the result, (40) is transformed into where χ g (ζ) = exp q 0 q t ζ exp − 2g + 1 4 arc cosh ζ , Γ 2 and Γ 3 are the lower and the upper edges of the cut along the real axis in the complex plane of ζ from −∞ to 1 .
After corresponding reorganization of the integrals in (41) and obvious substitutions, we obtain the fundamental solution of the Cauchy problem in the final form ψ g (q, t ; q 0 ) = F g (q, t ; q 0 ) sin π 2g + 1 4 where If g = 0, the integrals can be evaluated explicitly. In this case,

VI. CONCLUDING REMARKS
In this paper, we demonstrate that, due to the existence of the Dif f 3 + -invariant (8), there is the one-to-one correspondence Having in mind the analogy with the form of the Riemann-Lebesgue measure on twodimensional plane in polar co-ordinates, we give the name "polar decomposition of the Wiener measure" to the representation (27). Elements of the group ϕ ∈ Dif f 1 + play the role of angles, and values of the invariant ρ 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) In [51], we have already noticed the apparent violation of Markov property by the function ϕ(t) . Although x(t) and ξ(τ ) 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