Beyond Complex Langevin Equations: positive representation of a class of complex measures

A positive representation for a set of complex densities is constructed. In particular, complex measures on a direct product of U(1) groups are studied. After identifying general conditions which such representations should satisfy, several explicit realizations are proposed. Their utility is illustrated in few concrete examples representing problems in abelian lattice gauge theories.

Unfortunately no theorems, involving only conditions on the original weight, exist which relate the large τ behaviour to the complex action. Consequently the old troubles [3,4], which had plagued the method, resurfaced again and, in spite of substantially better understanding [5], compared to the pioneering times , the approach still has serious difficulties and limitations [6][7][8]. For recent review see Ref. [9].

Avoiding the trouble -Beyond Complex Langevin
In view of the above problems, the natural question to ask is wether one can construct the positive distribution P(x, y) from the "matching conditions" (1) alone without any reference to the problematic complex stochastic process at all. The question has been addressed before. In 2002 Weingarten has shown that such a distribution always exists [10], Salcedo [11,12] constructed P for the gaussian weights with polynomial modifications. More recently, analytical properties of P in two variables were used to solve the matching conditions, again for a gaussian ( and also a specific quartic ) weight [13]. This time the construction was generalized to gaussian quantum mechanical path integrals providing, for the first time, a positive representation for a particle in an external magnetic field directly in the Minkowski time. Until then this text book quantum problem, did not have a positive representation, even after Wick rotation.
In this talk I would like to report on a general prescription how to construct the corresponding positive distributions for complex, weights on a torus U(1) N [14]. After a short presentation of the main principle, some applications ranging from one degree of freedom to small, low dimensional U(1) lattices, will be discussed. An extension for non-compact measures has been already constructed [15]. See also [16] for another approach. For generalization for nonabelian integrals see Ref. [17] in this volume.

This work: periodic weights
For periodic weights it is natural to rewrite matching conditions (1) in Fourier space (from now on we assume that ρ and P are normalized). Introducing the Fourier components of ρ(x) and partial Fourier transforms of P(x, y) ρ(x) = Σ n a n e inx and P(x, y) = Σ n P n (y)e inx one rewrites (1) as That is a n = ∞ −∞ e ny P n (y)dy (2) It is evident that P(x, y) is not uniquely defined by conditions (2). Actually, only one moment of each partial Fourier component P n (y) is fixed. This freedom is seen already at the level of the general matching equations (1). It would take at least the full set of two-dimensional moments M r,s = x r y s P(x, y)dxdy to define uniquely the two-dimensional distribution P(x, y). Instead, as a heritage of the Complex Langevin way of thinking, we imposed in (1) only moments in one, holomorphic variable 1 . Given the above freedom some Ansatz for y-dependence of the partial Fourier components is necessary. We take the simplest one and leave the shift y s as a free parameter. Then the matching equations (2) imply λ n e n·y s + µ n e −n·y s = a n , λ n e −n·y s + µ n e n·y s = a * −n , with the solutions λ n = e ny s a n − e −ny s a * −n 2 sinh(2ny s ) µ n = e ny s a * −n − e −ny s a n 2 sinh(2ny s ) .
Before we proceed, a few comments are in order.
• P(x, y) is real by the construction.
• However it is not positive in general. Positivity can be achieved by the dominance of the lowest mode. This can be realized by choosing large enough y s .
• Other Ansätze are possible, e.g. Gaussian. The gaussian prescription corresponds merely to smearing the point like distributions of the imaginary part y. Adjusting a width of these smearing can additionally help to satisfy positivity.
• Generalization to many variables is straightforward in principle. One basically replaces: x, n −→ x, n etc.

Examples
All our examples are built around the problem of the space structure of confining strings. This question attracted attention of lattice community almost since the formulation of lattice QCD [18][19][20]. The answer is given by the energy density of the colour field in the presence of external qq sources. The problem boils down to measuring correlations between an elementary plaquette and a large Wilson loop. Including a Wilson loop in the equivalent, positive measure would dramatically improve results obtained so far.

One DOF -a prototype of a Polyakov line
We seek for a positive distribution equivalent to the following complex, periodic weight 1 In this context notice that the starting point of all constructions in [13] is the general function P(z,z) of two variables.
Indeed the P ± (x) are real and positive for large y s . We check explicitly one average < sin 2 (x) >.
Integrating the complex weight gives immediately which is readily reproduced by the positive density integral.

Four DOF with gauge invariance -Wilson loop
Our second example contains four link angles x i with the miniscule gauge invariance: The unnormalized complex density reads For the corresponding positive distribution P P ( x, y) we take now P P ( x, y) = δ( y − y s )P + (x) + δ( y + y s )P − (x), y s = y s (1, 1, −1, −1), which essentally reproduces the previous example up to a simple rescaling of the shift parameter.

Tiny 2D abelian lattice
In the last example we put 2 Polyakov lines on a 2x2 U(1) lattice and construct equivalent positive distribution. Links and plaquettes are labeled as in Fig. 1. The complex density reads (for simplicity we denote link angles by their indices θ i → i).
There are only three independent variables and in this simple example. One can take them to be any three plaquette angles. We choose (φ I , φ II , φ III ) → (φ 1 , φ 2 , φ 3 ). Then Fourier components of ρ are again simple a n = m I m I m−n 2 I m−n 1 +1 I m−n 3 +1 , n = (n 1 , n 2 , n 3 ) and one can readily construct the corresponding positive density ( φ = x + i y).
+ n 0 e i n· x e n· y s a n − e − n· y s a − n 2 sinh (2 n · y s ) δ( y − y s ) + e n· y s a − n − e − n· y s a n 2 sinh (2 n · y s ) δ( y + y s ) To avoid singularities introduced by zeroes of n · y s , we took y s = y s (1, √ 2, √ 3). It is a simple matter to check that indeed P reproduces all moments of the complex weight e iφ 1 r 1 e iφ 2 r 2 e iφ 3 r 3 ρ( φ) = e i(x 1 +iy 1 ) r 1 e i(x 2 +iy 2 ) r 2 e i(x 3 +iy 3 ) r 3 as it should. It is also instructive to examine directly the effect of complex phases, cf. Fig.2.
The influence of the complex phases is dramatic. The effective distribution differs essentially from the original Boltzmann density. This confirms explicitly the common sense expectations, that The second good news is that the variation of P + is substantial.This means that the dominance of the first, i.e. (0, 0, 0) mode, required in the proof of positivity, does not preclude the importance of other modes. Consequently the effective positive distribution reveals an interesting structure.

Generalizations, summary and an outlook
Extension of this example to larger lattices is straightforward, c.f. Fig.3. Therefore, at least for these simple models, the present construction provides positive solutions, even for many variables. This generalizes readily to arbitrary, abelian (2D) lattices with the complex density in the form (ignoring, or choosing suitable boundary conditions).
Of course, some practical questions remain. For example inversion of the multidimensional Fourier transform will be expensive for more variables. Using the Fast Fourier Transforms might alleviate the problem. Also for the separable systems, like in (12), this is not an issue, since the complexification procedure can be carried variable by variable and resulting positive distributions would also factorize.
In the general case, however, even with local nearest neighbour interactions in a complex weight, positive distributions do not have to be local. At the same time an interesting possibility appears. As explained earlier, matching conditions do not determine the positive distribution uniquely. There is a large freedom in constructing P(x, y).
Conceivably it can be used to satisfy additional requirements imposed on P, possibly locality.
Summarizing, one can avoid poorly convergent stochastic processes by constructing a positive distribution P(x, y) directly from the matching conditions (1). At first sight this task looks rather formidable, but after closer scrutiny it turns out to be underdetermined. One way to solve the problem