aa r X i v : . [ h e p - l a t ] N ov Quantum Mechanics à la Langevin andSupersymmetry
Stam Nicolis ∗ CNRS–Laboratoire de Mathématiques et Physique Théorique (UMR7350)Fédération de Recherche “Denis Poisson” (FR2964)Département de Physique, Université“François Rabelais” de ToursParc Grandmont, 37200 Tours, FranceE-mail:
We study quantum mechanics in the stochastic formulation, using the functional integral ap-proach. The noise term enters the classical action as a local contribution of anticommuting fields.The partition function is not invariant under N = k − symmetry. Thekinetic term for the fermions is a total derivative and can contribute only on the boundaries. Wedefine combinations that scale appropriately, as the lattice spacing is taken to zero and the latticesize to infinity and provide evidence, by numerical simulations, that the correlation functions ofthe auxiliary field do satisfy Wick’s theorem. We show, in particular, that simulations can becarried out using a purely bosonic action.The physical import is that the classical trajectory, f ( t ) , becomes a (chiral) superfield, ( f ( t ) , y a ( t ) , F ( t )) , when quantum fluctuations are taken into account. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ uantum Mechanics & SUSY
Stam Nicolis
1. Introduction
Supersymmetry is a symmetry that unifies internal and space-time symmetries. Within thecontext of quantum field theory it implies the existence of particles, “partners” to the known par-ticles. The absence of such partners, e.g. a spin-0 particle with the same charge and mass as theelectron, implies that supersymmetry must be broken at the energy scales that have been exploredso far. How it is broken, within the framework of the Standard Model, is unknown. It is, therefore,useful to explore this issue within a framework that is not sensitive to perturbation theory. Latticefield theory is such a framework.However there are challenges, technical and conceptual, in carrying out this program [1]. Onthe other hand, supersymmetry, as a symmetry of space-time, should have consequences beyondparticle physics. To date the driving force behind supersymmetry is that it provides the simplestway of stabilizing the hierarchy between the weak scale and the Planck scale and controlling thecorrections to the mass of the Higgs boson. This still leaves a large parameter space to explore. Anobstacle in our understanding of the role of supersymmetry is that we do not know how to explore“superspace”, where supersymmetry acts “naturally”.In an effort to acquire some intuition about how supersymmetry may be realized and broken,it might be helpful to study it in “simpler” situations. For the intuition so acquired to be useful, it isnecessary that the simplicity not eliminate the signal. In the Standard Model, the supersymmetry,that is relevant, is target space supersymmetry: the particles are target space fermions or bosons,regarding statistics and are described by spinors, vectors–and scalars–regarding spin; so are theirputative superpartners. In less than four space-time dimensions, where the rotation group is abelian,we may talk of commuting or anticommuting fields; but the statistics is not, necessarily, that offermions or bosons. And it might not be possible to define spinors.A conceptual stumbling block, in these situations, indeed, is, how we can define supersymme-try. The definition we shall use is the following: that the action is invariant under transformations,whose parameters are anticommuting variables and the anticommutator of two such transforma-tions closes on the generator of the translations.The situation that we will study is quantum mechanics, as a quantum field theory in one space-time dimension. There have been extensive studies of “supersymmetric quantum mechanics” [2],however these used a canonical formalism–i.e. worked in phase space, rather than configurationspace and the aim was rather to understand how supersymmetric field theoretic notions appearedin the simpler setting. Lattice investigations [3] have, also, focused in this direction.So it is useful to try to set up the framework for the opposite direction. In this way, in fact, wecan study superspace.The starting point of our investigation is the stochastic formulation [4]. In this formulation wewrite the Langevin equation ¶f¶ t = − ¶ U ¶f + h ( t ) (1.1)where t is the equilibration time–we are interested in the limit t → ¥ . The field, h ( t ) , is a Gaussian2 uantum Mechanics & SUSY Stam Nicolis stochastic process: h h ( t i = h h ( t ) h ( t ′ ) i = d ( t − t ′ ) h h ( t ) h ( t ) · · · h ( t n ) i = (cid:229) p (cid:10) h ( t p ( ) ) h ( t p ( ) ) (cid:11) · · · (cid:10) h ( t p ( n − ) ) h ( t p ( n ) ) (cid:11) (1.2)where the sum is over all perumutations.At equilibrium, t → ¥ , h = ¶ U ( f ) ¶f (1.3)If the fields, h and f , are random variables, we have a problem in probability theory, whose solutioncan provide interesting insights for physics [5]. If they are functions of other variables, then wehave a problem in quantum field theory.If U ( f ) is a local functional of f , in particular, if ¶ U ( f ) ¶f = d f ( t ) d t + ¶ W ( f ) ¶f ( t ) (1.4)where t ∈ R and W ( f ( t )) is an ultralocal functional of f ( t ) , we obtain the following stochasticequation for f ( t ) : d f ( t ) d t = − ¶ W ¶f ( t ) + h ( t ) (1.5)This is the Langevin equation that describes quantum mechanics, i.e. a quantum field theory in oneEuclidian dimension. The essential difference to eq. (1.1) is that we are not interested, only, in thelimit t → ¥ , but in the solution for all values of t . This assertion is meaningful only at the level ofthe correlation functions, of course.In this case, h ( t ) is a Gaussian stochastic process, whose correlation functions obey the sameidentities as in eq. (1.2), only the time is, now, the Euclidian time.We are interested in the correlation functions, h f ( t ) f ( t ) · · · f ( t n ) i , of the field f and theidentities that constrain them. We shall compute them from the following partition function Z = Z [ d h ( t ) d f ( t )] e − R d t h ( t ) d (cid:18) h ( t ) − d f d t − ¶ W ¶f ( t ) (cid:19) = Z [ d f ( t )] e − R d t (cid:16) d f d t + ¶ W ¶f ( t ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) d ( t − t ′ ) (cid:18) dd t + ¶ W ¶f ( t ) (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = Z [ d f ( t ) d y ( t )] e − R d t (cid:26) (cid:16) d f d t + ¶ W ¶f ( t ) (cid:17) − R d t d t ′ y a ( t ) e ab (cid:16) d ( t − t ′ ) (cid:16) dd t + ¶ W ¶f ( t ) (cid:17)(cid:17) y b ( t ′ ) (cid:27) (1.6)We have introduced the determinant of the local operator, ¶ U / ¶f ( t ) ¶f ( t ′ ) , in the action usinganticommuting fields, y a ( t ) . These are not ghosts! There isn’t any spin in one dimension, so thespin–statistics theorem is vacuous [6].This is a formal expression, since the measure, [ d f ( t ) d y ( t )] , is not well-defined and thedeterminant of the operator is not well-defined either. Indeed, the passage from the first line to thesecond only holds, if the determinant can’t vanish–if it can, then, already, the first line is ill-defined,since the delta function doesn’t fully constrain the function h ( t ) . So what we really are after is away to define the expression in the third line–and use that definition to define the expression in thefirst line. We shall use the lattice. But, first, let us look at its symmetries.3 uantum Mechanics & SUSY Stam Nicolis
2. Elusive supersymmetry
The action in eq. (1.6) is not invariant under transformations, that mix the commuting andthe anticommuting variables. If we introduce, by hand, a second pair, c a ( t ) , of anticommutingvariables, however, we may check that the action S = Z d t (cid:20) − F + F (cid:18) d f d t + W ′ ( f ) (cid:19) − y a e ab (cid:18) dd t + W ′′ ( f ) (cid:19) c b (cid:21) (2.1)is invariant, up to a total derivative, under the two transformations [7, 8]: d f = − z a e ab c b d y a = − z a (cid:0) ˙ f + F (cid:1) d F = d ˙ fd c a = d f = z ′ a e ab y b d c a = z ′ a (cid:0) − ˙ f + F (cid:1) d F = d ˙ fd y a = d A = , A = , { d , d } = − z a e ab z ′ b ¶¶t (2.3)They deserve, therefore, to be called supersymmetric. The target space of this theory is the realline. This supersymmetry is, of course, not “target space”, but “worldline”.However, we notice that this action doesn’t seem to stem from a Langevin equation. Also, thatwe’ve doubled the number of fermions and introduced a mismatch between fermions and bosons.In fact, one of the fermions is a gauge artifact, since only one appears through a derivative term andwe may choose which one at any moment, there is a redundancy, i.e. a gauge symmetry. The gaugesymmetry is “ k − symmetry” [10]. Furthermore, the fermionic kinetic term, in both actions, is atotal derivative, thereby, providing only a surface term. The fermions, thus, contribute ultra–localterms to the action, Y a , I ( t ) K IJ e ab W ′′ ( f ) Y b , J ( t ) , where K IJ is the “bulk flavor mixing matrix”,that projects on the I = , I = I = , I = I = I = , leaving a “Polyakov loop”, the (path ordered) product ofexp ( log W ′′ ( f )) , as an ultra-local contribution to the action, once we have gauge-fixed the partitionfunction. The partition function in eq. (1.6) is the result of gauge-fixing the action in eq. (2.1) [9].We end up with a local, bosonic action: S [ f ] = Z d t " (cid:18) d f d t + W ′ ( f ) (cid:19) − log W ′′ ( f ) = Z d t (cid:20) − f d d t f + (cid:2) W ′ ( f ) (cid:3) − log W ′′ ( f ) (cid:21) (2.4)The observable we shall study is the auxiliary field, F = d f d t + W ′ ( f ) (2.5)Supersymmetry implies that the auxiliary field has Gaussian correlation functions [4, 7, 8]; inparticular, vanishing 1–point function and an ultra–local propagator. These are the properties that Provided W ′′ ( f ) is of fixed sign. In ref. [9] we discuss how to treat the case when W ′′ ( f ) can vanish in this case. uantum Mechanics & SUSY Stam Nicolis we will try to check. We shall present results for the quartic superpotential, i.e. the anharmonicoscillator with “sextic” and “quartic” non-linearity: W ( f ) = m f + l f ⇒ W ′ ( f ) = m f + l f ⇒ W ′′ ( f ) = m + l f (2.6)with m > l >
0. Indeed, for one degree of freedom, if the potential is bounded from below,we can always write it as a perfect square.
3. The auxiliary field on the lattice
We discretize the action in the standard way and impose periodic boundary conditions on thescalar. The lattice action takes the following form S latt [ F ] = gm N − (cid:229) n = ( − F n + F n + F n + m (cid:18) F n + F n (cid:19) − g sm log (cid:20) gs (cid:18) + F n (cid:19)(cid:21)) (3.1)where we have introduced the lattice parameters m = m a and l latt = l a (3.2)and the scaling parameters l latt m = l m ≡ g , a m l latt = m l ≡ s and F n ≡ f n (cid:18) a m l latt (cid:19) / (3.3)The scaling limit, therefore, consists in taking m → , l latt →
0, keeping g and s fixed. If g < g ≥
1, it is strongly coupled. We remark that g appears to beindependent of the lattice spacing.The (rescaled) auxiliary field is given by the expression F n = F n + − F n − + m (cid:18) F n + F n (cid:19) (3.4)We see immediately that its 1–point function will vanish, h F n i =
0, so we can use this as a checkof the simulations. For the, preliminary, results, presented below, we find h F i = . × − ± . × − , which is consistent with zero. We haven’t studied the autocorrelation time here.The interesting quantities are the 2–point function and the ultra–local part of the connected4–point function (the Binder cumulant ). The former should be a d − function in the scaling limitand the latter should vanish to numerical precision. We display typical results of a Monte Carlosimulation in figs. 1 and 2
4. Conclusions
We have studied quantum mechanics of a point particle in the stochastic formulation. Usingthe path integral formalism the quantum fluctuations can be expressed locally through anticommut-ing variables. The physical import of these calculations is the following: the classical trajectory,5 uantum Mechanics & SUSY
Stam Nicolis -4e-05-2e-05 0 2e-05 4e-05 6e-05 8e-05 0.0001 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06’vevf2_d=0.out’’vevf2_d=5.out’’vevf2_d=9.out’’vevf2_d=17.out’’vevf2_d=31.out’
Figure 1:
Monte Carlo time series for the 2–point function for the auxiliary field, h F | n − n ′ | F i , for | n − n ′ | = , , , ,
31 for N =
64. The red curve corresponds to | n − n ′ | =
0, the others to | n − n ′ | 6 =
0. Due to periodicboundary conditions, the relative distance cannot exceed N / -5e-09 0 5e-09 1e-08 1.5e-08 2e-08 0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06’BiC.out’ Figure 2:
Monte Carlo time series for the Binder cumulant for the auxiliary field, h F i − h F i , for N = f ( t ) , becomes a superfield ( f ( t ) , y a ( t ) , F ( t )) . The anticommuting variables enter the effectiveaction in an ultra–local way, since their kinetic term is a total derivative, so the fermions are “con-fined” in bilinears.The effective action, in the stochastic formulation, is the result of gauge fixingthe k − symmetry of a manifestly supersymmetric action. It is local and “bosonic” (depends onlyon commuting variables) and its correlation functions may be computed by standard numericaltechniques. Supersymmetry is broken by the lattice, since the regularized auxiliary field does nothave an ultra–local propagator, but is recovered in the scaling limit.These results carry over to matrix quantum mechanics [11], as well, since the Dirac algebrais, still, trivial. With a higher dimensional target space, however, it becomes possible to define atarget space Dirac algebra, from the fermionic zeromodes [12, 13], and, thus, spinors and target6 uantum Mechanics & SUSY Stam Nicolis space supersymmetry. This is, also, how “emergent spin” [14] might be realized. Details will bepresented in future work.
Acknowledgements:
I’d like to acknowledge discussions with M. Axenides, C. Bachas, J.Bloch, E. G. Floratos, H. Giacomini, J. Iliopoulos, C. Kounnas, G. Linardopoulos and T. Wettig.
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