Introduction to Supersymmetric Quantum Mechanics and Lattice Regularization

Transcription

1 Introduction to Supersymmetric Quantum Mechanics and Lattice Regularization Christian Wozar Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz, Jena, Germany (Dated: In a very elementary way basic concepts of supersymmetry and lattice regularization are introduced. Supersymmetric quantum mechanics is exemplified starting from the supersymmetric harmonic oscillator in the operator formalism. A generalization thereof is then transferred to the path integral formalism where supersymmetry becomes manifest as a symmetry of the Lagrangian with a spinorial symmetry parameter. Based on the path integral the Euclidean action is discretized on a lattice. Firstly the discretization of scalar fields is discussed and applied to determine the mass gap of the harmonic oscillator. Secondly fermionic degrees of freedom are discretized where a remnant of the fermion doubling problem becomes visible. Here supersymmetry and discretization are considered as two seperate concepts. Nevertheless, results of a naive discretization are presented which show that a lattice treatment of supersymmetric theories is a non-trivial task. I. SUPERSYMMETRY Supersymmetry (SUSY is an ingredient of many modern ideas to describe physics beyond the standard model. In this way it is included in the formulation of string theories and gives highly symmetric theories. It is often quoted that it is so beautiful it must be true. Supersymmetry relates fermions (matter with bosons (carrier of forces either in flat space (SUSY or in curved space-time (supergravity. Bosonic and fermionic states are connected via a supersymmetry generator Q Boson = Fermion, Q Fermion = Boson. ( This implies that particles exist in multiplets where the SUSY generator connects the members of the multiplet. For unbroken SUSY all states in a multiplet must therefore have the same mass. One caveat arises here: In all known experiments there is no sign of these SUSY multiplets and degenerate masses for bosonic and fermionic states. So, if SUSY is part of the description of the laws of nature then SUSY exists in a broken form (for the experimentally accessible energy range. Nevertheless, with experiments at the LCH we hope to find signatures of SUSY parners of our known particles. A. Supersymmetric harmonic oscillator As an introductory example we supersymmetrize the harmonic oscillator. With m = = we write the Hamiltonian of the ordinary harmonic oscillator (with substracted zero point energy as With the annihilation and creation operators we get the commutation relations and the Hamiltonian H B = p + ω x ω ( a = ω (p iωx, a = ω (p+iωx (3 [a,a] = [a,a ] = 0, [a,a ] = (4 H B = ωa a = ωn B. (5 The ground state 0 is annihilated by a and we get the excited states by the application of a to the ground state, n = n! (a n 0. (6

2 In this way the number operator is given by N B n = a a n = n n. (7 We are able to supersymmetrize this Hamiltonian to arrive at a super-hamiltonian H = H B +ωb b = ω(a a+b b. (8 The new operators b and b are the fermionic annihilation and creation operators and fulfill anticommutation relations {b,b} = {b,b } = 0, {b,b } =. (9 Both operators a a and b b are positive semidefinite and the ground state is given by the state which is annihilated by both a and b, a 0 = b 0 = 0 H 0 = 0. (0 The Fock space of this quantum mechanical system ist generated by applying the creation operators a and b to the ground state. These also increase the number operators N B = a a and N F = b b by one due to [N B,a ] = a, [N F,b ] = b. ( The anticommutation relations and the Pauli principle allow for N F only the eigenvalues 0 and. So we call states with fermion number 0 bosonic and states with fermion number fermionic. A further feature is the pairing of excited states. One bosonic state (a n 0 and the fermionic partner b (a n 0 have the same energy. We are in analogy with supersymmetric field theories able to introduce a nilpotent supercharge Q and its adjoint, Q = ( 0 0 A 0 (, Q 0 A = 0 0, A = d dx +ωx, A = d +ωx. ( dx We therefore obtain {Q,Q } = ( ( A A 0 HB 0 0 AA = H. (3 0 H F The SUSY algebra is completed by the nilpotency of Q and Q and the commutation with H, {Q,Q} = 0, {Q,Q } = 0, [Q,H] = 0. (4 B. Pairing and ground states To go beyond the harmonic oscillator we introduce the nilpotent supercharges Q and Q with operators A = d dx +W(x, A = d +W(x (5 dx If the superpotential W is a linear function then A and A are the bosonic annihilation and creation operators of the supersymmetric harmonic oscillator. In this way the super-hamiltonian is defined by H B = A A = ( d dx +W (x W (x, H F = AA = (6 ( d dx +W (x+w (x. Alternatively the Hamiltonian may be written as H = H B +W (xb b. (7 Both Hamiltonians are non-negative. This implies that all energies are zero or positive. A bosonic zero-energy state is annihilated by A and a fermionic one is annihilated by A, H B 0 = 0 A 0 = 0, H F 0 = 0 A 0 = 0. (8

3 3 Now we look at the excited states. Take a bosonic eigenstate ψ B with energy E > 0, For the state A ψ B follows H B ψ B = A A ψ B = E ψ B. (9 H F (A ψ B = (AA A ψ B = A(A A ψ B = AH B ψ B = E(A ψ B, (0 which gives the same energy for the fermionic partner state Both states have identical norm, ψ F = E A ψ B. ( ψ F ψ F = E ψ B A A ψ B = ψ B ψ B. ( Altogether this gives the complete degeneracy of the excited spectrum. Accordingly the bosonic partner state of a fermionic excited state is ψ B = E A ψ F. (3 We can calculate the ground state(s of the super-hamiltonian explicitly (in position space. So we study the first order differential equations ( d Aψ B (x = dx +W(x ψ B (x = 0, ( A ψ F (x = d (4 dx +W(x ψ F (x = 0. The solutions are given by ψ B (x exp ( x ( x W(x dx, ψ F (x exp W(x dx. (5 If one of these functions is normalizable, then the supersymmetric ground state exists and SUSY is unbroken. Since the product ψ B ψ F is a constant, there is at most one normalizable state with zero energy. This implies that for a polynomial W(x = N n=0 c nx n with c N 0, N > 0 SUSY is unbroken iff N is odd. Then we obtain one normalizable zero energy state. E B E F Q Q N F = 0 N F = FIG. : Energy spectrum for supersymmetric quantum mechanics. Q and Q map between bosonic and fermionic sector.

4 4 C. Supersymmetry breaking and the Witten index SUSY requires the degeneracy of (excited bosonic and fermionic states. If SUSY were unbroken the superpartners would have the same mass. When SUSY is broken, then the masses need not be degenerate, even if the partner particles still exist. On the experimentally accessible energy scale there is no such degeneracy apparent and SUSY, if existent, is broken. Nevertheless, there would still exist a supercharge and a super-hamiltonian obeying the SUSY algebra. Only the vacuum state is not invariant under the SUSY. For an existing and unbroken SUSY we require to have a ground state 0 with H B 0 = H F 0 = 0. So we get SUSY unbroken normalizable 0 with Q 0 = Q 0 = 0. (6 Witten defined an index to determine whether SUSY can be broken in supersymmetric field theories by There can be two cases: = Tr( NF. (7 For broken SUSY there is no normalizable zero energy state. Then all eigenstates of H have positive energies and must be paired. So all contributions to cancel and we obtain = 0. For unbroken SUSY there are n B bosonic and n F fermionic ground states with zero energy. They contribute with n B n F to the Witten index. All contributions from the excited states cancel and we obtain = n B n F. So by considering the Witten index we can decide if SUSY can be broken, 0 SUSY is unbroken. (8 However, it can be that SUSY is unbroken but there are the same number of bosonic and fermionic zero energy states. For one-dimensional SUSY QM this result can be further specified and we find 0 SUSY is unbroken in SUSY QM. (9 D. Path integral and symmetries of the Lagrangian The supersymmetric quantum mechanics can be formulated with the Hamiltonian H = p + W (x W (x+w (xb b = p + W (x+ W (x[b,b]. (30 In the standard way this system is transferred to a path integral representation, Z = DφDψD ψ exp( S[φ,ψ, ψ], (3 with Euclidean action S = dτ [ ( φ + ] W (φ+ ψ( +W (φψ. (3 Expectation values are computed via A = Z DφDψD ψa[φ,ψ, ψ]exp( S[φ,ψ, ψ]. (33 On the level of the action we can directly compute the invariance under SUSY transformations. We start with the transformation δ ( φ = εψ, δ ( ψ = ε( φ+w(φ, δ ( ψ = 0. (34

5 5 It follows δ ( S = = = dτ φ ε ψ +WW εψ ε( φ+w( +W ψ + ψ(w εψψ }{{} dτ φ ε ψ +WW εψ φ ε ψ WW εψ ε(w φψ +W ψ }{{} =0 dτ [ ( εwψ] = 0. =0 (35 In the same way the action allows for a second SUSY transformation δ ( φ = ψε, δ ( ψ = 0, δ ( ψ = ( φ Wε. (36 For the above supersymmetries to hold it is necessary that the fields vanish at infinity. In the case of a thermal path integral at inverse temperature β with the above action the fields obey boundary conditions given by Since the fields need not vanish anymore we get φ(0 = φ(β, ψ(0 = ψ(β, ψ(0 = ψ(β. (37 δ ( S = [ εwψ] β τ=0 = [ εwψ] τ=0 (38 which can be nonvanishing. So in this case SUSY is broken by the finite temperature. For the investigation of a supersymmetric system (with unbroken SUSY we prefer to choose periodic boundary conditions also for the fermions, ψ(0 = ψ(β, ψ(0 = ψ(β, (39 which amounts to have Z periodic = DφDψD ψ( NF exp( S[φ,ψ, ψ] = Z antiperiodic. (40 II. LATTICE REGULARIZATION At this point there are different approaches to gain information from the quantum system. For the quantum mechanical case Hamiltonian methods have been proven to be useful. A diagonalization with a discretized derivative can give the complete spectrum (up to a bound determined through details of the discretization. For quantum field theories this diagonalization procedures are not efficient anymore and Monte-Carlo methods based on the Euclidean path integral representation have provided great successes. Modern lattice field theories often deal with gauge theories, and in the last time there have been big achievments for QCD with dynamical fermions (e.g. the meson spectrum. In this introduction we do not cover the topic of gauge fields. We will consider only the basics to regularize the path integral by a lattice description for the case of the above described quantum mechanical system. The path integral as it stands, Z = DφDψD ψ exp( S[φ,ψ, ψ], (4 can show divergencies (at least in higher dimensions and has ambiguities with the definition of the integral measure (e.g. continuous functions are a null set in the set of all functions. Due to these problems it is necessary to regularize the path integral and remove the regularization in a well defined way. A. Scalar fields on the lattice To regularize the scalar part of the quantum mechanical system we start with the (thermal partition function β Z = Dφ exp( S[φ] with S = dτ [ ( φ +W (φ ]. (4 0

6 A regularization is then given by putting the fields (formally defined on the whole interval [0,β on a discrete lattice t n = n N β,n {0,...,N } with N lattice points. So the lattice spacing is given by a = β N. The regularized partition function naturally arises as Z N = N n=0 N dφ n exp( S N [φ] with S N [φ] = a n=0 [ ( φ n +W (φ n ], (43 where φ n = φ(t n and ( φ n is the derivative of φ at position t n. To perform the continuum limit we take Z = lim N Z N. In the same way also expectation values are defined via insertions into the regularized path integral. At this point we see an ambiguity of the lattice regularization. We must define ( φ n in a proper way. The canonical way for scalar fields is to use the nearest neighbour (forward or backward differences ( f φ n = a (φ n+ φ n or ( b φ n = a (φ n φ n (44 In the continuum limit these definitions lead to the derivative for continuous fields. Nevertheless, there are other definitions of derivatives on the lattice which may be of some use in specific situations. 6 B. Energies/masses for scalar fields One aim of our calculations is to gain information about the spectrum of the quantum system. The most fundamental part of the spectrum is the mass gap m which can be extracted from the correlator of fields (assume φ = 0, This mass gap m is related to the pole of the propagator C(t = φ(0φ(t. (45 through m = ip pole and shows up in the long range behaviour of the correlator as G(p = (FC(p (46 C(t t exp( m t. (47 In an example we want to compute the mass gap on the lattice for the harmonic oscillator with action S = a [ ] ( b φ n + ω φ n = S mn φm φn, n m,n S mn = ω a δ mn +(δ mn δ m,n δ m,n+, (48 φ n = φ n a. We define the partition function with external source as ( Z(J = D φ exp S + n J n φn, D φ = n d φ n (49 We use a transformation of variables φ n φ n + m G nm J m (50 with G as the inverse of S, S mn G nk = δ mk. (5 This brings Z(J to the form ( Z(J = Z(0exp G mnj m J n, (5

7 7 where Z(0 is just a Gaussian integral Z(0 = D φ exp ( S mn φ m φn =. (53 dets This implies that the correlator is given by φm φn = J m J n lnz(j = G mn. (54 The propagator can be constructed in momentum space. Therefore we determine the Fourier transform of S, S pq = m,ne ipam+iqan S mn The inverse of S in momentum space is given by = S p δ pq, S p = ω a + ( cos(pa = ω a +4sin ( pa. (55 So in the continuum limit the correct result is reproduced with G pq = G p δ pq, G p = ω a +4sin ( pa. (56 G(p a 0 = We can also compute the pole of the propagator and get and the mass gap is therefore given by In the continuum limit this gives ω +p +O(a. (57 p pole = ia arsinh m = a arsinh ( ωa, (58 ( ωa. (59 ( ωa m = a (ωa3 +O(a 5 a 0 = ω. (60 48 C. Fermions on the lattice The fermionic part of the partition function is (for fixed φ given by Z = DψD ψexp( S[φ,ψ, ψ] with S = dτ ψ( +W ψ. (6 Similarly as for the bosonic system we restrict the fermions to the lattice ψ(τ ψ n and let the derivative act as a matrix mn. Again an ambiguity in defining the derivative on the lattice arises. In the standard case this derivative is unrelated to the one chosen in the bosonic sector. The naive choice is an antisymmetric matrix. Is has the advantage that it has also an imaginary spectrum like the free derivative for continuous fields. Canonically we choose the symmetric derivative ( S ψ n = ψ n+ ψ n. (6 a

8 8 To analyze this derivative we investigate the free theory with W = M and the fermionic action S = a m,n ψ m ( S mn +Mδ mn ψ n. (63 Similar to the bosonic case we can calculate the propagator and get G(p = M +ia sin(pa. (64 This correlator has poles at p ( pole = ia arsinh(ma and p ( pole = π ia arsinh(ma. In this case there are two particles in the spectrum where only one is physical. One way to circumvent this problem is to remove the unphysical pole from the spectrum. Wilson has introduced a momentum dependent mass into the Dirac operator by setting ψ m Mδ mn ψ n m,n m,n ψ m Mδ mn ψ n + ar with a parameter r. For r = this construction gives in one dimension the action S = a m,n ( f ψn ( f ψ n (65 n ψ m ( b mn +Mδ mn ψ n (66 with the propagator In this case the pole is at G(p = M +a ( e ipa. (67 p pole = ia ln(+ma, (68 with a mass gap ( m = a Ma (Ma + (Ma3 +O(a 4 a 0 = M. (69 3 D. Monte-Carlo simulations To gain information in terms of expectation values and masses even in the non-perturbative regime of the system considered Monte-Carlo methods have proven to be very useful. With powerful algorithmical improvements during the last years even problems including dynamical fermions can be simulated. For this statistical method the path integral is interpreted as a probability distribution. We start with Z = DφDψD ψ exp( S[φ,ψ, ψ]. (70 The action is then splitted into a bosonic and fermionic part according to S[φ,ψ, ψ] = S B [φ]+ ψ m Q mn [φ]ψ n. (7 m,n }{{} =S F Due to the rules of Grassmann integration the fermionic part of the path integral can (for fixed φ be integrated out and gives Z = Dφ det(qexp( S B [φ] = Dφ exp( S B [φ]+lndetq[φ] }{{} S eff [φ] (7

9 In this way (bosonic expectation values are computed according to A[φ] = Z DφA[φ]exp( S eff [φ]. (73 9 In a Monte-Carlo simulation fields φ are sampled according to the distribution ρ[φ] = exp( S B [φ]+lndetq[φ]. (74 After a number of Q samples we get a time series φ (q and expectation values are evaluated using A Q Q = Q A[φ (q ]. (75 For the model of supersymmetric quantum mechanics we have set W(φ = mφ + gφ 3 (a super potential without SUSY breaking. For g = 0 we end up at the free theory (harmonic oscillator. The discretization has been done with the backward derivative b (which is nothing but the Wilson derivative for the fermions in one dimension with r =. As a cross check we also computed the mass (the energy difference between ground state and first excited state with a diagonalization of the Hamiltonian. q= 9 8 m exact m B m F 7 m a FIG. : Results for the naive discretization of the SUSY quantum mechanics with m = 0 and g = 00. We find that this naive discretization does not reproduce the correct (supersymmetric continuum limit. So for supersymmetric theories there must be further constraints on the used lattice regularization.