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Lattice Quantum Field Theory Lattice quantum field theory Lecture in SS 2020 at the KFU Graz Axel Maas 2 Contents 1 Introduction 1 2 Scalar fields on the lattice 4 2.1 Thepathintegral................................ 4 2.2 Euclidean space-time lattices . 7 2.3 Discreteanalysis ................................ 10 2.4 Thefreescalarfield............................... 11 2.5 φ4 theoryonthelattice............................. 14 2.6 Latticeperturbationtheory . 15 2.6.1 Particleproperties ........................... 15 2.6.2 Interactions ............................... 17 2.6.3 Hoppingexpansion ........................... 20 2.7 The phase diagram and the continuum limit . 22 2.8 Detectingphasetransitions. 26 2.9 Internal and external symmetries on the lattice . .... 28 2.9.1 Externalsymmetries .......................... 28 2.9.1.1 Translation symmetry . 29 2.9.1.2 Reflection positivity . 29 2.9.1.3 Rotationsymmetry and angularmomentum . 31 2.9.2 Internalsymmetries........................... 33 3 Measurements 35 3.1 Expectationvalues ............................... 35 3.2 Spectroscopyandenergylevels. 36 3.3 Boundstates .................................. 38 3.4 Resonances, scattering states, and the L¨uscher formalism . ......... 39 3.5 Generalcorrelationfunctions. 44 i ii Contents 4 Monte-Carlo simulations 48 4.1 ImportancesamplingandtheMarkovchain . 48 4.2 Metropolisalgorithm .............................. 52 4.3 Improvingalgorithms.............................. 54 4.3.1 Acceleration............................... 54 4.3.2 Overrelaxation ............................. 54 4.4 Statisticalerrors................................. 55 4.4.1 Determination and signal-to-noise ratio . 55 4.4.2 Autocorrelations, thermalization, and critical slowing down . .. 57 4.5 Systematicerrors ................................ 60 4.5.1 Latticeartifacts............................. 60 4.5.2 Overlapandreweighting . .. .. 61 4.5.3 Ergodicity................................ 62 4.6 Spectroscopy .................................. 63 4.6.1 Time-slice averaging . 63 4.6.2 Smearing ................................ 64 4.6.3 Variational analysis . 65 4.6.4 Resonances ............................... 67 4.7 Phase transitions revisited . 68 4.8 Spontaneous symmetry breaking in numerical simulations . .... 69 5 Gauge fields on the lattice 71 5.1 Abeliangaugetheories ............................. 72 5.2 Non-Abeliangaugetheories .......................... 76 5.3 Gauge-invariant observables and glueballs . 78 5.4 Numerical algorithms for gauge fields . 82 5.5 Gauge-fixing................................... 85 5.6 Perturbationtheoryrevisited . 90 5.7 Scaling and the continuum limit . 92 5.8 Thestrong-couplingexpansion. 94 5.8.1 Construction .............................. 94 5.8.2 Wilson criterion . 96 5.9 Improvedactions ................................ 98 5.10Topology..................................... 100 5.11 WeakinteractionsandtheHiggs. 101 Contents iii 6 Fermions 104 6.1 Naive fermions and the doubling problem . 104 6.2 Wilsonfermions................................. 106 6.3 Staggeredfermions ............................... 108 6.4 Ginsparg-Wilson fermions . 109 6.5 QCD....................................... 111 6.6 Perturbationtheoryforfermions. 113 6.7 Fermionic expectation values . 114 6.8 Hadronspectroscopy .............................. 117 6.9 Algorithmsforfermions ............................ 119 6.9.1 Molecular dynamic algorithms . 120 6.9.2 Molecular dynamics for gauge fields . 122 6.9.3 UpdatingQCD ............................. 123 7 Finite temperature and density 126 7.1 Finitetemperature ............................... 126 7.2 Finite density and the sign problem . 128 Index 130 Chapter 1 Introduction Lattice quantum field theories are essentially quantum field theories which are not defined in continuum Minkowski time, but rather in a finite Euclidean volume on a discrete set of points, the lattice1. It is at first sight very dubious that both formulations should have anything to do with each other, or that this may be useful. But this is not true. As will be discussed at length, a lattice quantum field theory can, in a very precise sense, be taken as an approximation of an ordinary continuum2 field theory. In fact, the lattice can be taken as a regulator of the theory, both in the ultraviolet, due to the discrete nature of the set of points there is a maximum energy given the smallest distance between points, as well as in the infrared, due to the finite volume. Thus, a lattice theory can be taken as a regularized version of the corresponding continuum field theory. Because the number of points is finite, there is only a finite number of degrees of freedom. Therefore, it is an approximation of a quantum field theory by a quantum mechanical system. Because of this, many statements in lattice field theory can be made on a quite rigorous level, because they have only to cover a finite number of degrees of freedom. There are therefore many powerful and rigorous statements on lattice theories. Whether they also hold for the continuum theory is actually rarely known. Taking the limit of an infinite number of degrees of freedom usually plays havoc with various assumptions. It is therefore usually unknown whether the approximation by a lattice version of the theory is good. Fortunately, for many theories this seems to be the case, though rather than proofs there is usually only circumstantial evidence. 1Note that a lattice field theory can also be defined in Minkowski space-time, and with various things being infinite rather than finite. However, this is beyond the mainstream lattice to be presented in this lecture. 2Though precisely continuum does say nothing about the different metric nor the finite volume, it has become customary to just use the statement continuum to distinguish between both versions of the theory. This is somewhat an abuse of language. 1 2 This problem is somewhat alleviated by the fact that ultimately this question is some- what academic. Since the universe is (likely) finite and quantum gravity appears at some ultraviolet scale, any theory on flat Minkowski space-time is anyhow only expected to be relevant over a finite distance and up to a finite energy in the sense of a low-energy effective theory. Therefore, the lattice version of a quantum field theory may in the end be actually even be a better approximation of nature than a continuum theory, if both should not coincide. These remarks show the conceptual importance of lattice quantum field theories. There is also a technical important one. As will be seen, the lattice version of a theory is accessible to numerical evaluations of a very particular kind: It is possible to approximate the path integral for any observable with, in principle, arbitrary precision by a numerical calculation. Especially, in many cases the required computational time grows only like a (small) power of the number of points of the lattice. Therefore, such calculations are actually feasible on nowadays computers. Since these numerical evaluations are, up to certain error sources which can be improved, exact, this implies that all information is sampled, including non-perturbative effects. This possibility makes lattice quantum field theories nowadays to a mainstay tool for non-perturbative calculations in lattice quantum field theory, though there are several areas where the numerical cost is actually still too large. This is probably an even more important reason to have a look a lattice quantum field theories. All of this will be discussed during this lecture. The particular emphasis is on the tech- niques and concepts of lattice quantum field theory. The phenomenology of the theories investigated using these methods is not the subject of this lecture, but rather of various other ones. It is thus a lecture about techniques. The advantage is that these techniques can be applied to essentially any quantum field theory as a powerful tool. The number of books on this topic is somewhat limited. The ones which have been used to prepare this lecture areas B¨ohm et al. “Gauge theories” (Teubner) DeGrand et al. “Lattice methods for quantum chromodynamics (World Scientific) Lang et al. “Quantum chromodynamics on the lattice” (Springer) Montvay et al. “Quantum fields on a lattice” (Cambridge) Rothe “Lattice gauge theories” (World Scientific) Seiler “Gauge theories as a problem of constructive quantum field theory and statis- tical mathematics” (Springer) Chapter 1. Introduction 3 As the titles in this list shows, one of the central subjects where lattice quantum field theory has shown its most spectacular successes in the past has been quantum chromo- dynamics. Due to the strength of its interactions, QCD exhibits a large number of hard to control nonperturbative features3. It is here where the ability to deal numerically with the nonperturbative features excelled. 3Note that strong interactions and nonperturbative features are not equivalent. The best example is the rich solid state physics, which is entirely due to non-perturbative interactions of the weakly interacting QED. Chapter 2 Scalar fields on the lattice 2.1 The path integral To define a lattice theory the path-integral formulation is the method of choice. Since defining the path integral itself is usually done using a lattice approximation, it is useful to consider this in more detail1. Since the path integral formulation is as axiomatic as is canonical quantization, it cannot be deduced. However, it is possible to motivate it. A heuristic reasoning is the following. Take a quantum mechanical particle which moves in time T
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