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3.1 WightmanAxioms ...... 15

3.2 Gaussian Random Processes, Q-Space and Fock Space ...... 17

3.2.1 FockSpace ...... 20

3.2.2 Wick Products and Wiener-Itˆo-Segal Isomorphism ...... 22

3.3 Osterwalder-SchraderAxioms ...... 23

4 Quantization of Yang-Mills Equations and Positive Mass Gap 27

4.1 Quantization ...... 28

4.2 GaugeInvariance...... 44

4.3 SpectralBounds ...... 47

4.4 MainTheorem ...... 53

4.5 RunningoftheCouplingConstant ...... 57

5 Conclusion 58

1 Introduction

Yang-Mills fields, which are also called gauge fields, are used in modern physics to describe physical fields that play the role of carriers of an interaction (cf. [EoM02]). Thus, the electromagnetic field in electrodynamics, the field of vector bosons, carriers of the weak interaction in the Weinberg-Salam theory of electrically weak interactions, and finally, the gluon field, the carrier of the strong interaction, are described by Yang-Mills fields. The gravitational field can also be interpreted as a Yang-Mills field (see [DP75]).

The idea of a connection as a field was first developed by H. Weyl (1917), who also attempted to describe the electromagnetic field in terms of a connection. In 1954, C.N. Yang and R.L. Mills (cf. [MY54]) suggested that the space of intrinsic degrees of freedom of elementary particles (for example, the isotropic space describing the two degrees of freedom of a nucleon that correspond to its two pure states, proton and neutron) depends on the points of space-time, and the intrinsic spaces corresponding to different points are not canonically isomorphic.

2 Gauge Theory Fundamental Forces Structure Group Quantum Electrodynamics Electromagnetism U 1 (QED) p q Electroweak Theory Electromagnetism SU 2 U 1 (Glashow-Salam-Weinberg) and weak force p qˆ p q Quantum Chromodynamics Strong force SU 3 (QCD) and electromagnetism p q Standard Model Strong, weak forces SU 3 SU 2 U 1 and electromagnetism p qˆ p qˆ p q Georgi-Glashow Grand Strong, weak forces SU 5 Unified Theory (GUT1) and electromagnetism p q Fritzsch-Georgi-Minkowski Strong, weak forces SO 10 Grand Unified Theory (GUT2) and electromagnetism p q Grand Unified Strong, weak forces SU 8 Theory (GUT3) and electromagnetism p q Grand Unified Strong, weak forces O 16 Theory (GUT4) and electromagnetism p q

Table 1: Gauge Theories

In geometrical terms, the suggestion of Yang and Mills was that the space of intrinsic degrees of freedom is a vector bundle over space-time that does not have a canonical trivialization, and physical fields are described by cross-sections of this bundle. To describe the differential evolution equation of a field, one has to define a connection in the bundle, that is, a trivialization of the bundle along the curves in the base. Such a connection with a fixed holonomy group describes a physical field, usually called a Yang-Mills field. The equations for a free Yang-Mills field can be deduced from a variational principle. They are a natural non-linear generalization of Maxwell’s equations.

Field theory does not give the complete picture. Since the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics, where classical observables correspond to typically non commuting self-adjoint operators on a Hilbert space, and classic notions as “the trajectory of a particle” do not apply. Since fields interact with particles, it became clear by the late 1920s that an internally coherent account of nature must incorporate quantum concepts for fields as well as for particles. Under this approach components of fields at different points in space-time become non-commuting operators.

The most important Quantum Field Theories describing elementary particle physics are gauge the- ories formulated in terms of a principal fibre bundle over the Minkowskian space-time with particular choices of the structure group. They are depicted in Table 1.

In order for Quantum Chromodynamics to completely explain the observed world of strong interac- tions, the theory must imply:

3 • Mass gap: There must exist some positive constant η such that the excitation of the vacuum state has energy at least η. This would explain why the nuclear force is strong but short-ranged, by providing the mathematical evidence that the corresponding exchange particle, the gluon, has non vanishing rest mass.

• Quark confinement: The physical particle states corresponding to proton, neutron and pion must be SU 3 -invariant. This would explain why individual quarks are never observed. p q • Chiral symmetry breaking: In the limit for vanishing quark-bare masses the vacuum is in- variant under a certain subgroup of the full symmetry group acting on the quark fields. This is required in order to account for the “current algebra” theory of soft pions.

The Seventh CMI-Millenium prize problem is the following conjecture.

Conjecture 1. For any compact simple Lie group G there exists a nontrivial Yang-Mills theory on the Minkowskian R1,3, whose quantization satisfies Wightman axiomatic properties of Constructive Quantum Field Theory and has a mass gap η 0. ą

The conjecture is explained in [JW04] and commented in [Do04] and in [Fa05]. To our knowledge this conjecture is unproved.

The first rigorous program of study of this problem is the one by Balaban ([Ba84], [Ba84Bis], [Ba85], [Ba85Bis], [Ba85Tris], [Ba85Quater], [Ba87], [Ba88], [Ba88Bis], [Ba89] and [Ba89Bis]). This program defines a sequence of block-spin transformations for the pure Yang-Mills theory in a finite volume on the lattice, a toroidal Euclidean space-time, and shows that, as the lattice spacing tends to 0 and these transformations are iterated many times, the resulting effective action on the unit lattice remains bounded. From this result the existence of an ultraviolet limit for gauge invariant observables such as “smoothed Wilson loops” should follow, at least through a compactness argument using a subsequence of approximations; but the limit is not necessarily unique. From this point of view, one must verify the existence of an ultraviolet limit of appropriate expectations of gauge-invariant observables as the lattice spacing tends to zero and the volume tends to infinity.

Magnen, Rivasseu and S´en´eor provide in [MRS93] the basis for a rigorous construction of the Schwinger functions of the pure SU 2 Yang-Mills field theory in four dimensions (in the trivial topo- p q logical sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. They check the validity of the construction by showing that Slavnov identities (which express infinitesimal gauge invariance) do hold non-perturbatively.

This paper is organized as follows. Section 2 presents the Yang-Mills equations and their Hamil- tonian formulation for the Minkowskian R1,3. Section 3 depicts Wightman axioms of Constructive

4 Quantum Field Theory, which are verified in Sections 4, where Yang-Mills Equations are quantized and the existence of a positive mass gap proven. Verifying the Osterwalder-Schrader axioms and the recon- struction theorem for quantum mechanics, the QFT Hamiltonian for the continuum theory constructed via quantization of the classical Hamiltonian, is proved to be a selfadjoint operator on the Hilbert Space of L2-Hida distributions. This operator has a continuous spectrum, which can be expressed as the direct limit of the continuous spectrum of another selfadjoint operator, the QFT Hamiltonian with cut off, when the cut off tends to infinity. For both operators strictly positive lower bounds of the spectra can be inferred. These depend on the running coupling constant for both Hamiltonians and on the cut off magnitude for the Hamiltonian with cut off. The gap for the cut off case “survives” then in the continuum limit for a strictly positive running coupling constant. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. The whole construction is proved to be invariant for those gauge transforms which preserve the Coulomb gauge. These results are first proved for the model with the bare coupling constant depending on the energy scale only, and then extended by means of renormalization. The renormalized model has a positive finite mass gas, which survives the removal of the ultraviolet cut off. Section 5 concludes.

2 Yang-Mills Connections

2.1 Definitions, Existence and Uniqueness

A Yang-Mills connection is a connection in a principal fibre bundle over a (pseudo-)Riemannian manifold whose curvature satisfies the harmonicity condition, i.e. the Yang-Mills equation.

Definition 1 (Yang-Mills Connection). Let P be a principal G-fibre bundle over a pseudoriemannian m-dimensional manifold M,h , and let V be the vector bundle associated with P and RK, induced by p q the representation ρ : G GL RK . A connection on the principal fibre bundle P is a Lie-algebra G Ñ p q valued 1-form ω on P , such that the following properties hold:

(i) Let A G and A˚ the vector field on P defined by P d A˚ : p exp tA . (1) p “ dt p p qq ˇt:“0 ˇ ˇ Then, ω A˚ A. ˇ p p q“

5 (ii) For g G let P

´1 Adg : G G, h Adg h : Lg R ´1 h ghg Ñ ÞÑ p q “ ˝ g p q“ (2) d ´1 adg : G G, A Adg A : g exp tA g Ñ ÞÑ p q “ dt p p q q ˇt:“0 ˇ ˇ be the adjoint isomorphism and the adjoint representation,ˇ respectively. ˚ Then, R ω ad ´1 ω. g “ g

The connection ω on P defines a connection ∇ for the vector bundle V , i.e. an operator acting on the space of cross sections of V . The vector bundle connection ∇ can be extended to an operator d :Γ p M V Γ p`1 M V , by the formula p p q q Ñ p p q q Ź Â Ź Â d∇ η v : dη v 1 pη ∇v. (3) p b q “ b ` p´ q b

The operator δ∇ :Γ p`1 M V Γ p M V , defined as the formal adjoint to d, is equal to p p q q Ñ p p q q Ź Â Ź Â δ∇η 1 p`1 d∇ , (4) “ p´ q ˚ ˚ where denotes the Hodge-star operator. ˚ A connection ω in a principal fibre bundle P is called a Yang-Mills field if the curvature F : “ dω ω ω, considered as a 2-form with values in the Lie algebra G, satisfies the Yang-Mills equations ` ^

δ∇F 0, (5) “ or, equivalently, δ∇R∇ 0, (6) “

∇ where R X, Y : ∇X ∇Y ∇Y ∇X ∇ denotes the curvature of the vector bundle V , and is a p q “ ´ ´ rX,Y s 2-form with values in V .

Remark 2.1 (Local Representations of Connections on Vector and Principle Fibre Bundles). The local section σ : U M P is defines the local representation of the connection given on the Ă Ñ open U M by A : ω σ : U G a Lie-algebra G valued 1-form on U, the fields Aj x : Ă “ ˝ Ñ p q “ K k K A x ej A x tk define by means of the tangential map Teρ : G L R of the representation p q “ k“1 j p q Ñ p q K ρ : G GLř R , with fields of endomorphisms TeρA1,...,TeρAm L Vx for the bundle V . Given a Ñ p q P p q K basis of the Lie-algebra G denoted by t1,...,tK , the endomorphisms ws : Teρ.ts s 1,...,K in L R t u t “ u “ p q have matrix representations with respect to a local basis vs x s 1,...,K denoted by ws . Since t p qu “ r stvspxqu

6 ρ is a representation, Teρ has maximal rank and the endomorphisms are linearly independent. Given

a local basis ej x j 1,...,m for x U M, the Christoffel symbols of the connection ∇ are locally t p qu “ P Ă defined by the equation K r ∇ej vs Γj,svr, (7) “ r“1 ÿ holding true on U, and they ,satisfy the equalities

K r r a Γj,s wa sAj . (8) “ a“1r s ÿ

K s Given a local vector field v f vs in V U and a local vector field e in TM U , the connection ∇ “ s“1 | | has a local representation ř K s s ∇ev df e .vs f ω e .vs , (9) “ s“1p p q ` p q q ÿ ˚ where ω is an element of T U U L V U , i.e. an endomorphism valued 1 form satisfying | p | q ´

 K r ω ej vs Γj,svr. (10) p q “ r“1 ÿ Remark 2.2. The curvature 2-form reads in local coordinates as

K M K K k 1 k k k a b F F tk dxi dxj jA iA C A A tk dxi dxj , (11) “ i,j ^ “ 2 B i ´B j ´ a,b i j ^ 1ďiăjďM k“1 i,j“1 k“1 ˜ a,b“1 ¸ ÿ ÿ ÿ ÿ ÿ

c where C C a,b,c 1,...,K are the structure constants of the Lie-algebra G corresponding to the basis “r a,bs “ t1,...,tK , which means that for any a,b t u

K c ta,tb Ca,btc. (12) r s“ c“1 ÿ

The existence and uniqueness of solutions of the Yang-Mills equations in the Minkowski space have been first established by Segal (cf. [Se78] and [Se79]), who proves that the corresponding Cauchy problem encoding initial regular data has always a unique local and global regular solution. He proves

as well that the temporal gauge (A0 0) chosen to express the solution does not affect generality, “ because any solution of the Yang-Mills equation can be carried to one satisfying the temporal gauge. This subject has been undergoing intensive research, improving the original results. For example in [EM82] and in [EM82Bis] the Yang-Mills-Higgs equations, which generalize (5) and are non linear PDEs of order two, have been reformulated in the temporal gauge as a non-linear PDE of order one, satisfying a constraint equation. This PDE can be written as an integral equation solving (always and uniquely,

7 locally and globally) the Cauchy data problem with improved regularity results. Existence, uniqueness and regularity of the Yang-Mills-Higgs equations under the MIT Bag boundary conditions have been investigated in [ScSn94] and [ScSn95].

2.2 Hamiltonian Formulation for the Minkowski Space

The Hamiltonfunction describes the dynamics of a physical system in classical mechanics by means of Hamilton’s equations. Therefore, we have to reformulate the Yang-Mills equations in Hamiltonian mechanical terms. We focus our attention on the Minskowski R4 with the pseudoriemannian structure of special relativity h dx0 dx0 dx1 dx1 dx2 dx2 dx3 dx3. The coordinate x0 represents “ b ´ b ´ b ´ b the time t, while x1, x2, x3 are the space coordinates.

We introduce Einstein’s summation notation, and adopt the convention that indices for coordinate variables from the greek alphabet vary over 0, 1, 2, 3 , and those from the latin alphabet vary over t u the space indices 1, 2, 3 . For a generic field F Fµ µ 0,1,2,3 let F : Fi i 1,2,3 denote the “space” t u “ r s “ “r s “ component. The color indices lie in 1,...,K . Let t u

1 (π is even) ` εa,b,c : $ 1 (π is odd) (13) “ ’ ’ ´ & 0 (two indices are equal), ’ ’ and any other choice of lower and upper% indices, be the Levi-Civita symbol, defined by mean of the 1 2 3 permutation π : in S3. “ ¨ a b c ˛ ˝ ‚ Remark 2.3. If the Lie-group G is simple, then the Lie-Algebra is simple, and the structure constants can be written as Cc gεc , (14) a,b “ a,b for a positive constant g called (bare) coupling constant, (see f.i. [We05] Chapter 15, Appendix A). The components of the curvature then read

k 1 k k k a b F ν A µA gε A A . (15) µ,ν “ 2pB µ ´B ν ´ a,b µ ν q

We will consider only simple Lie groups. As we will see, it is essential for the existence of a mass gap for the group G to be non-abelian.

We need to introduce an appropriate gauge for the connections we are considering.

8 Definition 2 (Coulomb Gauge). A connection A over the Minkowski space satisfies the Coulomb gauge if and only if a a A 0 and j A 0 (16) 0 “ B j “ for all a 1,...,K and j 1, 2, 3. “ “ Definition 3 (Transverse Projector). Let F be the Fourier transform on functions in L2 R3, R . p q The transverse projector T : L2 R3, R3 L2 R3, R3 is defined as p q Ñ p q

´1 pipj T v i : F δi,j F vj , (17) p q “ ´ p 2 p q ˆ„ | |  ˙

and the vector field v decomposes into a sum of a transversal (vK) and a longitudinal (vk) component:

K k K k vi v v , v : T v i, v : vi T v i. (18) “ i ` i i “ p q i “ ´ p q

Remark 2.4. The Coulomb gauge condition for the space part of a connection A is equivalent to the vanishing of its longitudinal component: Aak t, 0 (19) i p ¨q “ for all i 1, 2, 3, all a 1,...K and any t R. The time part of the connection A vanishes by “ “ P definition of Coulomb gauge.

Proposition 2.1. For a simple Lie-group as structure group let A be a connection over the Minkowskian R4 satisfying the Coulomb gauge, and assume that Aa t, C8 R3, R L2 R3, R for all i 1, 2, 3, i p ¨q P p qX p q “ all a 1,...K and any t R. The operator L on the real Hilbert space L2 R3, RK defined as “ P p q

a,b a,b R3 a,c,b c L L A; x L A; x : δ ∆ gε A t, x k (20) “ p q“r p qs “r x ` kp qB s is essentially self adjoint and elliptic for any time parameter t R. Its spectrum lies on the real line, P and decomposes into discrete spec L and continuous spectrum spec L . If 0 is an eigenvalue, then it dp q cp q has finite multiplicity, i.e. ker L is always finite dimensional . p q The modified Green’s function G G A; x, y Ga,b A; x, y S1 R3, RKˆK for the operator L “ p q“r p qs P p q is the distributional solution to the equation

N a,b b,d a,d a d L A; x G A; x, y δ δ x y ψn A; x ψn A; y , (21) p q p q“ p ´ q´ n“1 p q p q ÿ

2 where ψn A; n is an o.n. L -basis of N-dimensional ker L . In equation (21) x is seen as variable, t p ¨qu p q while y is considered as a parameter. This modified Green’s function can be written as a Riemann-

9 Stielties integral: For any ϕ S R3, RK L2 R3, RK P p qX p q 1 G A; x, ϕ d Eλϕ x , (22) p ¨qp q“ λ p qp q żλ‰0 where Eλ λ R is the resolution of the identity corresponding to L. p q P Remark 2.5. In [Br03] and [Pe78] the modified Green’s function is constructed assuming that the

operator L has a discrete spectral resolution ψn A; , λn n 0 as p p ¨q q ě

1 : G A; x, y ψn A; x ψn A; y . (23) p q“ λn p q p q n:λ ‰0 ÿn In particular, we have the symmetry property

G A; x, y : G A; y, x (24) p q “ p q

for all x, y, A for which the expression is well defined. Since the discontinuity points of the spectral

resolution Eλ λ R are the eigenvalues, i. e. the elements of spec L (cf. [Ri85], Chapter 9), the p q P dp q solution (22) extends (23) to the general case.

Remark 2.6. Rigged Hilbert spaces have been introduced in mathematical physics to utilize Dirac calculus for the spectral theory of operators appearing in quantum mechanics (see [Ro66], [ScTw98] and [Ma08]). Within that framework the role of distributions to provide a rigorous foundation to generalized eigenvectors and eigenvalues is highlighted by the Gel’fand-Kostyuchenko spectral theorem (see [Ze09] Chapter 12.2.4 and [GS64] Section I.4).

Theorem 2.2 (Gel’fand-Kostyuchenko). Let A : S RN S RN be a linear sequentially contin- p q Ñ p q uous operator, which is essentially selfadjoint in the Hilbert Space L2 RN . Then, the operator A has p q a complete system Fm m M of eigendistributions. This means that there exists a non empty index set p q P 1 N M such that FM S R for all m M and P p q P

(i) Eigendistributions: there exists a function λ : M R such that Ñ

Fm Aϕ λ m Fm ϕ , (25) p q“ p q p q

for all m M and all test functions ϕ S RN . P P p q N (ii) Completeness: if Fm ϕ 0 for all m M and a fixed test function ϕ S R , then ϕ 0. p q“ P P p q “

The spectrum of A then reads spec A λ M , where for all m M the real number λ m is either an p q“ p q P p q 2 N 2 N eigenvalue, i.e. Fm L R , or an element of the continuous spectrum, i.e. Fm L R . P p q R p q

10 Proof of Proposition 2.1. As long as the connection satisfies the Coulomb gauge condition, the operator L is symmetric and essentially selfadjoint on the appropriate domain, as a direct computation involving integration by parts can show. As its leading symbol is elliptic, the operator L is elliptic and restricted to R , R 3, under the Dirichlet boundary conditions it has a discrete spectral resolution (cf. [Gi95], r´ 2 ` 2 s Chapter 1.11.3). Every eigenvalue has finite multiplicity. The dimension of the eigenspaces is an integer- valued, continuous, and hence constant function of R. For R the discrete Dirichlet spectrum of L Ñ `8 on R , R 3 clusters in the spectrum of L on R3, which decomposes into a discrete and a continuous r´ 2 ` 2 s spectrum. Therefore, the eigenvalues must have finite multiplicity and, in particular, ker L is finite p q dimensional. Equation (22) gives the modified Green’s function as it can be verified by the following computation, which holds true for any ϕ S R3, RK L2 R3, RK : P p qX p q 1 1 L A; x G A; x, ϕ L d Eλϕ x Ld Eλϕ x p q p ¨qp q“ λ p qp q“ λ p qp q“ żλ‰0 żλ‰0 0` d Eλϕ x d Eλϕ x φ x PkerpLqϕ x (26) “ R p qp q´ ´ p qp q“ p q´ p q“ ż ż0 N : δ x ϕ ψn A; x ψn A; ϕ . “ p ´ ¨qp q´ n“1 p q p ¨qp q ÿ

The existence of eigenvalues of L depends on the additive perturbation to the Laplacian given by a,c,b c gε A t, x k. For example, if g 0 or A 0, the operator L has no eigenvalues and spec L kp qB “ “ p q “ spec L , 0 . In general, the spectrum depends on the choice of the connection A. In [BEP78] cp q“s´8 s and in [Pe78] special cases comprising pure gauges and Wu-Yang monopoles are computed explicitly. We are interested in a reformulation of the general solution (22), where the dependence on the connection becomes explicit. Inspired by [Ha97] we find

Proposition 2.3. For a simple Lie-group as structure group, if we assume that the coupling constant g 1, then a Green’s function K K A; x, y Ka,b A; x, y for the operator L in Proposition 2.1, ă “ p q“r p qs that is, a distributional solution to the equation

La,b A; x Kb,d A; x, y δa,dδ x y , (27) p q p q“ p ´ q

11 is given by the convergent series in S1 R3, RKˆK p q

b,d b,d δ d,e,b 3 1 e 1 K A; x, y gε d u1 Ak u1 k p q“4π x y ` R3 4π x u1 p qB 4π u1 y ` | ´ | ż | ´ | ˆ | ´ |˙ `¨¨¨` n n´1 1 n b,e1,s1 sn´1,en,d 3 1 e1 1 ` g ε ε d u1 Ak u1 (28) ` p´ q 2 ¨ ¨ ¨ R3 4π x u1 p q¨ ż | ´ | 3 1 3 1 en 1 k d u2 ... d un Al un l ¨B R3 4π u1 u2 R3 4π un 1 un p qB 4π un y ` ż | ´ | ż | ´ ´ | ˆ | ´ |˙ ... `

Note that x is the variable and y a parameter.

Proof. The series (28) converges because of the integrability of the connection A and the fact that g 1. ă Recall that 1 L1 R3 S1 R3 for any fixed y R3. We now check that it represents a Green’s |x´y| P locp qĂ p q P function for L: a,b b,d a,d L A; x K A; x, y δ δ x y lim Restn, (29) p q p q“ p ´ q` nÑ`8 where, after having evaluated a “telescopic sum”, the remainder part reads

n x n´1 1 n`1 b,e1,s1 sn´1,en,d a,c,b c 3 k e1 Restn 1 ` g ε ε ε Ak x d u1 ´ 3 Ak u1 “p´ q 2 ¨ ¨ ¨ p q R3 4π x u1 p q¨ ż | ´ | (30) 3 1 3 1 en 1 k d u2 ... d un Al un l . ¨B R3 4π u1 u2 R3 4π un 1 un p qB 4π un y ż | ´ | ż | ´ ´ | ˆ | ´ |˙ Because of the integrability of the connection, there exists a constant C 0 such that for any ϕ ą P S R3, R p q n`1 Restn ϕ Cg ϕ 2 R3 R 0 n , (31) | p q| ď } }L p , q Ñ p Ñ `8q and the proposition follows.

Remark 2.7. For the Minkowskian R4 the assumption of a (dimensionless) bare coupling constant g 1 is well posed: for the Yang-Mills theory one is basically allowed to choose any value g 0 by ă ą appropriate rescaling of the energy scale (see [SaSc10]). Moreover, we will later have to analyze the case where g 0`. Ñ Corollary 2.4. Under the same assumptions as Proposition 2.3 the distribution

1 : : G A; x, y K A; x, y K A; x, y ψn A; x ψ A; y (32) p q“ 2p p q` p q q´ p q np q n:λ “0 ÿn is a (symmetric) modified Green’s function for the operator L.

12 After this preparation we can turn to the Hamiltonian formulation of Yang-Mills’ equations, following the results in [Br03] and [Pe78], which just need to be adapted for the generic case that L has a mixture of discrete and continuous spectral resolution.

Theorem 2.5. . For a simple Lie-group as structure group and for canonical variables satisfying the Coulomb gauge condition, the Yang-Mills equations for the Minkowskian R4 can be written as Hamilton equations dE BH A, E dt “´ BA p q $ (33) ’ ’ & dA BH A, E dt “` BE p q ’ for the following choices: %’

• a Position variable: A Ai t, x a“1,...,K also termed potentials , “r p qs i“1,2,3 • a Momentum variable: E Ei t, x a“1,...,K , whose entries are termed chromoelectric fields, “r p qs i“1,2,3 • Hamilton function: defined as a function of A and E as

1 3 a 2 a 2 a a c c H H A, E : d x Ei t, x Bi t, x f t, x ∆f t, x 2ρ t, x A0 t, x , (34) “ p q “ 2 R3 p q ` p q ` p qq p q` p q p q ż ` ˘ where B Ba , whose entries are termed chromomagnetic fields, is the matrix valued-function “ r i s defined as

a 1 j,k a a a b c B : ε jA kA gε A A , (35) i “ 4 i B k ´B j ` b,c j k ` ˘ and ρ ρa t, x , termed charge density, is the vector valued function defined as “r p qs

ρa : gεa,b,cEbAc, (36) “ i i

and where

f a t, x : g d3y Ga,b A; x, y ρb t,y Ga,b A; x, ρb t, p q “´ R3 p q p q“ p ¨qp p ¨qq ż (37) a 3 a,b b a,b b A0 t, x : d y G A; x, y ∆f t,y G A; x, ∆f t, p q “ R3 p q p q“ p ¨qp p ¨qq ż for the modified Green’s function G A; x, y for the operator L A; x . p q p q

We consider position A and momentum variable E as elements of S R3, RKˆ3 depending on the time p q parameter t, so that the RHSs of equations (37) are well defined distributions applied to test functions.

13 Remark 2.8. As shown in [Pe78] the ambiguities discussed by Gribov in [Gr78] concerning the gauge fixing (see also [Si78] and [He97]) can be traced precisely to the existence of zero eigenfunctions of the operator L.

Corollary 2.6. The Hamilton function (34) for the Yang-Mills equations can be written as

H HI HII V, (38) “ ` ` where

1 3 a 2 HI d x Ei t, x “ 2 R3 p q ż

2 2 g 3 3 a,b b,c,d d c HII d x d y iG A; x, y ε Ak t,y Ek t,y (39) “ 2 R3 R3 B p q p q p q ż „ż 

1 3 j,k p,q a a a,b,c b c a a a,b,c b c V d xεi εi jAk kAj gε Aj Ak pAq qAp gε ApAq t, x . “ 16 R3 rpB ´B ` qpB ´B ` qsp q ż

Proof. The expressions for the functions HI and V are obtained by a straightforward calculation. For the function HII some more work is needed. Inserting (37) in the last addendum of (34) we obtain

3 1 a a c d x f t, x ∆f t, x ρc t, x A0 t, x R3 2 p q p q` p q p q “ ż ˆ ˙ 3 1 3 a,b 3 a,b d x d y G A; x, y ρb t,y d y¯ G A; x, y¯ ρb t, y¯ “ R3 2 ´ R3 p q p q ´ R3 p q p q ` ż „ ˆ ż ˙ ˆ ż ˙ 3 c,b 3 b,d ρc t, x d y G A; x, y d y¯∆G A; y, y¯ ρd t, y¯ ` p q R3 p q R3 p q p q “ ż ż  3 3 3 1 a,b a,d d x d y d y¯ G A; x, y ρb t,y ∆G A; x, y¯ ρd t, y¯ “ R3 R3 R3 2 p q p q p q p q ` ż ż ż „ c,b ` b,d ˘ (40) ρc t, x G A; x, y ∆G A; y, y¯ ρd t, y¯ ´ p q p q p q p q “ 3 3 3 1 a,b a,d ‰ d x d y d y¯ G A; x, y ∆G A; x, y¯ ρb t,y ρd t, y¯ “ R3 R3 R3 2 p q p q p q p q ` ż ż ż „ a,b `a,d ˘ G A; y, x ∆G A; x, y¯ ρb t,y ρd t, y¯ ´ p q p q p q p q “ 1 3 3 3 a,b ‰ a,d d x d y d y¯ iG A; x, y ρb t,y iG A; x, y¯ ρd t, y¯ “ 2 R3 R3 R3 B p q p q B p q p q “ ż ż ż “ 2 ‰ “ ‰ 1 3 3 a,b d x d y iG A; x, y ρb t,y , “ 2 R3 R3 B p q p q ż „ż 

where in the last transformation formula we have utilized integration by parts in the variables x1, x2, x3, and the fact that G A; x, y G A; y, x . Inserting expression (36) for the charge density completes p q “ p q

14 the proof.

3 Axioms of Constructive Quantum Field Theory

3.1 Wightman Axioms

In 1956 Wightman first stated the axioms needed for CQFT in his seminal work [Wig56], which remained not very widely spread in the scientific community till 1964, when the first edition of [SW10] appeared. We will list the axioms in the slight refinement of [BLOT89] and [DD10].

Definition 4 (Wightman Axioms). A scalar (respectively vectorial-spinorial) quantum field theory consists of a separable Hilbert space E, whose elements are called states, a unitary representation U of 4 the Poincar´egroup P in E, an operator valued distribution Φ (respectively Φ1,..., Φd) on S R with p q values in the unbounded operators of E, and a dense subspace D E such that the following properties Ă hold:

(W1) Relativistic invariance of states: The representation U : P U E is strongly continuous. Ñ p q

(W2) Spectral condition: Let P0, P1, P2, P3 be the infinitesimal generators of the one-parameter 2 2 2 2 groups t U teµ, I for µ 0, 1, 2, 3. The operators P0 and P P P P are positive. ÞÑ p q “ 0 ´ 1 ´ 2 ´ 3 4 This is equivalent to the spectral measure E¨ on R corresponding to the restricted representation R4 a U a, I having support in the positive light cone (cf. [RS75], Chapter IX.8). Q ÞÑ p q

(W3) Existence and uniqueness of the vacuum: There exists a unique state Ω0 D E such P Ă 4 that U a, I Ω0 Ω0 for all a R . p q “ P 4 (W4) Invariant domains for fields: The maps Φ : S R O E , and, respectively Φ1,..., Φd : p q Ñ p q S R4 O E , from the Schwartz space of test functions to (possibly) unbounded selfadjoint p q Ñ p q operators on the Hilbert space, satisfy following properties

(a) For all ϕ S R4 and all field maps, the domain of definitions D Φ ϕ , D Φ ϕ ˚ , and P p q p p qq p p q q ˚ respectively D Φj ϕ , D Φj ϕ , all contain D and the restrictions of all operators to D p p qq p p q q agree.

(b) Φ ϕ D D, and, respectively Φj ϕ D D. p qp qĂ p qp qĂ

(c) For any ψ D fixed, the maps ϕ Φ ϕ ψ, and, respectively ϕ Φj ϕ ψ, are linear. P ÞÑ p q ÞÑ p q

(W5) Regularity of fields: For all ψ1, ψ2 D, the map ϕ ψ1, Φ ϕ ψ2 , and, respectively the P ÞÑ p p q q 1 4 maps ϕ ψ1, Φj ϕ ψ2 , are tempered distributions, i.e. elements of S R . ÞÑ p p q q p q

15 (W6) Poincar´einvariance: For all a, Λ P, ϕ S R4 , and ψ D, the inclusion U a, Λ D D p q P P p q P p q Ă must hold and

(Scalar field case): The following equation must hold for all Λ O 1, 3 P p q

U a, Λ Φ ϕ U a, Λ ´1ψ Φ a, Λ ϕ ψ. (41) p q p q p q “ pp q q

(Vectorial/Spinorial field case): There exists a representation of SL 2, C on Cd denoted p q by ρ, and satisfying ρ I I or ρ I I, such that for all Λ SO` 1, 3 and Φ : p´ q “ p q “ P p q “ : Φ1,..., Φd r s U a, Λ Φ ϕ U a, Λ ´1ψ ρ s´1 Λ Φ a, Λ ϕ ψ, (42) p q p q p q “ p p qq pp q q where s denotes the spinor map s : SL 2, C SO` 1, 3 defined as below. The vector p q Ñ p q spaces of Hermitian matrices H in C2 and R4 are isomorphically mapped by

X : R4 H ÝÑ x0 x3 x1 ix2 (43) x x0, x1, x2, x3 X x : ` ´ . “ p q ÞÑ p q “ » x1 ix2 x0 x3 fi ` ´ – fl The group SL 2, C acts on H by p q

SL 2, C H H p qˆ ÝÑ (44) P,X P.X : P XP ˚. p q ÞÑ “

The spinor map is defined as

s : SL 2, C SO` 1, 3 p q ÝÑ p q (45) P s P : x s P x : X´1 PX x P ˚ . ÞÑ p q ÞÑ p q “ p p q q

(W7) Microscopic causality or local commutativity: Let ϕ, χ S R4 , whose supports are space- P p q like separated, i.e. φ x χ y 0, if x y is not in the positive light cone. Then, p q p q“ ´ (Scalar field case): The images of the test functions by the map field must commute

Φ ϕ , Φ χ 0. (46) r p q p qs “

16 (Vectorial/Spinorial field case): For any field maps j, i 1,...,d either the commutations “

˚ Φj ϕ , Φi χ 0, Φ ϕ , Φi χ 0, (47) r p q p qs “ r j p q p qs “

or the anticommutations

˚ Φj ϕ , Φi χ 0, Φ ϕ , Φi χ 0 (48) r p q p qs` “ r j p q p qs` “

hold.

(W8) Cyclicity of the vacuum:

(Scalar field case): The set

4 D0 : Φ ϕ1 Φ ϕn Ω0 ϕj S R ,n N0 (49) “t p q¨¨¨ p q | P p q P u

is dense in E.

(Vectorial/Spinorial field case): The set

4 ˚ ˚ D0 : Ψ ϕ1 Ψ ϕn Ω0 ϕj S R , Ψ Φ1,..., Φd, Φ ,..., Φ , n N0 (50) “t p q¨¨¨ p q | P p q Pt 1 d u P u

is dense in E.

3.2 Gaussian Random Processes, Q-Space and Fock Space

The Hilbert spaces utilized in quantum field theory are realized as L2 spaces over the tempered dis- tributions, the latter seen as probability space. This construction turns out to be isomorphic to that of the Fock space. We follow chapter 1 of [Sim15], chapter 5 of [BHL11] and appendix A.4 of [GJ87]. For a general overview see Kuo ([Ku96]), and the framework for functional integration developed by Cartier/DeWitt-Morette as in [Ca97], [La04] and [CDM10]. Although Feynman’s integral can be pro- vided a rigorous foundation by mean of functional integration, we prefer to use the Feynman-Kaˇc approach, since we will be working with Euclidean fields.

Let Ω, A,µ be a probability space, and M Ω, A : f : Ω R f measurable the algebra of p q p q “ t Ñ | u ´1 random variables. A random variable f on Ω has a probability distribution function µf U : µ f U p q “ p p qq defined on the measurable sets U of R, and a characteristic function

itx Sf t : e dµf x , (51) p q “ R p q ż

17 1 defined as the Fourier inverse transform of the probability density ρf : µf . A real valued random 2 “ ´ a t variable f has mean 0 and variance a 0 if and only if Sf t e 2 . A well known result (see f.i. ě p q “ [RS75]) is the following

Theorem 3.1 (Bochner). A function S : R C is the characteristic function of a random variable Ñ f :Ω R if and only if the following conditions are satisfied: Ñ

(i) S 0 1. p q“ (ii) t S t is continuous. ÞÑ p q n (iii) ti i 1,...,n R, zi i 1,...,n C ziz¯jS ti tj 0. t u “ Ă t u “ Ă ñ i,j“1 p ´ qě ř The construction of Bochner’s theorem can be lifted to generating functionals of random variables over the space of tempered distributions.

Definition 5. A random process indexed by a real vector space V is a linear map Φ : V M Ω, A . Ñ p q It is termed Gaussian random process if

(i) Φ v is a Gaussian random variable for all v V . p q P (ii) Φ v v V is full, i.e. A is the smallest σ-algebra for which this is a family of measurable t p q | P u functions.

Theorem 3.2. Let H be a real Hilbert space. Up to isomorphism there exists exactly one Gaussian random process indexed by H such that

1 Φ v , Φ w 2 : Φ v Φ w v, w H. (52) p p q p qqL pΩ,dµq “ p q p q“ 2p q żΩ Proof. See Theorem I.9 (page 20) in [Sim15] or Lemma 5.4 (page 258) and Proposition 5.6 (page 260) in [BHL11].

Given a real Hilbert space there are different but isomorphic models for the probability space Ω, A,µ admitting a Gaussian process. We choose the tempered distributions S1 RN as model space. p q p q Remark that any fixed test function f S RN and variable Φ S1 RN , the expression Φ f defines P p q P p q p q a random variable over S1 RN . As in appendix A.6 of [GJ87] Bochner’s theorem generalizes to p q Theorem 3.3 (Milnos). Let S : S RN C be a function. There exists a probability measure µ p q Ñ satisfying S f eiΦpfqdµ Φ , (53) p q“ 1 RN p q żS p q for all f S RN , if and only if P p q

18 (i) S 0 1. p q“ (ii) f S f is continuous in the S RN C topology. ÞÑ p q p q Ñ N n (iii) fi i 1,...,n S R , zi i 1,...,n C ziz¯j S fi fj 0. t u “ Ă p q t u “ Ă ñ i,j“1 p ´ qě Remark 3.1. The σ-algebra A is generated byř the cylinder sets in S1 RN , i.e. subsets of the tempered p q distribution space of the form

1 N Φ S R Φ f1 ,... Φ fn U , (54) P p q | p p q p qq P ( n n where U is a fixed Borel set in R , and f1,...,fn fixed test functions in S R for a n N0. p q P

We utilize Minlos’ theorem to define Gaussian measures on the tempered distributions. Let c be a positive semidefinite quadratic form on S RN . Applying Theorem 3.3 to the functional S f : 1 p q p q “ e´ 2 cpf,fq we can construct a measure µ on S1 RN such that p q

1 eiΦpfqdµ Φ e´ 2 cpf,fq (55) 1 RN p q“ żS p q

Therefore, Φ f is a Gaussian random variable with variance c f,f , because for all t R p q p q P

2 t eitΦpfqdµ Φ e´ 2 cpf,fq (56) 1 RN p q“ żS p q holds true. If H is a real Hilbert space such that the embedding S ã H is continuous and dense, Ñ then the inner product in H restricts to a positive definite bilinear form on S, which can be written as c f,g Cf,g H for all f,g S for a positive definite operator C on H with domain of definition S. p q “ p q P Definition 6 (Gaussian measures). A measure µ on S1 RN defined by Milnos’s theorem satisfying p q

1 eiΦpfqdµ Φ e´ 2 pCf,fqH , (57) 1 RN p q“ żS p q where C is a positive semidefinite operator C on H with domain of definition S, termed covariance operator, is called Gaussian.

The operation of derivation can be defined for functionals of random variables as well.

Definition 7 (Functional Derivative). Let F L2 S1 RN ,µ . Its functional directional deriva- P p p q q tive in the direction Υ S1 RN is defined as Gˆateaux derivative as P p q δF d Φ .Υ: F Φ ǫΥ . (58) δΦ p q “ dǫ p ` q ˇǫ“0 ˇ ˇ 19ˇ The special case Υ δ x for x RN arises often and is thus given a special notation “ p¨ ´ q P δ δF F Φ : Φ .δ x . (59) δΦ x p q “ δΦ p q p¨ ´ q p q

3.2.1 Fock Space

To extend a quantum mechanical model accounting for a fixed number of particles to one accounting for an arbitrary number, the following procedure is required.

Definition 8 (Second Quantization). Let H be the Hilbert space whose unit sphere corresponds to the possible pure quantum states of the system with a fixed number of particles. The Fock space F H : 8 Hpnq where, Hp0q : C and Hpnq : H H (n times tensor product) is the p q “ n“0 “ “ b¨¨¨b vector spaceÀ representing the states of a quantum system with a variable number of particles. The pnq vector Ω0 : 1, 0 ... F H is called the vacuum vector. Given ψ F H , we write ψ for the “ p q P p q P p q pnq pnq orthogonal projection of ψ onto H . The set F0 consisting of those ψ such that ψ 0 for all “ sufficiently large n is a dense subspace of the Fock space, called the space of finite particles. The symmetrization and anti-symmetrization operators

1 Sn ψ1 ψn : ψ σ p b¨¨¨b q “ n! σp1q b¨¨¨b σpnq σPSn ÿ (60) 1 sgnpσq An ψ1 ψn : 1 ψ σ p b¨¨¨b q “ n! p´ q σp1q b¨¨¨b σpnq σPSn ÿ

pnq pnq extend by linearity to H and are projections. The state space for n fermions is defined as Ha : “ n pnq n An H and that for n bosons as Hs : Sn H . The Fermionic Fock space is defined as p q “ p q

8 pnq Fa H : Ha , (61) p q “ n“0 à and the Bosonic Fock space as 8 pnq Fs H : Hs . (62) p q “ n“0 à A unitary operator U : H H can be uniquely extended to a unitary operator Γ U : F H F H Ñ p q p q Ñ p q as n Γ U Hpnq : U. (63) p q| “ j“1 â A selfadjoint operator A on H with dense subspace D A H can be uniquely extended to a selfadjoint p qĂ

20 operator dΓ A on F H as closure of the essentialy selfadjoint operator p q p q n dΓ A pnq : 1 ... 1 A 1 ... 1, (64) p q|DpdΓAqXH “ b b b j“1 j à loomoon where n pnq D dΓ A : ψ F0 ψ D A for each n . (65) p p qq “ # P ˇ P j“1 p q + ˇ ˇ â The operator dΓ A is called the second quantizationˇ of A. p q ˇ

It is easy to prove that the the spectrum of the second quantization can be infered from the spectrum of the first.

Proposition 3.4. Let A be a selfadjoint operator with a discrete spectral resolution, i.e. Aϕj λj ϕj , “ where λj j 0 R and ϕj j 0 is a o.n.B in H. Then, dΓ A pnq has a discrete spectral resolution t u ě Ă t u ě p q|H given by n

dΓ A Hpnq ϕi1 ϕin λij ϕi1 ϕin ij 0, 1 j n . (66) p q| b¨¨¨b “ ˜j“1 ¸ b¨¨¨b p ě ď ď q ÿ

If A is a selfadjoint operator with continuous spectrum spec A , then dΓ A pnq has a continuous cp q p q|H spectrum given by

n

specc dΓ A Hpnq λj λj specc A for 1 j n . (67) p p q| q“ # ˇ P p q ď ď + j“1 ˇ ÿ ˇ ˇ Definition 9 (Segal Quantization). Let f ˇH be fixed. For vectors in Hpnq of the form η P “ ´ pnq pn´1q ψ1 ψ2 ψn we define a map b f : H H by b b¨¨¨b p q Ñ

´ b f η : f, ψ1 ψ2 ψn. (68) p q “ p q b¨¨¨b

The expression b´ f extends by linearity to a bounded operator on F H . The operator N : dΓ I p q p q “ p q is termed number operator and a´ f : ?N 1 b´ f (69) p q “ ` p q

´ ˚ is called the annihilation operator on Fs H . Its adjoint, a f is called the creation operator. p q p q Finally, the real linear (but not complex linear) operator

1 ´ ´ ˚ ΦS f : a f a f (70) p q “ ?2 p q` p q ` ˘

21 is termed Segal field operator, and the map

ΦS : H O Fs H Ñ p p qq (71) f ΦS f ÞÑ p q the Segal quantization over H.

Theorem 3.5. Let H be a complex Hilbert space and ΦS the corresponding Segal quantization. Then:

(a) For each f H the operator ΦS f is essentially selfadjoint on F0. P p q

(b) The vacuum Ω0 is in the domain of all finite products ΦS f1 ΦS f2 ... ΦS fn and the linear span p q p q p q of ΦS f1 ΦS f2 ... ΦS fn Ω0 fi H,n 0 is dense in Fs H . t p q p q p q | P ě u p q

(c) For each ψ0 F0 and f,g H P P

ΦS f ΦS g ψ ΦS g ΦS f ψ ıIm f,g ψ p q p q ´ p q p q “ p q ı (72) exp ıΦS f g exp Im f,g exp ıΦS f exp ıΦS g . p p ` qq “ ´2 p q p p qq p p qq ´ ¯

(d) If fn f in H, then: Ñ

ΦS fn ΦS f p q Ñ p q (73) exp ıΦS fn exp ıΦS f p p qq Ñ p p qq

(e) For every unitary operator U on H, Γ U : D ΦS f D ΦS Uf and for ψ D ΦS Uf and p q p p qq Ñ p p qq P p p qq for all f H P ´1 Γ U ΦS f Γ U ψ ΦS Uf ψ. (74) p q p q p q “ p q

Proof. See Theorem X.41 in [RS75].

3.2.2 Wick Products and Wiener-Itˆo-Segal Isomorphism

Definition 10 (Wick Products of Random Variables). Let Ω, A,µ be a probability space and p q n f : Ω R a random variable with finite moments. Then, for n N0, the random variable : f :, Ñ P termed nth Wick power of f is defined recursively by

: f 0 : 1, “ (75) B : f n : n : f n´1 : and Eµ : f n : 0 for n 1. f “ r s“ ě B

22 n1 nk Let f1,...,fk : Ω R be random variables with finite moments. The Wick product : f ...f : is Ñ 1 k defined recursively in n n1 nk by “ `¨¨¨`

0 0 : f1 ...fk : 1, “ (76) n1 nk n1 ni´1 nk Eµ n1 nk B : f1 ...fk : ni : f1 ...fi ...fk : and : f1 ...fk : 0 for n 1. fi “ r s“ ě B Definition 11 (Wick Products of Segal Fields). Let H be a Hilbert space, and for any f H P let ΦS f be the Segal field on the bosonic Fock space Fs H . Then, for f,f1,...,fk H the Wick p q p q P product :ΦS f1 ... ΦS fk : is defined recursively by p q p q

:ΦS f : ΦS f , p q “ p q k k 1 k (77) :Φ f Φ f : Φ f : Φ f : f,f : Φ f : S S j S S j 2 j H S j p q j“1 p q “ p q j“1 p q ´ j“1p q q i‰j p q ź ź ÿ ź

2 1 N Proposition 3.6 (Wiener-Itˆo-Segal Isomorphism). The space Fs H is isomorphic to L S R , dµ p q p p q q 2 1 N and the isomorphism θW : Fs H L S R , dµ , termed Wiener-Itˆo-Segal isomorphism, satisfies p q Ñ p p q q the following properties:

(i) θW Ω0 1, “

pnq 2 1 N (ii) θW Hs L S R , dµ , “ np p q q

´1 1 N N (iii) θW ΦS f θ Φ Φ f for all Φ S R and f S R , p q W “ p q P p q P p q

2 1 N where Ω0 1, 0, 0,... Fs H and L S R , dµ is the closure of all linear combinations of Wick “ p qP p q np p q q products of random variables over S1 RN up to order n. p q

Proof. See Proposition 5.7 in [BHL11].

3.3 Osterwalder-Schrader Axioms

Osterwalder and Schrader (see [OS73], [OS73Bis]) utilized the Wick rotation technique to pass from the Minkowskian to the Euclidean space and formulate axioms equivalent to Wightman in terms of Euclidean Green functions. In [OS75] they defined free Bose and Fermi fields and proved a Feynman- Kaˇcformula for boson-fermion models.

The dynamics of the quantized system satisfying Wightman axioms is given by the Schr¨odinger, or, equivalently by the heat equation, where an H unbounded, selfadjoint Hamilton operator H on

23 a physical Hilbert space H appears. The Hamilton operator can be extracted from the Osterwalder- Schrader axioms and ideally coincides with an operator obtained by a quantization procedure of the Hamiltonfunction in the classical description of the physical system.

Following chapter 6 of [GJ87] we introduce the

Definition 12 (Osterwalder-Schrader Axioms). The primitive of a quantum field model is a Borel probability measure dµ on S1 RN , whose inverse Fourier transform gives the generating functional p q

S f : eiΦpfqdµ Φ , (78) p q “ 1 RN p q żS p q where f S RN . Formally, we write Φ f Φ x f x dN x. P p q p q“ RN p q p q ş (OS0) Analyticity: The functional S f is entire analytic. For every finite set of test functions p q N n fj S R , j 1,...,n and complex numbers z : z1,z2,...,zn C , the function P p q “ “ p qP

n z S zj fj (79) ÞÑ ˜j“1 ¸ ÿ is entire on Cn.

(OS1) Regularity: For some p 1, 2 , some constant c and for all f S RN Pr s P p q

p cp}f} 1 N `}f} q S f e L pR q LppRN q . (80) | p q| ď

(OS2) Euclidean Invariance: The functional S f is invariant under Euclidean symmetries E of p q RN . This means that for any translation, rotation and reflection, for all f S RN P p q

S Ef S f , (81) p q“ p q

where Ef x : f E´1 x . p q “ p p qq (OS3) Reflection Positivity: Let

n Φpfj q N L : ψ ψ Φ cje ,cj C,fj S R , j 1,...,n (82) “ # ˇ p q“ P P p q “ + ˇ j“1 ˇ ÿ ˇ be the algebra of exponentialˇ functionals on tempered distributions and

n Φpfj q N N L` : ψ ψ Φ cj e ,cj C,fj S R , supp fj t, x R t 0 (83) “ # ˇ p q“ P P p q p qĂtp qP | ą u+ ˇ j“1 ˇ ÿ ˇ ˇ 24 the subalgebra of exponential functionals whose defining functions are supported in the positive time half space. Euclidean transforms E on RN act on S1 RN via p q

Eψ Φ : ψ EΦ and EΦ f : Φ Ef , (84) p qp q “ p q p q “ p q

for all ψ L, Φ S1 RN , f S RN . P P p q P p q We assume that the time reflection

t, x θ t, x : t, x (85) p q ÞÑ p q “ p´ q

satisfies θΨ Φ Ψ¯ Φ dµ Φ 0. (86) 1 RN p q p q p qě żS p q for all Ψ L . P `

(OS4) Ergodicity: The Euclidean time translation subgroup T t t 0 acts ergodically on the mea- t p qu ě sure space S1 RN , dµ , i.e. p p q q 1 t lim T s Ψ Φ T s ´1ds Ψ Φ dµ Φ , (87) tÞÑ`8 t p q p q p q “ 1 RN p q p q ż0 żS p q

for all Ψ L1 S1 RN , dµ . P p p q q

From the Osterwalder-Schrader axioms we can reconstruct quantum field theory as described by the Wightman axioms and derive the quantum mechanical dynamics. Table 2 depicts the formal scheme of this construction. The proper definition of the quantum mechanical Hilbert space relies on the reflection positivity axiom. The exponential functionals L are dense in E : L2 S1 RN , dµ . Let E “ p p q q ` be the closure of L` in E, termed as positive time subspace of E, and define the bilinear form on E E ` ˆ ` b Ψ, Υ : Ψ Φ Υ Φ dµ Φ . (88) p q “ 1 RN p q p q p q żS p q By OS3 the bilinear form b is positive semidefinite. Let p q

N : ψ E b Ψ, Ψ 0 (89) “t P ` | p q“ u

the subspace of vectors for which the bilinear form degenerates, and define the quantum mechanical Hilbert space as H : E N , (90) “ `{

25 Path space S1 RN p q

Configuration space S1 RN´1 p q

Measure on path space dµ

Measure on configuration space dν dµ t: 0 “ | “

Path space for quantum operators E : L2 S1 RN , dµ “ p p q q

Physical Hilbert space H L2 S1 RN´1 , dν – p p q q

Table 2: Quantum field theory from Osterwalder-Schrader axioms

with hermitian scalar product defined by (88) independently of the representative of the equivalence class. We denote the canonical embedding of the positive time subspace unto the quantum mechanical Hilbert space as

:E` H N Ñ (91) Ψ ΨN : Ψ N , ÞÑ “ `

N and transfer operators S acting on E` to operators S acting on H by

SNΨN : SΨ N, (92) “ p q

which is well defined for all Ψ E , if P `

S : D S E E and S : D S N N . (93) p qX ` Ñ ` p qX Ñ

Theorem 3.7 (Reconstruction of quantum mechanics). If the probability measure µ on S1 RN satisfies p q the reflection and time translation invariance axiom (OS2), and the reflection positivity axiom (OS3), then for all t 0 the time translation T t satifies (93) and ě p q

T t N e´tH , (94) p q “

26 ˚ N where H H 0 is a (possibly unbounded) selfadjoint operator on H with ground state Ω0 : 1 , “ ě “ i.e. HΩ0 0. “

Proof. See Theorem 6.13 in [GJ87].

The reflection positivity axiom is not easy to verify. The following proposition provides a useful criterion.

Proposition 3.8. The probability measure µ on S1 RN satisfies the reflection positivity axiom (OS3) p q if and only if the matrix M : S θ fi fj has positive eigenvalues for all choices of fi i 1,...,n “ r p p q´ qs p q “ Ă S RN with support in the time positive half space. p q

Proof. See Corollary 3.4.4 in [GJ87].

Gaussian measures play a prominent role in quantum field theory, because they originate free fields.

Definition 13. A linear operator C defined on RN satisfies the reflection positivity property if and only if

θf,Cf 2 RN N 0, (95) p qL p ,d xq ě for all f RN supported at positive times. P Theorem 3.9. A Gaussian measure on the space of tempered distributions satifies reflection positivity if and only if its covariance operator does.

Proof. See Theorem 6.22 in [GJ87].

2 ´1 2 Proposition 3.10. The covariance operator C : ∆RN m is reflection positive for all m 0. “ p´ ` q ą

Proof. See Proposition 6.2.5 in [GJ87].

4 Quantization of Yang-Mills Equations and Positive Mass Gap

There are several methods of designing a quantum theory for non-Abelian gauge fields. The Hamilto- nian formulation is the approach used in the original work by Yang-Mills ([MY54]), which was later abandoned in favour of an alternative method based on Feynman path integrals ([FP67]). When it became clear that the Faddeev-Popov method must be incomplete beyond perturbation theory, Hamil- tonian formulation enjoyed a partial renaissance. More recent, didactically accessible, examples of the Hamiltonian approach in the physical literature can be found in [Sch08].

27 In the first section of this chapter we construct a quantized Yang-Mills theory in dimension 3 1 and ` in the second we compute spectral lower bounds for the Hamilton operator. In the third we summarize our findings and prove the main result, Theorem 4.13.

4.1 Quantization

In the Yang-Mills 3 1 dimensional set up, in order to account for functionals on transversal fields ` as required by the Coulomb gauge, we introduce the configuration space S1 R4, RKˆ3 and the path Kp q space S1 R3, RKˆ3 . These are the duals in the sense of nuclear spaces of the test functions satisfying Kp q the transversal condition:

L2 R3, RKˆ3, d3x : A L2 R3, RKˆ3, d3x Ak 0 , Kp q “t P p q “ u S R3, RKˆ3 : S R3, RKˆ3 L2 R3, Rˇ Kˆ3, d3x (96) Kp q “ p qX Kp ˇ q S R4, RKˆ3 : f S R4, RKˆ3 f t, L2 R3, RKˆ3, d3x for all t R Kp q “t P p q p ¨q P Kp q P u ˇ Not that we do not need to bother about the timeˇ component of the connection, because it vanishes in the Coulomb gauge. We define the Hilbert space E : L2 S1 R4, RKˆ3 , dµ and the physical Hilbert “ p Kp q q space as H : L2 S1 R3, RKˆ3 , dν for appropriate probability measures µ and ν. “ p Kp q q We introduce the Hamilton operator originated by the quantization of the Hamiltonian formulation of Yang-Mills equations. This operator is the infinitesimal generator of a time inhomogenoeus Itˆo’s diffusion, whose probability density solves the heat kernel equation. We construct a probability measure on the tempered distributions using this Itˆo’s process as integrator and utilizing the Feynman-Kaˇc formula. We prove that the necessary Osterwalder-Schrader axioms for the Hamilton operator to be selfadjoint on the probability space of the time zero tempered distributions are fulfilled. Then, we verify that the Wightman axioms are satisfied.

When we try to introduce the canonical quantization for the Hamilton function H in (39), we face the

problem for HII and V that the classical fields A t, x cannot be directly interpreted as multiplication p q operators in L2 S1 R3, RKˆ3 , dν , because the multiplication of distributions cannot be properly p Kp q q defined. To circumvent this issue, following the idea expressed f.i. in [Sim05] (page 257) and applied to Nelson’s model in [BHL11] (page 297) in the case of ultraviolet divergence, we introduce a cut off test function to regularize fields.

Definition 14 (Ultraviolet Cut Off). A test function ϕΛ S R4, RKˆ3 is called ultraviolet cut P Kp q off for the cut off level Λ 0 if and only if for all t R the Fourier transform of ϕΛ x : ϕΛ t, x ě P t p q “ p q P

28 S R3, RKˆ3 satisfies Kp q

ϕˆΛ p 0, 1 for all p R3 t p qPr s P ϕˆΛ p 1 p Λ (97) t p q“ p| |ď q supp ϕˆΛ R3. p t q ĂĂ

1 Λ 1 3 Kˆ3 Note that for all t R in the limit we have S limΛ ϕ δ S R , R and that P ´ Ñ`8 t “ P Kp q A ϕΛ x is a polynomially bounded function in x R3. Theorem 2.5 can be now quantized as p t p¨ ´ qq P follows.

Proposition 4.1. Let H H A, E be the Hamilton function of the Hamilton equations equivalent to “ p q the Yang-Mills equations as in Theorem 2.5 for a simple Lie-group as structure group. Let us consider a cut off Λ 0 and the cut off test function ϕΛ S R4, RKˆ3 . For a probability measure ν on ě P Kp q S1 R3, RKˆ3 the canonical quantization of the position variable A, of the momentum variable E and Kp q of the Hamilton function H means the following substitution:

A C8 R3, RKˆ3 A O L2 S1 R3, RKˆ3 , dν P p q ÝÑ P p p Kp q qq 8 3 Kˆ3 1 δ 2 1 3 Kˆ3 E C R , R O L SK R , R , dν (98) P p q ÝÑ ı δA P p p p q qq 1 δ H C8 R2pKˆ3q, R HΛ : H A ϕΛ , O L2 S1 R3, R , dν , P p q ÝÑ “ p t p¨ ´ ¨qq ı δA ϕΛ P p p Kp q qq ˆ p t p¨ ´ ¨qq ˙ where O denotes the set of all linear operators over the corresponding underlying vector space. The p¨q cut off Hamilton operator HΛ for the quantized Yang-Mills equations reads

HΛ HΛ HΛ V Λ, (99) “ I ` II `

29 where

2 Λ 1 3 δ HI Ψ A d x a Λ Ψ A p qp q“´2 R3 δA ϕ x p q ż „ i p t p¨ ´ qq 

2 2 Λ g 3 3 a,b Λ b,c,d d Λ δ HII Ψ A d x d y iG A ϕt y ; x, y ε Ak ϕt y c Λ Ψ A p qp q“´ 2 R3 R3 B p p p¨ ´ qq q p p¨ ´ qqδA ϕ y p q ż ż „ kp t p¨ ´ qq

V ΛΨ A d3x : V Λ t, x, A :Ψ A , p qp q“ R3 p q p q ż

Λ 1 j,k p,q a Λ a Λ V t, x, A : ε ε jA ϕ x kA ϕ x p q “ 16 i i rpB kpp t p¨ ´ qq´B j pp t p¨ ´ qq` gεa,b,cAb ϕΛ x Ac ϕΛ x ` j pp t p¨ ´ qq kpp t p¨ ´ qqq a Λ a Λ pA ϕ x qA ϕ x pB q pp t p¨ ´ qq´B ppp t p¨ ´ qq` gεa,b,cAb ϕΛ x Ac ϕΛ x , ` ppp t p¨ ´ qq q qpp t p¨ ´ qqs (100)

with the domain of definition

D HΛ : Ψ L2 S1 R3, R , dν HΛΨ L2 S1 R3, R, dν . (101) p q “ P p Kp q q P p Kp q ˇ ( ˇ Note that the time t is considered as a parameter in the expressions (98)and (100). The ground state N Kˆ3 Ω0 : 1 1RKˆ3 H, where 1RKˆ3 is the matrix in R with ones as entries everywhere, satisfies “ P

Λ Λ Λ H Ω0 HI Ω0 H Ω0 V Ω0 0. (102) “ “ II “ “

Proof of Proposition 4.1. It is a straightforward consequence of Corollary 2.6 where we introduce the Λ quantization specified by (98). The ground state Ω0 is the eigenvector of V for the eigenvalue 0 because the expectation of any Wick power vanishes.

Remark 4.1. The derivative

δ d δ Ψ A Ψ A εϕΛ x Ψ A Λ (103) δAa ϕΛ x p q“ dε p ` t p¨ ´ qqq ÝÑ δAa t, x p q p Ñ `8q i t ˇε:“0 i p p¨ ´ qq ˇ p q ˇ a Λ ˇ a 1 Λ in line with fact that A ϕ x tends to A t, x , because S limΛ ϕ x δ x for i p t p¨ ´ qq i p q ´ Ñ`8 t p¨ ´ qq “ p¨ ´ q all t.

30 Remark 4.2. The operator HΛ tends for Λ to the Laplace operator in infinite dimensions for I Ñ `8 functionals of the potential fields. The operator V Λ is a multiplication operator corresponding to the Λ Λ fibrewise multiplication with the square of the connection curvature. Both operators HI and V do not vanish if g 0 and do not contribute to the existence of a mass gap. The operator HΛ vanishes if “ II g 0, f.i. when G is an abelian groups, and, as we will see, is responsible for the existence of a mass “ gap.

Remark 4.3. In the physical literature (see f.i. [Sch08]) the operator C C A; x, y “ p q

Ca,b A; x, y : d3yG¯ a,c A; x, y¯ ∆Gc,b A; y, y¯ (104) p q “´ R3 p q p q ż is termed Coulomb operator and G G A; x, y is also called the Faddeev-Popov operator. Note that “ p q they make sense only after the substitution

A t, x A ϕΛ x , (105) p q Ñ p t p¨ ´ qq

and passing to the limit Λ . Ñ `8 Theorem 4.2. If the bare coupling constant g 0 is small enough, there exists a probability measure ě Λ 1 4 Kˆ3 Λ Λ Λ µ on S R , R such that the Hamilton operator H is selfadjoint for the choice ν : µ t: 0. Kp q “ | “ If the coupling constant g vanishes, both measures µ and ν are Gaussian, otherwise not.

To prove this theorem we need a result about infinitesimal generators of time inhomogeneous Itˆo’ s diffusions.

Proposition 4.3. Let Wt t 0 be a M-dimensional standard Brownian motion with respect to the p q ě N N N NˆM filtration At t 0. Let b : 0, R R , σ : 0, R R be Borel measurable and p q ě r `8rˆ Ñ r `8rˆ Ñ locally bounded functions, satisfying

b t, x b t,y RN σ t, x σ t,y RNˆM K x y RN (106) | p q´ p q| ` | p q´ p q| ď | ´ |

for a positive constant K, meaning Lipschitz-continuity with respect to x uniform in t. The solution of the SDE

dXt b t,Xt dt σ t,Xt dWt “ p q ` p q (107) $ N X0 x0 R , & “ P is a time inhomogeneous Itˆo’s diffusion,% whose infinitesimal generator is given by the PDO with variable

31 coefficients

1 N 2 N Lt ai,j t, x B bi t, x B “ 2 p q x x ` p q x i,i“1 i j i“1 j (108) ÿ B B ÿ B 8 N N 2 N N N 2 N N N D Lt : C R , R L R , R , d x L R , R , d x , p q “ 0 p qĂ p q Ñ p q where a t, x : σ t, x σ t, x :. (109) p q “ p q p q

1 Conversely, if the operator Lt is elliptic for all t 0, then for σ t, x : a 2 t, x and M N, there ě p q “ p q “ exists an Itˆo’s diffusion as (114) whose transition density κt x0, x is the heat kernel of Lt, i.e. the p q solution of B u t, x Ltu t, x Bt p q“ p q (110) $ u 0, x δ x S1 RN , RN . & p q“ p qP p q It follows that the solution of %

B u t, x Ltu t, x Bt p q“ p q (111) $ u 0, x f x C8 RN , RN & p q“ p qP 0 p q

is given by % N u t, x0 E f Xt At f x κt x0, x d x (112) p q“ r p q| s“ RN p q p q ż Proof. See chapters 8.3-8.5 of [CCFI11] and chapters VII.1-VII.2 [RJ99].

What is the situation in the infinite dimensional case?

Proposition 4.4. Let F and G be a real separable Hilbert spaces and LHS G, F denote the vector space p q of all Hilbert-Schmidt operators from G to F. Let Wt t 0 be a standard Wiener process with respect to p q ě the filtration At t 0 taking values G and assume that b : 0, F F, σ : 0, F LHS G, F p q ě r `8rˆ Ñ r `8rˆ Ñ p q be Borel measurable and locally bounded functions, satisfying

b t, x b t,y σ t, x σ t,y K x y (113) | p q´ p q|F ` | p q´ p q|LHSpG,Fq ď | ´ |F for a positive constant K, meaning Lipschitz-continuity with respect to x uniform in t. The (strong) solution of the SDE

dXt b t,Xt dt σ t,Xt dWt “ p q ` p q (114) $ X0 x0 F & “ P is a time inhomogeneous Itˆo’s diffusion.%

32 Proof. It is a special case of Theorem 7.4 of [DZ92] for identity covariance of the Wiener process and vanishing linear operator in the drift.

We can now proceed with the

Proof of Theorem 4.2. Inspired by the treatment of the quantum field associated to a particle in a potential as depicted in chapter 3 of [GJ87], we adapt the ideas therein to the Yang-Mills fields with a cutoff. Using the Feynman-Kaˇcformula, we construct probability measures on E and H satisfying those Osterwalder-Schrader axioms implying the reconstruction theorem of quantum mechanics, and thus the selfadjointness and non-negativity of the cut off Hamilton operator. If we exclude for the moment the part of the Hamiltonian containing the potential, from Proposition 4.1 formula (100) we can write

Λ Λ 3 Λ,g HI HII d xh0 t, x ` “ R3 p q ż 2 Λ,g 1 δ h t, x : (115) 0 p q “´2 δAa ϕΛ x ` „ i p t p¨ ´ qq  2 2 g 3 a,b Λ b,c,d d Λ δ d y iG A ϕt y ; x, y ε Ak ϕt y c Λ . ´ 2 R3 B p p p¨ ´ qq q p p¨ ´ qqδA ϕ y „ż kp t p¨ ´ qq Since

δ 2 δAa ϕΛ x “ „ i p t p¨ ´ qq  (116) ¯ ¯ δ δ lim d3y d3y¯ ψΞ y x ψΞ y¯ x δdδk , d k d Λ d¯ Λ “´ ΞÑ`8 R3ˆR3 p ´ q p ´ q δA ϕt y δA ϕ y¯ ż kp p¨ ´ qq k¯p t p¨ ´ qq

Ξ 3 1 Ξ 1 3 Ξ where ψ Ξ 0 S R is a delta sequence, i.e. S limΞ ψ δ S R , for which ψ 0 for p q ě Ă p q ´ Ñ`8 “ P p q ą all Ξ, and the limit is pointwise on the domain of definition of the operatoronthel. h. s. of(116), we can express the operator hΛ,g t, x as 0 p q

Λ,g 3 d Λ δ h0 t, x lim d ybk Ξ,x,g; A ϕt y d Λ p q“´ ΞÑ`8 R3 p p p¨ ´ qqqδA ϕ y ` „ż kp t p¨ ´ qq 3 3 d,d¯ Λ Λ δ δ d y d ya¯ ¯ Ξ, x, y, y,g¯ ; A ϕt y , A ϕt y¯ ¯ , 3 3 k,k d Λ d Λ ` R ˆR p p p¨ ´ qq p p¨ ´ qqqδAk ϕt y δA¯ ϕt y¯ ff ż p p¨ ´ qq kp p¨ ´ qq (117)

for appropriate matrix a Ξ, x, y, y,g¯ ; A t,y , A t, y¯ and vector b Ξ,x,y,g; A t,y valued coefficient p p q p qq p p qq functions. The limit is pointwise on the domain of definition of hΛ,g t, x . Note that we have written 0 p q

33 the matrix A t,y A ϕΛ y (118) p q“ p t p¨ ´ qq in the equivalent column vector form. Furthermore, equation (117) becomes

Λ,g δ δ δ h0 t, x lim b Ξ,x,g; A , a Ξ,x,g; A , , (119) p q“´ ΞÑ`8  p q δA `  p qδA δA 

3 r r where hb1,b2i : 3 d y b y b y for a b, which a function of y and a a which is an operator on “ R 1p q 2p q the functions of y.ş Both b and a are functionals of A and depend on the parameters x and Ξ. There exists a g0 0, 1 not depending on Λ, such that, if the coupling constant g 0,g0 , then for all x P r r Ps r the expression a Ξ,x,g; A represents a positive definite operator valued functional of A A t,y seen p q “ p q as R3K -valued function of y in the separable Hilbert space L2 R3, R3K . By Proposition 4.4, we can p q construct the diffusion At At Ξ,x,g; y “ p q

1 2 dAt b Ξ,x,g; At dt a Ξ,x,g; At dWt, (120) “ p q ` p q

2 3 3K 1 where Wt tě0 is the standard Wiener process in the Hilbert space L R , R . Note that a 2 is p q p q 1 a Hilbert-Schmidt operator because a is a Hilbert-Schmidt integral operators. Both b and a 2 are Lipschitz-continuous with respect to A, because as C8-functions of A. The Lipschitz constant does 1 not depend on t, because b and a 2 do not either.

3 3K For fixed y R the R -valued process At Ξ,x,g; y is a Itˆodiffusion with transition probability P p q 1 2 1 1 3K 1 2 density κt Ξ,x,y,g; A ; A by Proposition 4.3, a function of A , A R , as well as κt x,y,g; A , A : p q P p q “ 1 2 limΞ κt Ξ,x,y,g; A , A . Therefore, Ñ`8 p q

1 2 3 1 2 κt x, g; A , A : d yκt x,y,g; A , A (121) p q “ R3 p q ż is a transition probability density function of A1, A1 R3K . P Let W A, A1,t,x be the set of continuous paths A s, x in R3K which take the values A t 2, x p q p q p´ { q“ A and A t 2, x A1 at their endpoints. The cylinder sets of Wx A, A1,t have the form p` { q“ p q

1 Z A, A , t, Ij j , x p t u q“ (122) 1 A s, x A t 2, x A, A t 2, x A , A tj , x Ij , for all j 1,...,n , “ p q p´ { q“ p` { q“ p qP “ ˇ ( ˇ 3K where t 2 t1 t2 tn t 2 and Ij are Borel subsets of R . On these cylinder sets we can ´ { ă ă 㨨¨ă ă {

34 define the measure given by

t,x,g 1 2 n 1 1 2 UA A1 Z : dA dA ... dA κ 1 x, g; A, A κt2 t1 x, g; A , A ... , p q “ t `t{2p q ´ p q żI1 żI2 żIn (123) ...κ x, g; An, A1 , t{2´tn p q which is countably additive and has a unique extension to the Borel subsets of W A, A1,t,x . Let p q ωΛ,t,x,g ωΛ,t,x,g A t, x be the ground state of hΛ,g t, x . The expression 0 “ 0 p p qq 0 p q

Λ,g 3 t,x,g Λ,t,x,g Λ,t,x,g 1 dξ : d x dUA,A1 ω0 A ω0 A (124) “ R3 R3K R3K p q p q ż ˆ ˆ defines a measure on S R4, RKˆ3 , which pushes forward to a measure ξΛ,g A on S1 R4, RKˆ3 . Kp q p q Kp q Now, as : V Λ : : V Λ,g : is non-negative, by the Feynman-Kaˇctheorem we can define “

t ` 2 Λ,g Λ,g Λ,g Zt : exp : V s, x, A : ds dξ “ S1 pR4,RKˆ3q ´ ´ t p q ż K ˜ ż 2 ¸ t (125) ` 2 Λ,g 1 Λ,g Λ,g dµt : Λ,g exp : V s, x, A : ds dξ “ Z ´ t p q t ˜ ż´ 2 ¸

as a measure on S1 R4, RKˆ3 with generating functional Kp q

Λ,g iApfq Λ,g St f e dµt A , (126) p q“ S1 pR4,RKˆ3q p q ż K for f S R4, RKˆ3 . The measures µΛ,g are absolutely continuous with respect to ξ for all t 0 and P Kp q t ě converge to a µΛ,g on every measurable subset of S1 for t , because Ñ `8

t `8 Λ,g ` 2 V s, x, ds 1 exp ´8 : A : lim exp : V Λ,g s, x, A : ds ´ p q . Λ,g `8 tÑ`8 Z ´ ´ t p q “ ´ ş Λ,g ¯ Λ,g t ˜ ż 2 ¸ S1 pR4,RKˆ3q exp ´8 : V s, x, A : ds dξ K ´ p q ş ´ ş ¯ (127) By the Vitali-Hahn-Saks theorem we conclude that µΛ,g is a probability measure on S1 R4, RKˆ3 E, Kp q“ and its Fourier transform, the functional SΛ,g f satifies (i)-(iii) of Milnos’ Theorem (Theorem 3.3). p q Λ,g Λ,g 1 3 Kˆ3 Now we have to show that ν : µ t: 0 is a probability measure on S R , R such that “ | “ Kp q HΛ,g is a selfadjoint operator on H. The invariance of the generating functional SΛ,g under time translation and time reflection follows directly from the definition of µΛ,g. The proof that SΛ,g satisfies

reflection positivity for g 0,g0 requires more work. We have to show that the complex matrix P r r M Λ,g : M Λ,g , where “r i,j s Λ,g Λ,g M : S θfi fj , (128) i,j “ p ´ q

35 4 3 is positive definite for all choices of fi i 1,...,n S R , R , such that supp fi 0, R , and p q “ Ă p q p qĂr `8rˆ θf t, x : f t, x denotes the time reflection. p qp q “ p´ q We remark that, according to the representation (100),

Λ,g 3 Λ,g H d xHx , (129) “ R3 ż where HΛ,g : hΛ,g t, x : V Λ,g s, x, A : (130) x “ 0 p q` p q is a selfadjoint unbounded operator on the Hilbert space L2 R3, RKˆ3 with fundamental state ΩΛ,g p q x “ Λ,g Λ Λ 3 Ω t , i.e. H Ω 0. Note that t is seen as parameter. Let x R be fixed and tk k 1,...,N a x p q x x “ P p q “ T ´τ partition of the interval τ,T defined as tk : τ k . By Theorem 3.4.1 in [GJ87], which holds true r s “ ` N beyond the Gaussian case, we can write

´pt2´t1qHx ´pt3´t2qHx N Ωx,B1e B2e B3 BN Ωx Πk“1Bk A tk, x dµ A , x , (131) x ¨¨¨¨¨ y“ R p p qq p p¨ qq ż where:

• the expression µ A , x represents the probability measure on S1 R1, RKˆ3 induced by µΛ,g p p¨ qq Kp q along the timeline,

3K • the scalar product h , i is defined as hϕ, ψi : 3 d a ϕ a ψ a , ¨ ¨ “ RKˆ p q p q

2 ş Kˆ3 • the functions Bk Bk a k 1,...,N are in L R , C . p “ p qq “ p q

We now choose the functions Bk as

ı arθpfiqptk,xq´fj ptk ,xqs Bk a : e N , (132) p q “ leading for the r.h.s. of (131) to

N ´ıAp¨,xqpθpfiqqp¨,xq´fj p¨,xqq lim Πk“1Bk A tk, x dµ A , x e dµ A , x , (133) NÑ`8 R p p qq p p¨ qq “ S1 pR1,RKˆ3q p p¨ qq ż ż K when τ and T . We have approximated the time integration in A f by means of a Ñ ´8 Ñ `8 p q Riemann sum for the partition tk k 1,...,N . For the l.h.s of (131) we obtain in the limit p q “

´pt2´t1qHx ´pt3´t2qHx ıa dt rθpfiq´fj spt,xq lim Ωx,B1e B2e B3 BN Ωx Ωx,e R Ωx NÑ`8x ¨¨¨¨¨ y“x ş y“ (134) `8 8 ´ıa dtθpfiqpt,xq ´ıa dt fj pt,xq ´ıa dt fipt,xq ´ıa dt fj pt,xq e R Ωx,e R Ωx e 0 Ωx,e 0 Ωx , “x ş ş y“x ş ş y

36 because the supports of fi and fj lie in the time positive half space. Putting (131) with (133) and (134) together shows that the matrix M Λ,g x with entries r i,j p qs

Λ,g ´ıAp¨,xqppθpfiqqp¨,xq´fj p¨,xqq Mi,j x : e dµ A , x p q “ S1 pR1,RKˆ3q p p¨ qq “ ż K (135) `8 8 ´ıa dtfj pt,xq ´ıa dtfipt,xq e 0 Ωx,e 0 Ωx , “x ş ş y is positive definite, (providing an explicit proof of Corollary 3.4.4 in [GJ87] for x fixed). g,Λ By Fubinis’s theorem, the matrix entries of Mi,j can be written as

Λ,g iApfq Λ,g Mi,j e dµ A “ S1 pR4,RKˆ3q p q“ ż K A d3x ei p¨,xqpθpfiq´fj qp¨,xqdµΛ,g A , x (136) “ R3 S1 pR3,RKˆ3q p p¨ qq “ ż ż K 3 Λ,g d xMi,j x . “ R3 p q ż

Λ,g Λ,g Hence, being M continuous, the matrix M must be positive definite for all g 0,g0 , and, by i,j p¨q Pr r Proposition 3.8 (or Corollary 3.4.4 in [GJ87]), the reflection positivity axiom (OS3) is fullfilled.

We conclude that all the assumptions of Theorem 3.7 are satisfied and, hence, the time translation operator T t satisfies p q Λ ˜ Λ T t N e´tH , (137) p q “ H˜ Λ is a selfadjoint operator on the Hilbert space L2 S1 R3, RKˆ3 , dνΛ . Note that the canonical Kp q embedding is defined utilizing the measure µΛ, which` is highlighted by˘ the superscript Λ in the N notation Λ. To conclude the proof we have to show that HΛ H˜ Λ. Let x R3 be fixed. A slight N “ P reformulation of Theorem 3.4.1 in [GJ87] provides the equality

Λ Λ ´sHx Λ Λ Ωx , Ψ1e Ψ2Ωx Ψ1 A , x δ0 Ψ2 A , x δs dµx A , x , (138) “ S1 pR1,RKˆ3q p p¨ qp qq p p¨ qp qq p p¨ qq D E ż K where:

• the distribution A , x S R1, RKˆ3 depends on the parameter x R3, p¨ qP Kp q P • the notation n A , x δs : lim A , x ϕs , (139) p¨ qp q “ nÑ`8 p¨ qp q

n 1 Kˆ3 1 n where ϕ n 0 is a delta-sequence in S R , R , i.e. S limn ϕ δs. is a practical p s q ě Kp q ´ Ñ`8 s “ shorthand.

37 2 1 3 Kˆ3 Λ • the functionals Ψ1, Ψ2 are in L S R , R , dν . p Kp q q

We now integrate both sides of equation (138) with respect to x and obtain on the right hand side

3 Λ d x Ψ1 A , x δ0 Ψ2 A , x δs dµx A , x R3 S1 pR1,RKˆ3q p p¨ qp qq p p¨ qp qq p p¨ qq “ ż ż K Λ Ψ1 A δ0 Ψ2 A δs dµ A “ S1 pR4,RKˆ3q p p qq p p qq p q“ ż K Λ Ψ1 A δ0 Ψ2 A δs dµt A dλ A “ S1 pR4,RKˆ3q p p qq p p qq p q p q“ ż K Λ ´tH˜ Λ Ψ1 A δ0 Ψ2 A δs e dν A dλ A (140) “ S1 pR4,RKˆ3q p p qq p p qq p q p q“ ż K Λ Λ ´tH˜ ´tH˜ Λ Ψ1 e A δ0 Ψ2 e A δs dν A dλ A “ S1 pR4,RKˆ3q p q p q p q p q“ ż K ´ ¯ ´ ¯ Λ ´sH˜ Λ Ψ1 A 0, Ψ2 e A 0, dν A 0, “ S1 pR3,RKˆ3q p p ¨qq p ¨q p p ¨qq “ ż K ´ ¯ Λ ´sH˜ Λ Ψ1 A e Ψ2 A dν A , “ S1 pR3,RKˆ3q p q p q p q ż K where we have utilized the integration mean value theorem for the dλ integration and the fact that dλ 1, being λ a probability measure. “ Weş now take the derivative d on both sides of the integrated version of (138) and obtain ´ ds s:“0 ˇ ˇ Λ Λ Ψ1 A H˜ Ψ2 A dν A S1 pR3,RKˆ3q p q p q p q“ ż K 3 Λ Λ Λ d x Ωx , Ψ1Hx Ψ2Ωx “ R3 “ ż 3 3K Λ Λ Λ d x d a Ωx a Ψ1 a Hx Ψ2 a Ωx a “ R3 RKˆ3 p q p q p q p q“ ż ż (141) 3K 3 Λ Λ 2 d a d x Ψ1 a Hx Ψ2 a Ωx a “ RKˆ3 R3 p q p qp p qq “ ż ż 3 Λ 3 Λ 2 3K Ψ1 a d xHx Ψ2 a d x Ωx a d a “ RKˆ3 p q R3 p q R3 p p qq “ ż „ż  „ż  Λ Λ Ψ1 A H Ψ2 A dν A , “ S1 pR3,RKˆ3q p q p q p q ż K

3 Λ 2 Λ because 3 d x Ω a is the density of the probability measure ν . This can be seen by choosing R p x p qq s 0, Ψş1 1A, (the characteristic function of the set A) and Ψ2 1 in in the integrated version of “ “ ” (138) leading to Λ 3 Λ 2 3K ν A 1A a d x Ωx a d a. (142) p q“ RKˆ3 p q R3 p p qq ż „ż  Hence, by (141), we conclude that HΛ H˜ Λ and the proof is completed. “

38 Now we remove the ultra violet cut off by letting Λ and making sure that certain properties are Ñ `8 maintained.

Corollary 4.5 (Cut off removal). There exists a g0 0, 1 such that, if the bare coupling constant g Ps r P Λ Λ 0,g0 , then, for any choice of the cut off test function the probability measures µ and ν converge in the r r 1 4 Kˆ3 1 3 Kˆ3 limit of a sequence Λj to probability measures µ on S R , R , and ν on S R , R . The Ñ `8 Kp q Kp q cut off Hamiltonian HΛ converges to a selfadjoint non negative operator H on L2 S1 R3, RKˆ3 , dν . p Kp q q If the coupling constant g vanishes, both measures µ and ν are Gaussian, otherwise not. The domain of definition is D H : Ψ L2 S1 R3, R , dν HΨ L2 S1 R3, R, dν . (143) p q “ P p Kp q q P p Kp q ˇ ( Moreover, the operator H can be decomposed as ˇ

H HI HII V, (144) “ ` ` where

2 1 3 δ HI d x a “´2 R3 δA t, x ż „ i p q

2 2 g 3 3 a,b b,c,d d δ (145) HII d x d y iG A t,y ; x, y ε Ak t,y c “´ 2 R3 R3 B p p q q p qδA t,y ż ż „ kp q 

1 A V d3x R∇ t, x 2 “ 2 R3 | p q| ż

2 3 Kˆ3 3 for A L R , R , d x . The ground state Ω0 satisfies P Kp q

HΩ0 HI Ω0 HII Ω0 V Ω0 0. (146) “ “ “ “

4 Kˆ3 Proof. For any f S R , R and g 0,g0 as in the proof of Theorem 4.2, the generating P Kp q P r r functional C Λ iApfq Λ,g S f e dµ A B1 0 : z C z 1 , (147) p q“ S1 pR4,RKˆ3q p qP p q “t P | | |ď u ż K because the integrand lies on the unit circle and µΛ is a probability measure. We define now

C Λ S f : lim sup S f B1 0 , (148) p q “ ΛÑ`8 p qP p q

39 which can be see to satisfy properties (i)-(iii) in Milnos’ Theorem (Theorem 3.3). Therefore, if we can prove that S satisfies properties (i)-(iii) in Theorem 3.3, there exists a probability measure µ on S1 R4, RKˆ3 satisfying Kp q A S f ei pfqdµ A , (149) p q“ S1 pR4,RKˆ3q p q ż K for all f S R4, RKˆ3 . We proceed as follows to prove the assumptions of Milnos’ theorem: P Kp q

(i) : Since SΛ 0 1 holds true for all Λ 0, this property is maintained by passing to lim sup , p q“ ą ΛÑ`8 leading to S 0 1. p q“

4 Kˆ3 (ii) : Let fj j 0, and f in S R , R satisfy t u ě Kp q

S lim fj f, (150) ´ jÑ`8 “

meaning by this

lim f fj k 0, for all k N0, (151) jÑ`8 | ´ | “ P

where we have defined the family of norms k k N0 as p| ¨ | q P

k 2 2 β f k : sup 1 x f x . (152) | | “ x,|β|ďkp ` | | q |B p q|

For every Λ 0, the functional SΛ satisfies property (ii) of Milnos’ theorem ą

Λ Λ lim S fj S f . (153) jÑ`8 p q“ p q

In virtue of the mean value theorem of differential calculus, we can write

Λ Λ Λ δS S f S fj tf 1 t fj . f fj , (154) p q´ p q“ δf p ` p ´ q q p ´ q

Λ for a t 0, 1 , where δS is the Fr´echet derivative of SΛ, which can be computed utilizing the P r s δf property ıApfq δe A .g ıA f A g eı pfq, (155) δf “ p q p q holding true for any f,g S and A S1, because A is a linear functional of its argument. Hence, P P Λ δS A f .g ı A f A g eı pfqdµΛ A , (156) δf p q “ S1 pR4,RKˆ3q p q p q p q ż K

40 from which we infer the inequality

Λ Λ Λ S f S fj A f A fj A f fj dµ A . (157) p q´ p q ď S1 pR4,RKˆ3qp| p q| ` | p q|q| p ´ q| p q ż K ˇ ˇ ˇ ˇ For any positive smooth and monotone decreasing ϕ such that `8 dΛ ϕ Λ 1 the expression 0 p q“ ş `8 θ : dΛ ϕ Λ µΛ (158) “ p q ż0 defines a probability measure on S1. Weighting (157) with ϕ and integrating with respect to Λ leads to

`8 Λ Λ dΛ ϕ Λ S f S fj A f A fj A f fj dθ A . (159) 0 p q p q´ p q ď S1 pR4,RKˆ3qp| p q| ` | p q|q| p ´ q| p q ż ż K ˇ ˇ ˇ ˇ The integrand in the l.h.s (159) is a continuous function which converges to 0 pointwise in A by (150). Since θ is finite measure on S1, by Egorov’s theorem, the convergence is uniform on S1 but for a subset of arbitrary small θ-measure. This means that, for every ε 0 there exists a F 1 S1 ą Ă for which θ F 1 ε such that p qă

lim sup A f A fj A f fj 0. (160) jÑ`8 APS1zF 1p| p q| ` | p q|q| p ´ q| “

Therefore,

lim A f A fj A f fj dθ A 0. (161) jÑ`8 S1 pR4,RKˆ3qzF 1 p| p q| ` | p q|q| p ´ q| p q“ ż K The integrand over the small part has the upper bound

A f A fj A f fj sup A f A fj A f fj K A , (162) p| p q| ` | p q|q| p ´ q| ď jě0p| p q| ` | p q|q| p ´ q| “ p q where K is a positive continuous function on S, whose integral over F 1 can be computed by the mean value theorem for integration as

1 K A dθ A K A0 θ F K A0 ε, (163) 1 p q p q“ p q ď p q żF ` ˘ 1 for a fixed A0 F . Therefore, P

lim A f A fj A f fj dθ A 0, (164) jÑ`8 1 p| p q| ` | p q|q| p ´ q| p q“ żF

41 which, inserted with (161) into (159), leads to

`8 Λ Λ lim dΛ ϕ Λ S f S fj 0. (165) jÑ`8 p q p q´ p q “ ż0 ˇ ˇ ˇ ˇ Hence, for any ε 0 there exists a j0 0 such that for all j j0 ą ě ě

`8 Λ0`δ Λ Λ Λ Λ ε dΛ ϕ Λ S f S fj dΛ ϕ Λ S f S fj ą 0 p q p q´ p q ě Λ0 p q p q´ p q “ ż ż (166) ˇ Λ0`δ ˇ ˇ ˇ Λ1 Λ1ˇ ˇ Λ1 Λˇ 1 ˇ S f S fj dΛ ϕ Λ S f S fj ϕ Λ0 δ δ, “ p q´ p q p qě p q´ p q p ` q żΛ0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ for any Λ0 0 and δ 0 to be appropriately chosen. The parameter Λ1 Λ1 Λ0 Λ0 satisfying ą ą “ p qě the equality in (166) exists by the mean value theorem for integration. If the function ϕ converges fast enough to 0 at , for example choosing a ϕ such that `8

ϕ Λ o Λ´p1`αq Λ (167) p q“ p q p Ñ `8q

for α 0, then we can always find a δ 0 such that ą ą

ϕ Λ0 δ δ ?ε, (168) p ` q “

and we conclude that

Λ1pΛ0q Λ1pλ0q S f S fj ?ε. (169) p q´ p q ď ˇ ˇ ˇ ˇ By letting Λ0 we finally obtainˇ ˇ Ñ `8

S f S fj ?ε for all j j0, (170) | p q´ p q| ď ě

proving the continuity of S.

N Λ (iii) : Let fi i 1,...,n S R and zi i 1,...,n C. Then, since property (ii) holds true for S , we t u “ Ă p q t u “ Ă have for all Λ 0 ą n Λ ziz¯jS fi fj 0, (171) i,j“1 p ´ qě ÿ

and this property is maintained by passing to lim supΛÑ`8, leading to

n ziz¯jS fi fj 0. (172) i,j“1 p ´ qě ÿ

1 3 Kˆ3 Hence, ν : µ t: 0 is a probability measure on S R , R . By definition of lim sup there exists a “ | “ Kp q

42 sequence Λj j 0 with limj Λj , such that in the strong limit p q ě Ñ`8 “ `8

lim µΛj µ and lim νΛj ν. (173) jÑ`8 “ jÑ`8 “

4 Kˆ3 Let fi i 1,...,n S R , R , whose supports lies in the time positive half space. Then, the matrix p q “ Ă p q g M : S θ fi fj satisfies, for all g 0,g0 , “r p p q´ qs Pr r

g Λk,g Mi,j lim Mi,j (174) “ kÑ`8

and is positive definite because M Λk,g is positive definite for all k as we saw in (136), and S is r i,j s continuous. By Proposition 3.8 (or Corollary 3.4.4 in [GJ87]), the reflection positivity axiom (OS3) is fullfilled by µ. Moreover, µ clearly fulfills (OS2), too, because all µΛk s do. By Theorem 3.7 we can reconstruct a selfadjoint operator H, and, by (137) it follows that

lim HΛj H, (175) jÑ`8 “ where the pointwise convergence on the projective limit D HΛj is meant. jě0 p q For A L2 R3, RKˆ3, d3x we see that Ş P Kp q

A lim : V Λj t, x, A : R∇ t, x 2 (176) jÑ`8 p q “ | p q|

pointwise, and thus

A lim d3x : V Λj t, x, A : d3x R∇ t, x 2 : V. (177) jÑ`8 R3 p q “ R3 | p q| “ ż ż Taking the limit on both side for the equation

Λ Λ HΛj H j H j V Λj (178) “ I ` II `

leads to

H HI HII V, (179) “ ` `

43 with the definitions for A L2 R3, RKˆ3, d3x P Kp q

2 1 3 δ HI : d x a “´2 R3 δA t, x ż „ i p q  (180)

2 2 g 3 3 a,b b,c,d d δ HII : d x d y iG A t,y ; x, y ε Ak t,y c . “´ 2 R3 R3 B p p q q p qδA t,y ż ż „ kp q

The property (146) of the ground state Ω0 follows from (102).

Remark 4.4. Note that that H is selfadjoint for all choices of the coupling constant g 0,g0 . As P r r we will see, the non vanishing of the g-contribution in H is essential for the proof of the existence of a positive mass gap.

4.2 Gauge Invariance

We want to prove that the construction of the Hamiltonian in Subsection 4.1 is gauge invariant. That for we show that, if we repeat the construction for a principal fibre bundle subject to a gauge transformation preserving the Coulomb gauge, we obtain an Hamiltonian which is unitary equivalent with the original one and has, in particular, the same spectrum.

Definition 15 (Gauge Transformation). Let P be a principal fibre bundle over a manifold M and π : P M be the projection. An automorphism of P is a diffeomorphism f : P P such that Ñ Ñ f pg f p g for all g G, p P . A gauge transformation of P is an automorphism f : P P p q “ p q P P Ñ such that π p π f p for all p P . In other words f induces a well defined defined diffeomorphism p q“ p p qq P f¯ : M M given by f¯ π p π f p . Ñ p p qq “ p p qq

Following section 3.3 of [Bl05] we notice that the Lagrangian density on the principal fibre bundle P on which we define the Yang-Mills connection is a G-invariant functional on the space of 1-jets of maps from P to the fibre of the vector bundle V associated with P induced by the representation ρ : G GL R3K . Hence, the position variable A occurring in the Lagrangian density and its Legendre Ñ p q transform, the Hamiltonian density takes value in R3K , which is the fibre of the vector bundle V . We want to analyze how the the position variable behaves if the the principal fibre bundle is subject to a gauge transformation.

Proposition 4.6. Let f be a gauge transformation of the principal fiber bundle P and ω a connection.

44 Then, ωf : f ´1 ˚ω is a connection on P . They have the local representation on π´1 U “ p q p q

˚ ˚ ωp adζppq´1 π A ζ θ “ ˝ ` (181) f ˚ f ˚ ω ad ´1 π A ζ θ, p “ ζppq ˝ ` and f ˚ A adφ A φ θ , (182) “ ˝ p ´ q where:

• π : P M is the projection of the principal fibre bundle P onto its base space M, Ñ • U M is an open subset of the base space, Ă • ψ : π´1 U U G is a local trivialization of π´1 U P , that is a G-equivariant diffeomorphism p q Ñ ˆ p qĂ such that the following diagram commutes

ψ ´1 / π U Ut G (183) p q tt ˆ tt π tt tt pr1  ytt U

This means that ψ p π p , ζ p , where ζ : π´1 U G is a fibrewise diffeomorphism satisfy- p q “ p p q p qq p q Ñ ing ζ pg ζ p g for all g G. p q“ p q P • the trivialization map ψ f p π p , ζ f p let us define φ¯ : π´1 U G by φ¯ p : p p qq “ p p q p p qqq p q Ñ p q “ ζ f p ζ p ´1, whence φ¯ p φ π p for a well defined function φ in virtue of the equivari- p p qq p q p q “ p p qq ance of ψ and f.

• The Maurer-Cartan form is the G-valued 1-form defined by θg : L ´1 : TgG TeG G. “ p g q˚ Ñ “ • A and Af are G-valued 1-forms on M introduced in Remark 2.1.

Proof. These are collected results from Proposition 3.3 and Proposition 3.22 in [Bau14].

Remark 4.5. For matrix groups equation (182) becomes

Af φAφ´1 dφφ´1 (184) “ ´

45 For the Yang-Mills construction we denote the K 3 matrices of the local representation of A and ˆ its gauge transformation Af by A and Af , which is in line with the notation utilized so far for the position variable and introduced in Theorem 2.5.

Theorem 4.7. Let f be a gauge transform preserving the Coulomb gauge for the Yang-Mills construc- tion and let HΛ and HΛ,f be the cut off Hamilton operators for the quantized Yang-Mills equation, before and after the gauge transform, as shown in Proposition 98 and Theorem 4.2. Let U be the operator on L2 S1 R3, RKˆ3 , dνΛ induced by the gauge transform as p Kp q q

UΨ A : Ψ Af . (185) p q “ p q

Then, U is a unitary operator in L2 S1 R3, RKˆ3 , dνΛ and HΛ and HΛ,f are unitary equivalent: p Kp q q

HΛ,f UHΛU ´1. (186) “

Moreover, the same holds true for the operator H and Hf , where the cut off is removed as shown in Corollary 4.5. The operator U is unitary in L2 S1 R3, RKˆ3 , dν and p Kp q q

Hf UHU ´1. (187) “

Proof. First, we remark that U maps L2 S1 R3, RKˆ3 , dν onto itself, because it preserves the Coulomb p Kp q q gauge. Next, we prove that U is unitary. For all Ψ, Φ L2 S1 R3, RKˆ3 , dνΛ P p Kp q q

UΨ,UΦ Ψ Af Φ Af dνΛ A p q“ S1 pR3,Rq p q p q p q“ ż K f ´1 A Λ Ψ A Φ A B dν A (188) “ S1 pR3,Rq p q p q A p q“ ż K ˇ ˇ ˇ B ˇ ˇ “1ˇ ˇ ˇ Ψ, Φ looomooon “ p q

The change of variable is given by equation (182) which is affine in A because the adjoint representation

is linear in A, which means adg adg for all g G. Moreover, since p q˚ “ P

adg A Lg A R ´1 , (189) p q “ p q˚ p g q˚

the Jacobi determinant reads

det adg det adg det Lg 1R3K R ´1 det 1R3K 1. (190) pp q˚q“ p q“ p q˚ p g q˚ “ p q“ ` ˘

46 Note that the change of variable respects the fibre of the vector bundle V , and hence the change of variable formula for the integral is the one of finite dimensional analysis.

Next, we prove the unitary equivalence of the Hamilton operators before and after the gauge trans- form. Their definitions read

1 δ HΛ H A ϕΛ , “ p t p¨ ´ ¨qq ı δA ϕΛ ˆ p t p¨ ´ ¨qq ˙ (191) 1 δ HΛ,f H Af ϕΛ , “ p t p¨ ´ ¨qq ı δAf ϕΛ ˆ p t p¨ ´ ¨qq ˙ For appropriate Ψ, Φ L2 S1 R3, RKˆ3 , dνΛ we have P p Kp q q

HΛ,f Ψ, Φ p q“ 1 δ H f ϕΛ , dνΛ A t f Λ Ψ A Φ A A “ S1 pR3,Rq p p¨ ´ ¨qq ı δA ϕt p q p q p q“ ż K ˆ p p¨ ´ ¨qq ˙ f ´1 (192) Λ 1 δ ´1 ´1 A Λ H A ϕt , Λ U Ψ A U Φ A B dν A “ S1 pR3,Rq p p¨ ´ ¨qq ı δA ϕt p q p q A p q“ ż K ˆ ˙ ˇ ˇ p p¨ ´ ¨qq ˇ B ˇ ˇ “1ˇ ˇ ˇ HΛU ´1Ψ,U ´1Φ , looomooon “ p q

leading to HΛ,f UHΛU ´1 (193) “ on the corresponding domains. The proof for the Hamilton operators where the cut offs have been removed are formally the same.

We can therefore conclude that the spectrum of the Hamilton operator for the quantized Yang-Mills problem is gauge invariant.

4.3 Spectral Bounds

We prove now that the Hamilton operator has a mass gap, being the sum of three non negative selfadjoint operators, one of which has a mass gap and having all the same ground state, the vacuum.

47 Proposition 4.8. The spectra of HI , HII and V are:

spec HI 0, p q“r `8r

spec HII 0 η, , for a η 0 (194) p q“t uYr `8r ą spec V 0, . p q“r `8r

2pn`1q Moreover η O g for any n N0, where g is the bare coupling constant. “ p q P

To prove this proposition we need to prove some intermediate Lemmata beforehand.

Lemma 4.9. Let R 0 be a positive constant and let ą

N D : fk A A (195) “ p qB k k“1 ÿ

N N be a first order PDO on R for given functions f1,f2,...,fN on R . If there exists a diffeomeorphism B mapping some compact subset of RN (in the ”A”-space) to R , R N (in the ”B”-space), such that r 2 ` 2 s

N gj B : fk A A Bj (196) p q “ p qB k k“1 ÿ

2 ´1 R R N depends only on Bj , so that gj B gj Bj , then the Dirichlet eigenvalues of D on B , p q “ p q pr 2 ` 2 s q are N 2 2 π kj 2 , (197) ` R ´ j“1 2 ´1 R dBj gj Bj ÿ ´ 2 p q ” ı ˚ ş where kj Z for j 1,...,N, provided the integrals in the denominators of (197) exists. P “

Lemma 4.9 is proved by a direct computation utilizing second order ODE.

Lemma 4.10. Let us assume that for all k 1,...,N “ `8 ´1 dAk fk A (198) p q ă `8 ż´8 holds uniformly in A. Then, the function B : RN RN defined as Ñ

Aj ´1 ´1 Bj : Φ Lj dA¯j f A1,..., A¯j ,...,AN Kj A1,...,Aj 1, Aj 1,...,AN , (199) “ j p q ` p ´ ` q ˜ «ż´8 ff¸

48 where

Aj ¯ ´1 ¯ Kj A1,...,Aj´1, Aj`1,...,AN : inf dAj fj A1,..., Aj ,...,AN p q “´ Aj p qą´8 ż´8 ´1 Aj ´1 Lj : sup dA¯j f A1,..., A¯j ,...,AN Kj A1,...,Aj 1, Aj 1,...,AN (200) “ j p q` p ´ ` q ă `8 « A «ż´8 ffff v 1 2 Φ v : du e´u , p q “ ?π ż´8 is a diffeomorphism satisfying the assumptions of Lemma 4.9 for any constant R 0, and the functions ą gj read N B2 gj Bj ?π Lk e j . (201) p q“ ˜k“1 ¸ ÿ Proof. By definition (196), if

fk A Bj Ck,j Bj (202) B k “ p q

for a function Ck,j of one variable, then the function gj explicitly depends on the variable Bj only. This leads to the differential equation dB dA j k , (203) Cj,k Bj “ fk A p q p q which is fulfilled if

Bj Ak dB¯j dA¯k Kk A1,...,Ak´1, Ak`1,...,AN , (204) Cj,k B¯j “ fk A1,...,Ak 1, A¯k, Ak,...,AN ` p q ż´8 p q ż´8 p ´ q where Kk is a function of A not depending on Ak. The choice

u2 Cj,k u : ?πLj e (205) p q “ for Lj and Kj defined in (200) leads to the desired result.

Now we can compute the lower bound of the spectrum of the Hamilton operator.

Proof of Proposition 4.8. We will construct generalized eigenvectors to show that the spectra have only a continuous part depicted as in (194). First, we analyze the operator HI , which can be seen as

1 3 HI d x∆Apt,xq. (206) “´2 R3 ż

3 R R 3K Let x R and t R now be fixed. For any R 0 the Laplace operator ∆A on , under P P ą r´ 2 ` 2 s π2 Dirichlet boundary conditions has a discrete spectral resolution λk, ψk k 0, where λk 2 k 1 , p q ě “ ´ R p ` q

49 8 R R 3K and ψk ψk A C , , C . We can extend ψk outside the cube by setting its value to “ p q P 0 pr´ 2 ` 2 s q 0, obtaining an approximated eigenvector for the approximated eigenvalue λk, which is in line with the fact that the Laplacian on L2 R3K , C has solely a continuous spectrum, which is , 0 . The p q s´8 s functional A Ψ A¯ : δ A¯ A ψk A (207) k p q “ p ´ q p q

3K 1 1 3 Kˆ3 for k N and A R is a generalized eigenvector in E SK R , R , dν for the operator HI P P p p q q 2 2 1 3 Kˆ3 π on the rigged Hilbert space L S R , R , dν for the generalized eigenvalue 2 k 1 , which, by p Kp q q R p ` q Theorem 2.2, is an element of the continuous spectrum of the non negative operator HI . By varying the generalized eigenvalue over k and R, the claim about the spectrum follows.

Next, we analyze the operator

1 A V d3x R∇ t, x 2, (208) “ 2 R3 | p q| ż

A where R∇ is the curvature operator associated to the connection A. Let A L2 R3, RKˆ3, d3x now P Kp q be fixed. Any non zero ψ L2 R3K , C is eigenvector of the multiplication with the non negative real A P p q R∇ t, x 2. The functional | p q| A Ψ A¯ : δ A¯ A ψ A (209) p q “ p ´ q p q

1 1 3 Kˆ3 is a generalized eigenvector in E SK R , R , dν for the operator V on the rigged Hilbert space p p q q A L2 S1 R3, RKˆ3 , dν for the generalized eigenvalue R∇ t, x 2, which, by Theorem 2.2, is an element p Kp q q | p q| of the continuous spectrum of the non negative operator V . By varying the generalized eigenvalue over A, and taking into account that

A inf R∇ t, x 2 0, (210) A 2 R3 RKˆ3 3 PLKp , ,d xq | p q| “ the claim about the spectrum follows.

2 3 Kˆ3 3 Finally, we analyze the operator HII , which we can write for any A L R , R , d x as P Kp q

2 g 3 3 a 2 HII d x d y Di A; x, y , (211) “´ 2 R3 R3 r p qs ż ż for the operator D D A; x, y defined as “ p q

a a,b b,c,d d δ D A; x, y : iG A t,y ; x, y ε A t,y . (212) i p q “B p p q q kp qδAc t,y kp q

50 3 Let x0,y0 R now be fixed. We set P

a,c a,b b,c,d d f A; x0,y0 : iG A t,y0 ; x0,y0 ε A t,y0 (213) i,k p q “B p p q q kp q and apply Lemma 4.9 and Lemma 4.10. Assuming that for all indices c, k

`8 c a,c ´1 dA f A t,y0 ; x0,y0 (214) k i,k p p q q ă `8 ż´8 uniformly in A, we can find a diffeomeorphism B : R3K R3K in the form of formula (199), such Ñ a 2 ´1 R R 3K that for any R 0 the operator D A t,y0 ; x0,y0 on B , under Dirichlet boundary ą i p p q q pr´ 2 ` 2 s q a a conditions has a discrete spectral resolution λ x, y , ψ A t,y0 ; x0,y0 s 0, where p i,sp q i,sp p q qq ě

3 K 2 2 a π kj,c,s λi,s x0,y0 2 , (215) ` R a,c p q“´ j“1 c“1 2 c c ´1 R dBj gi,j Bj ; x0,y0 ÿ ÿ ´ 2 p q ”ş ı ˚ where kj,c,s Z for all indices s N0, j 1, 2, 3 and c 1,...,K , and, by Lemma 4.10 we defined P P Pt u Pt u

3 K 2 a,l l a,c Bl g B ; x0,y0 : L x0,y0 e j (216) i,j p j q “ i,k p q ˜k“1 c“1 ¸ ÿ ÿ for

c ´1 Aj a,c ¯c a,c ´1 a,c Li,j x0,y0 sup dAj fi,j A; x0,y0 Ki,j A; x0,y0 p q“ A p q ` p q « «ż´8 ffff (217) c Aj Ka,c A ; x ,y : inf dA¯c f a,c A; x ,y ´1. i,j 0 0 c j i,j 0 0 p q “´ Aj p q ż´8

´1 R R 3K 3K a Since B , R for R , we can extend ψ ; x0,y0 outside the cube by setting pr´ 2 ` 2 s q Ò Ò `8 i,sp¨ q a its value to 0, obtaining an approximated eigenvector for the approximated eigenvalue λ x0,y0 for i,sp q a 2 2 3K a a 2 the operator D A; x0,y0 on L R , C , which means that λ x0,y0 spec D A; x0,y0 . For i p q p q i,sp qP cp i p q q fixed i,s and a the functional

a,x0,y0,A a Ψ A¯ : δ A¯ A ψ A; x0,y0 (218) i,k p q “ p ´ q i,kp q is a generalized eigenvector in E1 S1 R3, RKˆ3 , dν for the operator p Kp q q

2 a g 3 3 a 2 Hi,II : d x d y Di A; x, y (219) “´ 2 R3 R3 r p qs ż ż

51 on the rigged Hilbert space L2 S1 R4, RKˆ3 ,µ for the strictly positive generalized eigenvalue p Kp q q

2 3 K π 2 a 2 2 kj,c,s λi,s x0,y0 g 2 , (220) ` R a,c p q“ j“1 c“1 2 c c ´1 R dBj gi,j Bj ; x0,y0 ÿ ÿ ´ 2 p q ”ş ı a which, by Theorem 2.2, is an element of the continuous spectrum of the operator Hi,II . We still have a to prove (214), and at the same time check that λ x0,y0 is bounded away from 0 uniformly in R. i,sp q By Proposition 2.3 and Corollary 2.4, for every n N0 there is a constant cn 0, bounded in n such P ą that

R ` 2 `8 1 dAc t,y Ga,b A; x ,y εb,c,dAd t,y ´1 c g2n dAc t,y . (221) j 0 i 0 0 j 0 n j 0 d 2n`1 R p qrB p q p qs ď p q 1 A t,y0 ż´ 2 ż´8 ` | j p q|

Therefore, inserting (221) into (217) and (215) leads to the spectral lower bound

a 2pn`1q λ x0,y0 Cn g (222) i,sp qě

for all i 1, 2, 3, s N0 and for all n N0, for an appropriate constant Cn bounded in n. Since the P P P collection of generalized eigenvectors obtained by varying (218) over k, x0, and y0 is complete in the sense of Theorem 2.2 (ii), by varying the generalized eigenvalue (220) over kj,c,s and R, the claim about the spectrum follows for η O g2pn`1q . The proof is complete. “ p q

Lemma 4.11. let A and B be two self adjoint operators on the Hilbert space H, such that D A B p ` q is dense in H, spec A 0, infty , spec B 0 η, infty for a η 0, and 0 is an eigenvalue p qĂr ` r p qĂt uYr ` r ą of finite multiplicity for the eigenvector Ω0 for both A and B. Then, spec A B 0 η, , i.e. p ` qĂt uYr `8r the spectral gap of B is maintained.

Proof. The spectral gap of A B reads ` ψ, A B ψ ψ,Bψ η A B inf p p ` q q inf p q p ` q“ ψPxΩ0yKXDpA`Bq ψ, ψ ě ψPxΩ0yKXDpA`Bq ψ, ψ “ p q p q (223) ψ,Bψ inf p q η B η 0. “ ψPxΩ0yKXDpAq ψ, ψ “ p q“ ą p q

Counterexample 4.1. Let H1 be a non negative selfadjoint operator on the Hilbert space H1, and H2

a non negative selfadjoint operator on the Hilbert space H2. Both operators have 0 a simple eigenvalue for the eigenvectors Ω1 H1 and Ω2 H2. Moreover let H2 have a spectral gap spec H2 0 η, P P p qĂt uYr `8r

52 for a η 0. Let us define ą

H : H1 H2 “ b (224) H : H1 1 1 H2. “ b ` b

The operator H is selfadjoint and non negative on the Hilbertspace H, and has 0 has eigenvalue. But it has no spectral gap. The reason is that 0, as an eigenvalue of H1 1 and 1 H2 has infinite multiplicity, b b leading to a clustering of elements of spec H near 0. Lemma 4.11 cannot be applied. p q Corollary 4.12. The spectrum of the Hamiltonian H is

spec H 0 η, , for a η 0, (225) p q“t uYr `8r ą

2pn`1q and η O g for any n N0, where g is the bare coupling constant. “ p q P

Proof. By Proposition 4.8 the operators HI , HI I and V are positive semidefinite and so is H. By

Proposition 4.5 the ground state Ω0 is the eigenvector of finite multiplicity for the eigenvalue 0 for all these four positive semidefinite operators. By Lemma 4.11 the spectral gap of H is bounded from below by the spectral gap of HII :

η H η HII : η, (226) p qě p q“

2pn`1q and η O g for any n N0 holds true by Proposition 4.5. “ p q P

4.4 Main Theorem

We describe the construction of field maps and verify Wightman axioms.

Definition 16. Quantum Field Setting for Yang-Mills Theory

Pseudoriemannian manifold: M : R1,3 with the Minkowski metric tensor g : dx0 dx0 dx1 “ “ b ´ b dx1 dx2 dx2 dx3 dx3. ´ b ´ b Positive light cone: C : x M g x, x 0 . ` “t P | p qě u Bundles over the manifold: P is a principle fibre bundle over M with simple structure group G.

Hilbert space: E : L2 S1 R4, RKˆ3 , dµ . “ p Kp q q

2 1 3 Kˆ3 2 3 Kˆ3 3 Physical Hilbert space: H : L S R , R , dν isomorphic to the Bosonic Fock space Fs L R , R , d x . “ p Kp q q p Kp qq

2 1 3 Kˆ3 Vacuum state: Ω0 : 1 L S R , R , dν . ” P p Kp q q

53 Dense subspace: The space of test functionals D : E S1 R4, RKˆ3 , dµ . “ p Kp q q Unitary representation of the Poincar´egroup:

U a, Λ Ψ A : Ψ A Λ a . (227) p q p q “ p p ¨` qq

It is defined on any Ψ E and extended to Fs E by Γ U . P p q p q Hamilton operator: H defined in Corollary (4.5), with domain of definition D H . p q Field maps: Let us consider a connection A for the principal fibre bundle P . Since the connection A is a Lie-algebra G valued 1-form on M, we can write

3 µ A x Aµ x dx (228) p q“ µ“0 p q ÿ

K a for the fields Aµ x : A y eµ A x ta, where t1,...,tK is a basis of G. We define now p q “ p q “ a“1 µp q t u following operators for j 1, 2, 3ř and a 1,...,K: “ “

Ea : S R4 E j p q Ñ (229) a a 4 a ϕ Ej ϕ , defined as Ej ϕ A : d x ϕ x Aj x ÞÑ p q p qp q “ R4 p q p q ż and for ϕ S R4 P p q a a a Φ ϕ : ΦS Re E ϕ ıΦS Im E ϕ , (230) j p q “ p p j p qqq ` p p j p qqq

where ΦS is the closure of the Segal operator.

Remark 4.6. The field maps are well defined. As a matter of fact, for any indices j 1, 2, 3, a “ “ 1,...,K and all ϕ S R4 the functional Ea ϕ is an element of E: P p q j p q

2 a 2 4 a dµ A Ej ϕ A dµ A d x ϕ x Aj x S1 pR4,RKˆ3q p q| p qp q| “ S1 pR4,RKˆ3q p q R4 p q p q “ ż K ż K ˇż ˇ ˇ ˇ ˇ 4 2 a ˇ 2 dµ A ˇ d x ϕ x Aj ˇx (231) ď S1 pR4,RKˆ3q p q R4 | p q| | p q| “ ż K ż 2 a 2 ϕ L2pR4,Cq dµ A Aj L2pR4,Cq , ď} } S1 pR4,RKˆ3q p q} } ă `8 ż K because µ is a probability measure and, hence, µ S1 R4, RKˆ3 1. Kp q “ ` ˘ Theorem 4.13. The construction of Definition 16 satisfies Wightman axioms (W1)-(W8) and the spectrum of the Hamilton operator H contains 0 as simple eigenvalue for the vacuum eigenstate. There

54 2pn`1q exists a constant η 0 such that spec H 0 η, . Moreover η O g for any n N0, ą p q“t uYr `8r “ p q P where g is the bare coupling constant.

Proof. First we prove that Wightman axioms are satisfied.

(W1): With the definition U a, Λ Ψ A : Ψ A Λ a the strong continuity follows by Lebesgue’s p q p q “ p p ¨` qq dominated convergence and the equality

1 1 2 1 1 2 U a, Λ Ψ U a , Λ Ψ L2 dµ A Ψ A Λ a Ψ A Λ a , (232) } p q ´ p q } “ S1 pR4,RKˆ3q p q| p p ¨` qq ´ p p ¨` qq| ż K

which holds for all Λ, Λ1 SO` 1, 3 and all a,a1 R R3. P p q P ` ˆ

1 B (W2): The computation of the infinitesimal generators shows that Pµ : “ ı Bxµ

1 d 1 d PµΨ A : U teµ, I Ψ A Ψ A teµ p q “ ı dt p q p q“ ı dt p p¨ ` qq ˇt“0 ˇt“0 ˇ 3 ˇ (233) 1 Ψˇ A 1 δΨ Ak ˇ B ˇ˝ B . ˇ “ ı xµ “ ı δAk xµ k“1 B ÿ B The operator ∆ : P 2 P 2 P 2 P 2 is the Laplacian for the Minkowskian metric and is positive “ 0 ´ 1 ´ 2 ´ 3 in the light cone. P0 is unitary equivalent to the multiplication operator x0 which is positive 4 on C`. Therefore, the spectral measure in R corresponding to the restricted representation R4 a U a, I has support in the positive light cone. Q ÞÑ p q (W3): The vacuum state satisfies for all a, x R4 P

Γ U a, I Ω0 A Ω0 A I a Ω0 1. (234) p p qq p q“ p p ¨` qq ” ”

(W4): follows from Theorem 3.5 (a)

(W5): follows from Theorem 3.5 (d).

(W6): For all a, Λ P the inclusion U a, Λ D D holds, because the test functional space is p q P p q Ă invariant under affine transformation of the domain. Since, by Theorem 3.5 (e) for any j 1, 2, 3, “ k 1,...,K, a, Λ P, ϕ S R4 and Ψ D “ p qP P p q P

Γ U a, Λ Φk ϕ Γ U a, Λ ´1Ψ Φk U a, Λ .ϕ Ψ, (235) p p qq j p q p p qq “ j p p q q

55 which can be rewritten as

Γ U a, Λ Φ ϕ Γ U a, Λ ´1Ψ ρ s´1 Λ Φ U a, Λ .ϕ Ψ, (236) p p qq p q p p qq “ p p qq p p q q

where s denotes the spinor map s : SL 2, C SO` 1, 3 , p q Ñ p q

Φ: Φ1,..., ΦK , Φ1,..., ΦK , Φ1,..., ΦK :, (237) “r 1 1 2 2 3 3 s

and

ρ 1 Cd (238) ” GLp q for d : 3K is a representation of SL 2, C on Cd. In particular, it satisfies ρ 1 1, which “ p q p´ q “ means that representation has integer spin, as expected in the bosonic case.

(W7): follows from Theorem 3.5 (c). More exactly, for two test functions ϕ and χ and any indices a,b 1,...,S , j, k 1, 2, 3 the commutator of the field maps can be computed as Pt u Pt u

Φa ϕ , Φb χ r j p q kp qs “ a b a b ΦS Re E ϕ , ΦS Re E χ ΦS Im E ϕ , ΦS Im E χ “r p p j p qqq p p kp qqqs ´ r p p j p qqq p p kp qqqs` a b a b ı ΦS Re E ϕ , ΦS Im E χ ΦS Im E ϕ , ΦS Re E χ (239) r p p j p qqq p p kp qqqs ` r p p j p qqq p p kp qqqs “ a b a b ı Re E ϕ , Re E χ 2 Im E ϕ , Im E χ 2 ( “ p p j p qq p kp qqqL ´ p p j p qq p kp qqqL ` a b a b Re E ϕ , Im E χ 2 Im E ϕ , Re E χ 2 (. ´ p p j p qq p kp qqqL ` p p j p qq p kp qqqL ( All the scalar products vanish because the supports of the test functions are space-like separated and hence disjoint. For example:

a b Re Ej ϕ , Re Ek χ L2pS1 pR4,RKˆ3q,dµq p p p qq p p qqq K “ 4 a 4 b dµ A Re d x ϕ x Aj x Re d yχ y Ak y (240) “ S1 pR4,RKˆ3q p q R4 p q p q R4 p q p q “ ż K „ż  „ż  4 4 a b dµ A d x d y Re ϕ x Re χ y Aj x Ak y 0. “ S1 pR4,RKˆ3q p q R4 R4 p p qq p p qq p q p q“ ż K ż ż

By Theorem 3.5 (a) the field map Φa ϕ is selfadjoint and thus j p q

Φa ϕ ˚, Φb χ Φa ϕ , Φb χ 0. (241) r j p q kp qs“r j p q kp qs “

(W8): follows from Theorem 3.5 (b).

56 Finally, for the spectrum of H on H Corollary 4.12 proves the existence of a positive lower spectral 2pn`1q bound η O g 0 for any n N0 such that “ p qą P

spec H 0 η, , (242) p q“t uYr `8r where 0 is a proper eigenvalue for the vacuum state Ω0, completing the proof.

Remark 4.7. The representation ρ has integer spin and leads therefore to connection fields Aj j 1,2,3 p q “ satisfying the Bose statistics, in accordance with the standard spin-statistics relation for gluons, which are bosons.

4.5 Running of the Coupling Constant

Till now all of our considerations referred to the bare coupling constant, which we now denote by g0. We can repeat the classical and quantum mechanical construction of Section 2 and Section 4 for the running

coupling constant g g µ , where µ is the energy scale, instead of the bare coupling constant g0. To “ p q treat the non-trivial behaviour of g µ we have to renormalize running fields A and constants g by an p q appropriate scaling of the bare quantities A0 and g0. Following [Ma03] we introduce renormalization

constants Z3 and Zg and the transform

a a ǫ A0 Z3A g0 Zgµ g, (243) j “ j “ a and choose the values of the constants as

1 αs 1 b0 Z3 : 1 CA CF Zg : 1 αs, (244) “ ´ p ` q ǫ 4π “ ´ ǫ 2

2 g 1 11 2 where αs : , b0 : CA nf TF , and CA, CF , nf , TF are positive constants. The parameter “ 4π “ 2π 6 ´ 3 b0 is positive if the number` of quark flavours˘ nf is not too large. The parameter ǫ will be let to converge to 0 at the end of the calculation. Since the bare coupling constant knows nothing about the energy scale µ,

dg 0 0, (245) dµ “

57 which, by mean of (243) leads to the Gellman-Low equation

ǫα β α s b α2 O ǫ, α ǫ, α 0 , (246) s ´ b0 0 s s s p q“ 1 αs “´ ` p q p Ñ q ´ ǫ where we have defined β g : dαs for t : log µ2 . With the choice ǫ : 0 we can easily solve the p q “ dt “ p q “ Gellman-Low equation and obtain the implicit

1 1 µ2 b0 log 2 , (247) αs µ ´ αs M “ M p q p q where M is an integration constant, which we choose such that lim ` αs µ , so that µÑM p q“`8

1 αs µ 2 µ M, , (248) p q“ µ p Ps `8rq b0 log M 2 which is in line with [We05] page 156. The running mass gap is

η µ O g2n`1 µ , (249) p q“ p p qq for any n N0, where P 4π g µ 2 . (250) p q“ µ db0 log M 2 We have therefore proved

Corollary 4.14. In the case of a running coupling constant g g µ the construction of Definition 16 “ p q satisfies Wightman axioms (W1)-(W8) and the spectrum of the running Hamilton operator H contains 0 as simple eigenvalue for the vacuum eigenstate. There exists a constant η η µ 0 such that “ p q ą 2pn`1qpµq spec H 0 η µ , . Moreover η O g for any n N0, where g µ is the running p q“t uYr p q `8r “ p q P p q coupling constant. The mass gap tends to 0 if the energy scale becomes arbitrary large.

5 Conclusion

We have quantized Yang-Mills equations for the positive light cone in the Minkowskian R1,3 obtaining field maps satisfying Wightman axioms of Constructive Quantum Field Theory. Moreover, the spectrum of the corresponding Hamilton operator is positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level. The construction is invariant under gauge transforms preserving the Coulomb gauge.

58 Acknowledgement

I would like to express my gratitude to J¨urg Fr¨ohlich, John LaChapelle, Edward Witten, Volker Bach, Gian-Michele Graf and to Lee Smolin for the challenging discussion leading to various important cor- rections of previous versions of this paper. The possibly remaining mistakes are all mine.

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