The pure Yang-Mills field, I: fixed time existence

1 Introduction

1.1 Main results

A pure Yang-Mills field is a Yang-Mills field that is lacking a matter field.

The main result is convergence established for the fixed time renormalized perturbation theory (RPT) expansion. The convergence is based on a weak concept of convergence: order by order to all finite orders, with convergence in a dual Schwartz-like space ^*o*S′.

For later use, the same proof extends to a Boson Yang-Mills field, namely, a Yang-Mills field coupled to a Boson matter field.

Construction of two distinct pure Yang-Mills quantum field, one using ultra violet (UV) asymptotics and the other infrared (IR). These distinctions are introduced in the choice of the vacuum state. The choice of the vacuum state is the initial step in the definition of RPT.

1.1.1 Simple non-Abelian Yang-Mills field

A general non-Abelian Yang-Mills field is allowed. Such fields are direct products of simple fields, and for simplicity the Yang-Mills field is assumed to be simple. Since the field is defined by a general gauge group, no discussion of color gauge transformation is possible. With no matter field, there is no discussion of color indices.

1.1.2 The space ^*o*S

A major result of this paper is convergence in the weak Schwartz space S′, to supplement the order by order cutoff convergence. The S′ convergence is accomplished in two steps, the first of which is ^*o*S′ convergence. ^*o*S′ convergence was introduced in [1] with the definition that the RPT approximation multiplied by the spherical harmonic basis functions converge in the S′ topology.

1.1.3 The larger picture

Based on [2], the order by order convergence occurs in the dual Schwartz space S′. This "extra" convergence is a key in all the results that follow.

The present paper is the first of a series of papers. The ultimate goal of this series is 2-fold: the solution of the Clay Foundation Yang-Mills Millennium Prize and a new solution of the Fermion sign problem.

In the present paper, we establish order by order (including S′) convergence for renormalized perturbation theory (RPT) at fixed times. Here, a factorization of the spherical and solid harmonics is used, with the fixed time spherical harmonics as one of the factors.

The next paper in this series [3] has the same order by order and S′ convergence, now defined for fixed space. In combination, the convergence of the two–fixed time and fixed space gives convergence of the full RPT expansion. The convergence is a major step in the mathematical understanding of Yang-Mills fields.

BRST laws are formulated in terms of the fixed space expansion only. There is one conservation law for each basis element T^*a* of the gauge Lie algebra 𝔤.

Here, the path forward splits. The new solution of the Fermion sign problem is attained in the standard manner using homology groups and thimbles. In the estimate of the residual sign problem after use of homology groups and thimbles, the S′ aspect of the order by order convergence controls the required remaining convergence. See [4].

Anomalies, missing with a Boson matter field, arise with a Fermion matter field. They are avoided, for physics-based theories, by a modification to the Lagrangian which enforces the chirality cancellation needed to ensure a zero anomaly. The Fermion sign problem, based on the modified Lagrangian, was solved [5].

Separately, in [6], the mass gap at low temperatures is addressed through the Dyson equation to study the low order moments involved in the phase transition which gives rise to the mass gap. The magnitude of the mass gap is given by the Dyson equation analysis and is unrelated to entropy.

The validity of the Dyson equation and the perturbative analysis requires a bound on the cumulative effect of high order moments, controlled by the S′ convergence and power counting. The S′ convergence and the power counting depend on entropy considerations.

The axioms [1] are not difficult to verify. All requirements for the Clay Foundation Yang-Mills Millennium Prize are fulfilled at this point.

1.1.4 The vacuum state

For the vacuum state (only), the maximum entropy principle is not used. In the solid harmonics expansion, the magnitude r = ±1 is assigned one of its two possible values. The UV expansion, with r = 1, is regular at the origin. The IR expansion, with r = −1, is regular at ∞.

1.1.5 Convergence modes

Three cutoffs are defined by the finite lattice, the finite lattice spacing, and the finite order RPT cutoff j. The fixed time estimates and convergence are uniform in both lattice cutoffs. The convergence to an infinite lattice is based on vector Fourier analysis. After the infinite lattice limit has been achieved, the remaining limit of zero lattice spacing is analyzed using concepts of spherical harmonics and solid harmonics.

The fixed time expansion terms are labeled by basis elements Y_{ℓ, *m*} of the spherical harmonic expansion.

By Hilbert space convergence, we refer to the Hilbert space convergence defined by the L₂ space of the Yang-Mills field, as well as the associated L₁ convergence. Hilbert space convergence implies convergence of the bare vacuum expectation values of Feynman diagrams.

Convergence related to any other Sobolev norm will be designated as Sobolev space convergence, with the power counting Sobolev indices indicating convergence relative to an L₂ or L₁ case.

1.1.6 Gauge independent and dependent

Two distinct Yang-Mills fields are constructed. One is based on gauge invariant functions of the field only. It is a scattering matrix theory. The other construction is based on the gauge dependent as well as gauge independent functions of the Yang-Mills field.

1.1.7 The bare lagrangian

The bare Lagrangian (order j = 1 of RPT) in the Fock space perturbative framework is constructed using Wick ordered products of annihilation and creation operators in the standard manner. These free fields become non-trivial through the introduction of the bare Lagrangian.

L=-∑a=1dim(14FaμνFμνa) .(1)

The superscript a is the component of F relative to a basis T^*a* of the gauge Lie algebra 𝔤 of the gauge group G, assumed to be compact, simple and non-Abelian. dim is the dimension of 𝔤.

In Equation 1, a summation over repeated indices convention (SRI) is assumed. The repeated indices are the over upper and lower μ or μν indices.

No SRI (NSRI) refers to terms of Equation 1 or to Feynman diagrams for which no SRI sum occurs. NSRI terms are needed in the convergence theorems of this paper.

The Lagrangian is not defined until a gauge is fixed.

Let 𝕋³ × 𝔻 denotes a space time cube with periodic spatial boundary conditions. The fixed time Lagrangian is a function of the rank 2 tensor fields F^*aμν*(x, t). These are defined to be elements of configuration space, that is rank 2 tensors defined as functions of

(x,t∈𝕋3×𝔻)(2)

including the fixed time space derivatives in the definition.

Other boundary conditions will be used as convenient.

More precise definitions of configuration space are given in Section 2, defined as a principal fiber bundle.

1.1.8 Examples of low order RPT diagrams

The bare vacuum energy has the Wick ordered diagrammatic formula

◯•◯.(3)

The mass diagram, to lowest order in RPT, is

•≡•.(4)

In lowest order RPT, a charge diagram is connected with three vertices and four external legs. The entropy production rate maximized charge diagram in lowest order RPT is

=•◯•◯•=.(5)

In this lowest order RPT, the charge is not uniquely determined by Equation 5.

The diagram formed from

=•-•◯•-(6)

with an extra line joining the first and last vertex is topologically inequivalent to Equation 5. Its vacuum expectation value is strictly smaller than Equation 5, and its entropy production rate is strictly smaller also. It is not entropy production maximized.

This is the first of many uses of the entropy principle throughout this paper.

1.1.9 Examples of order by order convergence

The Wick ordering subtraction in Equation 3 is sixth order divergent (UV; not listed in the power counting tables).

The Wick ordered line 2 bare vacuum diagram is quadratically divergent (UV) and linearly divergent (IR). These values are listed in the tables. The table values can be computed directly following methods of [7].

The bare mass diagram (Equation 4) is quadratically divergent (UV) and linearly convergent (IR), according to the tables.

The entropy production maximized bare charge diagram of Equation 5 is log divergent (UV and IR), according to the tables.

1.2 Prior results

The role of entropy in this paper was influenced by [8].

The RPT of QFT as presented, for example, in [7, 9], together with the theory of Feynman diagrams, provides the perturbative basis for this paper.

Essential use is made of tensor spherical harmonics [10, 11]. Overlapping subvariable divergences are resolved using ideas adapted from [12]. The extension of the gauge invariant fields to be defined as Schwartz space S′ functions is a consequence of [2].

The lattice cutoff is popular within the Wilson quantization methods [13]. The lattice, analyzed mathematically in [14], also defines a viable cutoff within the perturbative [7] approach to quantization; this is the route we follow.

The mathematically rigorous construction of quantum fields for space time dimension D = 2,3 is described in [15] with contributions of many authors cited. Balaban [16] and earlier papers and Dimock [17] and earlier papers extended this theory to include Yang-Mills fields for D = 2,3.

The methods of stochastic quantization have been developed by Chevryev and others in [18] and related papers and applied to D = 2,3 Yang-Mills fields.

AQFT [19] provides within its formalism the Gribov-related extension of the Hamiltonian.

The D = 4 case was considered [20] in a lattice cutoff study of the Higgs phenomena based on the Wilson loop construction.

The phase cell expansion construction of quantum fields, introduced for D = 2, 3 quantum fields and discussed in [15], has been extended to D = 4 Φ44 quantum fields, [21, 22]. The D = 4 phase cell methods are equivalent to RPT. We also mention a sequence of papers by Federbush, of which we cite the most recent, [23], on the use of phase cell methods to study Yang-Mills fields.

The QFT Millennium prize documentation reviews the D = 4 constructions of UV uniform limits for the partition function (grand canonical ensemble) by Balaban [24] and earlier papers based on a Wilson loop quantization and also the constructions of [22] with an outline of steps that could lead to a construction of the Yang-Mills field.

2 The lattice fiber bundle

Lattice cutoffs are developed mathematically in [14, 25]. The field operators are interpreted within the formalism of a principal bundle. The principal bundle refers to a fiber bundle in which each fiber is isomorphic to the gauge group G [26–29].

Explicit formulas are given for the field operators, the local gauge group Operators, and the Lagrangian.

The connection A^μ, the curvature F^μν, the gauge group, the Lie algebra elements, the exterior calculus, and the Hodge duality are identified.

This use of the lattice cutoff is distinct from the Wilson quantization.

The lattice approximation, as used here, is a specific realization of the cutoffs. The theory remains perturbative, as in [7].

The intrinsically defined inner product on 𝕋³ × 𝔻 extends to an inner product between the principal bundle P and the adjoint bundle ad(P).

In [14], the adjoint bundle is defined on the lattice displaced by a 1/2 unit from the lattice for the principle bundle. In this way, the inner product between the principal bundle and the adjoint bundle is uniquely defined.

An associated bundle admits an automorphism group acting on the gauge group and on the fibers of the bundle. The special case of the (left) adjoint automorphism group defines the adjoint bundle for a pure Yang-Mills quantum field.

The covariant derivative D_νA_μ is defined as an exterior 2-form on the adjoint bundle. The exterior forms are linear combinations of one, two, three, and four forms for 𝕋³ × 𝔻.

The exterior algebra (also called the Grassman algebra) is defined by the associative product ∧ on the space of exterior forms. The wedge product satisfies the identity ∧a ∧ a = 0 for all a in the exterior algebra.

For elements s, t of the exterior algebra, write 〈s, t〉_L₂ as the integral over 𝕋³ × 𝔻 of 〈s, t〉. The identity 〈DAs,t〉L2=〈s,DAt〉L2 follows, with DA defined as the adjoint of D_*A*.

The Hodge star, denoted ★, is defined as the adjoint, with a sign convention to preserve the natural orientation of space.

Consider the connection A_μ and its covariant first derivative, the curvature F^μν. A is a rank 3 tensor field, and F is a rank 2 tensor field.

Following the notation and formalism of [25], we assume a compact simple non-Abelian gauge group G with a simple Lie algebra 𝔤.

The covariant derivative is defined as

Dμ=∂μ-igTaAμa(7)

where the T^*a* are generators for the Lie algebra 𝔤.

They satisfy the equations

tr(TaTb)=12δab, [Ta,Tb]=ifabcTc.(8)

with f^*abc* the structure constants of 𝔤.

A^*a* is the vector potential (connection). In the lattice fiber bundle, it is defined as a function on the lattice. Its location within a single cell is on one edge of the cell.

F is related to the connection A by the formula

Fμνa=∂μAνa-∂νAμa+gfabcAμbAνc(9)

derived from its definition

Faμν=DνAaμ(10)

as the covariant derivative of the connection A and from the commutator formula [Dμ,Dν]=-igTaFμνa. As a function on the lattice, its values are located along two consecutive sides of each lattice cell.

2.1 Particle states

• Proposition 2.1.1 A connected Feynman diagram is associated with a unique representation of the gauge group G. As a consequence, it carries a unique index a.

Proof. For a connected Feynman diagram, any node (point vertex) j₁ is connected to some other point vertex vertex j₂. The link connecting j₁ to j₂ is subject to the variational restrictions on the probability measure which defines the solution. Gauge invariance of the probability measure as it defines this link, forces j₁ and j₂ to share a common spherical harmonics representation of O(3). A consequence is that the entire connected diagram is described by a single tensor spherical harmonic representation of O(3).

Since this is an important observation, an additional and independent proof is given. Start with the SRI Lagrangian, with X=FμνaFaμν. Both of the F factors have a common index a, as in Equation 1. Additional point vertices can be added by covariant variational derivatives of the form

(D2)nX.(11)

The paired derivatives D² act on the common space variable of the two factors of X. A more compact notation is to set X = E as an energy variable and with a single covariant variation derivative, so that Equation 11 becomes D^*n*E. The n covariant derivatives D^*n* act on distinct variables of E = FF within an RPT expansion term or Feynman diagram.

Tensor indices are understood in Equation 11 but not indicated explicitly.

Assuming that all point vertices in X have a common a label, a non-zero value for Equation 11 requires the same label a for the new point vertices introduced by Equation 11. If internal lines are broken to form a new diagram or external lines are joined, the diagrams still have a common a. From these operations, we recover all connected Feynman diagrams, arriving at a second proof of the proposition. □

The formula (Equation 11) defines n particle states.

The fixed time Lagrangian E=FμνaFaμν is formed by restriction of the μν indices to be purely spatial. Additional fixed time point vertices can be added by covariant variational derivatives as in Equation 11: (D²)^*n*E.

With a fixed time restriction, the formula (Equation 11) defines fixed time n particle states.

The number of external legs of a Feynman diagram is denoted as k. The diagram contributes to the j-particle state with j ≥ k/2.

2.2 Connected Feynman diagrams

Connected Feynman diagrams are called prime, in that they cannot be written as disconnected products of two or more factor diagrams.

• Proposition 2.1 A Feynman diagram has a unique representation as a sum of products of prime factor Feynman diagrams.

Proof. Obvious. □

• Proposition 2.2 For X a quadratic pair from the Lagrangian,

〈Xj〉=∑Pj∏P∈Pj〈XP〉.(12)

Proof. The sum runs over all partitions P_*j*; each partition is a decomposition of the set {1, ⋯ , j} into a disjoint union of |P| subsets P ∈ P_*j*. With X_*P* = ∏_{p_*i* ∈ P}X_{p_*i*}, each X_{p_*i*} is a prime sequence.

□

3 Functional analysis

3.1 Spatial and temporal analysis

The space time formalism is Euclidean in that the metric is ++++, as opposed to the relativistic metric −+++.

The cutoff study is restricted to the finite lattice based on 𝕋³ × 𝔻.

The fixed time Hilbert space H is defined as the space of L₂ rank 2 tensor valued fields defined on 𝕋³ × 𝔻. It is called the configuration space.

3.2 Harmonic analysis

The rotational and translation groups combine to form a semidirect product. The rotational and translation degrees of freedom of a random field, all but its absolute magnitude, are defined by its invariance properties under this semidirect product. The translation analysis is Fourier analysis; the rotational analysis is spherical harmonics. The missing variable, the magnitude, is included within the theory of solid harmonics. Detailed treatments [10, 11] describe this theory.

3.3 Regularity at 0 versus (vs.) ∞

When the finite lattice Yang-Mills field is expanded as a series in vector solid harmonics, a spatial inversion interchanges the origin with the point at ∞.

With the spatial inversion, the choice of the configuration space r mode is reversed between regularity at r = 0 and at r = ∞.

Spatial inversion, repeated twice, is the identity operation. Repeated three times is identical to a single inversion. For this reason, there is a single Yang-Mills quantum field regular at 0 and a single Yang-Mills quantum field regular at ∞. The entropy maximizing field is the average of the two.

3.4 Function spaces

Let E denotes the Schwartz space of C^∞ tensor fields defined on the space volume 𝕋³ or ℝ³ with Dirichlet temporal boundary conditions. Let E′ be the dual space, both E and E′ having the customary Schwartz space topologies. The choice of Schwinger test functions f ∈ E makes the space E′ natural for their study. Tensor indices as needed are included in these definitions.

H is the L₂ space of rank 2 tensor fields, and 〈·, ·〉 is the inner product in this L₂ space.

W^{*n, p*} is the space of n order derivatives as elements of L^*p*. W0=W0,1 denotes the corresponding L₁ space, with the expectation value denoted 〈·〉.

H_*s* is the Sobolev space of rank 2 tensor fields which are asymptotically s powers of momentum convergent relative to H when all variables become UV infinite in a common proportion. W_*s* has a similar L₁ definition. Observe the sign reversal from convention, so that convergence indices s are positive.

3.5 The entropy principle

Entropy is defined as the log volume of a surface of constant energy.

3.5.1 Entropy to resolve non-uniqueness

The maximization of entropy production is a fundamental law of physics. Yang-Mills fields admit the possibility of non-physical solutions. The principle of entropy production maximization excludes many, probably all but one, of the non-unique solutions.

Thermodynamically, energy dissipation and entropy production are related by a Legendre transformation. For this reason, when useful, the entropy principle of maximal production of entropy is replaced by the energy principle of maximal dissipation of energy.

3.6 Feynman diagrams and entropy

Feynman diagrams provide a graphical description of products of the quadratic pairs (gauge invariant terms) from the Lagrangian. These diagrams describe terms in the gauge invariant RPT expansion. A Feynman diagram with 2k external lines defined by n elementary steps (Equation 14) has j + k point vertices, with

n=2j+k.(13)

3.6.1 Vertex insertion

The concept of vertex insertion serves as an introduction to the broader topic of manipulation of Feynman diagrams, known as graphology. A gauge invariant point vertex is a product of a pair F_μν of field operators with matched upper and lower indices evaluated at a common point.

Consider a Feynman diagram G and a subdiagram G₀ ⊆ G. An insertion consists of replacing G₀ by another graph G₁. For this to make sense, the external legs of G₁ must match in type, number, and order with the lines of G that join G₀ to G\G₀.

A simple gauge invariant vertex insertion (a point vertex) is a single Wick ordered quadratic pair from the Lagrangian. If the inserted diagram has no external legs (k = 0), it is the insertion of a disconnected vacuum diagram. If it has two external legs (k = 2), it is a mass insertion. If it has four external legs, it is a charge insertion. Step (Equation 15) in the inductive definition of RPT allows for a reduction in k with n and j fixed.

3.6.2 The integrand

Feynman diagrams with j point vertices are constructed at fixed time with four lines attached to each vertex. The opposite end of each line may or may not be attached to a point vertex, with a maximum of four lines at each point vertex. The diagram is multiplied by a combinatorial factor which is defined as the number of distinct occurrences of the diagram in the RPT series.

A Feynman moment is the bare vacuum expectation value of a Feynman diagram. Feynman moments are described by power counting rules with Sobolev convergence indices tabulated in Tables 1 and 2. The tables are SRI. They show a rate of energy dissipation, related to entropy production by a Legendre transformation. The two tables are distinct, showing distinction of the UV and IR limits.

Table 1. UV SRI convergence rates for divergent prime diagrams within the fixed time RPT expansion are listed with the convergence expressed in terms of indices of Sobolev norms.

Table 2. IR SRI convergence rates for divergent prime diagrams within the fixed time RPT expansion are listed with the convergence expressed in terms of indices of Sobolev norms.

The NSRI vertex factors are addressed separately.

The diagrams also describe the gauge dependent Schwinger functions and tadpole contractions of gauge dependent Feynman diagrams. The external field (Schwinger function) is represented as a vertex with two legs attached. A tadpole diagram is formed when one or both legs of a Schwinger function are connected to gauge independent variable(s).

3.6.3 The role of r

r, as a configuration space variable, has the value ±1, with positive sign selected by regularity at the origin and the negative sign by regularity at ∞.

The role of r is to select the origin vs. ∞ for regularity. The maximum entropy choice is the equally weighted sum of the two choices. This maximum entropy, after the vacuum has been established, is used within both the UV and IR RPT expansions. Maximum entropy is not used within the definitions that yield distinct vacuums and distinct UV and IR expansions.

3.6.4 The roles of ℓ, m

The roles of ℓ, m are addressed in a fixed time spherical harmonics analysis. The convergence proof for fixed time is completed in Theorem 5.2.1.

3.6.5 Entropy production maximization

The maximization of the production of entropy is an important tool, used often throughout this paper. This maximization is called the entropy principle for short. The precise formulation of this principle requires

• (a) identification of the functional of the Yang-Mills field whose entropy is maximized and
• (b) the function space in which this maximization occurs.

Given (a) and (b), the entropy principle states the existence of a Yang-Mills field X for which the entropy of (a) in the space (b) is maximal relative to the entropy of (a) in the space (b) for any Yang-Mills field Y.

The functionals are the vertex factors (external legs k = 0), propagators (external legs k = 2), and more generally Feynman diagrams defined by the Yang-Mills field X.

The function spaces are specified as the vacuum energy power counting SRI tables with addition of the NSRI terms.

3.6.6 Existence of an entropy production maximum

• Theorem 3.5.1 Consider SRI limits of the fixed time Yang-Mills fields and function spaces defined by the tables, with k external legs, whose Feynman diagrams satisfy the entropy principle.

• There exist entropy principle limits, that is, limits which maximize entropy production of the Feynman diagrams relative to the entropy production of the Feynman diagrams for any Yang-Mills field.

• Entropy principle limits exist for the NSRI diagrams as well.

Proof. The proof has two steps: cutoffs and convergence.

Lattice cutoffs make the problem finite dimensional. The existence of an entropy principle cutoff Yang-Mills field is classical due to the finite dimensionality.

Limits for the Feynman diagrams as cutoffs are removed, exist within the specified function spaces. An entropy principle cutoff field is compared to the same cutoff removal limit of any Yang-Mills field. The entropy principle, satisfied for each value of the cutoff, remains true for the limit. □

3.6.7 The energy dissipation principle

In many uses of the maximum entropy principle, it is more convenient to work with the energy and a maximum energy dissipation principle. The energy principle states that there exists a Yang-Mills field which has maximum dissipation relative to other Yang-Mills fields. The energy dissipation principle, as with the entropy principle, depends on a specified function space and on a specified component of the Yang-Mills field.

• Theorem 3.5.2 For any limit under cutoff removal, there exists fixed time Yang-Mills fields in the table defined SRI function spaces whose connected Feynman diagrams satisfy the energy dissipation principle. The energy dissipation principle limits exist for the NSRI fields also.

Proof. The energy dissipation field follows from the entropy principle by multiple Legendre transforms. The proof is by introduction of cutoffs and convergence as the cutoffs are removed.

The Legendre transform of the entropy (in the thermodynamic space of entropy and energy) depends on convexity of the energy. As proofs employ regularizing cutoffs, convexity is not only a necessary condition. It is also sufficient. Convexity starts with the logarithm of the energy and convexity of the logarithm function. As enstrophy is defined as the log volume of a surface of constant energy, convexity of the enstrophy as a function of the energy follows.

Due to the sum over repeated Lie algebra basis labels a, all Lie algebra defined energies are included in the entropy production maximization, with an iteration over the convexity for each individual T^*a*. □

4 RPT

4.1 Elementary steps and Schwinger functions

The Schwinger functions are added to RPT with the same two elementary steps, adding a Schwinger function vertex as in (Equation 14) and contraction of a Schwinger function vertex, as in Equation 15.

4.2 Wave function, field strength, and charge renormalizations

At fixed time, the wave function, field strength, and charge renormalizations are finite and can be ignored.

4.3 Overlapping subvariable divergences

Removal of overlapping subvariable divergences in the entropy production maximizing RPT expansion depends on the prime sequence factorization expansion, Propositions 2.1, 2.2. Fixed time overlaps are removed here.

The analysis of [12, 30] for the derivation of bounds for overlapping divergences is simplified here in that exact formulas with counterterms are not required.

The analysis of [12] is illustrated by considering the two vertex Wick constant Feynman diagram 〈FμνFμνFγδFγδ〉.

This factor has two variables and a ten possible subvariable divergences. Four subvariable divergences are formed from any selection of one of its internal lines, and six formed by any selection of two of its internal lines.

These overlap. Removal of one line leaves three overlapping subvariable Divergences, and the removal of two lines leaves a diagram with no divergences, and hence no overlapping subvariable divergence.

We proceed with a systematic inductive development of this reasoning.

Recall the prime sequence expansion of a Feynman diagram, Proposition 2.1. The diagram defined by the sequence {1, ⋯ , n} of vertex labels is the sum of vertex labels over all decompositions of {1, ⋯ , n} into a disjoint union of non-overlapping subsets.

• Definition 4.1.1 A prime sequence has a subvariable divergence if its UV limiting behavior depends on the ratios of the momenta as the UV limit is approached. The divergent subvariables are defined as any minimum set of variables which exhibit stronger divergence that the power counting divergence of the diagram as a whole. Distinct sets of divergent variables overlap if they have at least one variable in common.

• Proposition 4.1.1 A Feynman diagram can be written as a sum of products of non-divergent and non-overlapping divergent diagrams and a remainder, denoted Itr (iterative remainder).

Proof. This is an elementary consequence of the definitions. □

The essence of the analysis [12] of overlapping divergences is the iterative analysis of Itr.

• Definition 4.1.2 Consider a Feynman diagram and the Itr term from its prime factor expansion. Fix one Lagrangian vertex X₁ occurring in the Itr term and consider a second prime factor expansion in the remaining Lagrangian vertex variables. This expansion is a called a double expansion of the original diagram because it is built on top of the original expansion that defines the prime sequence. The double expansion involves one fewer Lagrangian vertex variables than the original diagram.

• Proposition 4.1.2 The expansion of an Itr term from any iterated order of expansions can be written as a sum of products of non-divergent diagrams, non-overlapping divergent diagrams, and a remainder, defined as the Itr (iterative remainder) of the next iterated order.

• After a number of iterated Itr steps (at most equal to the number of vertices in the Feynman diagram), the Itr remainder is empty.

Proof. This is also an elementary consequence of the definitions. □

4.4 The RPT inductive step

A goal for the RPT is to define the exponent of various bare quantities such as 〈exp(↖L)〉 with X=FaμνFμνa a co-located quadratic pair from the Lagrangian L.

RPT starts with the Lagrangian and its bare phase space evaluation in the bare vacuum state, denoted by angle brackets 〈.〉.

The inductive RPT step accomplishes these goals relative to a single stochastic degree of freedom, represented by a single bond in a Feynman diagram, and thus relative to the correlations between two specific variables within the stochastic field.

4.4.1 Two elementary steps

Consider the pure gauge invariant RPT expansion with Sj=Xj, so that S₀ = I (the identity operator). We define a quantity Sjren inductively. The induction is initialized with the definition I^ren = I, so that S0ren=Iren=I=S0. Define S_{*j, k*} to be an RPT S_*j* expansion term with j vertices and k uncontracted (external) variables.

A new factor X = F^*μν*F_*μ*]*nu* from the Lagrangian is added to the S_{*j, k*} expansion term by the formula

Sj+1,k+3=XSj,k-〈XSj,k〉.(14)

with an increment of j to j + 1 and of k to k + 3. The angle brackets are defined as the bare vacuum expectation value. One internal line and three external lines are added by Equation 14.

In Equation 14, there is no need for S and X to share basis element T^*a* of the Lie algebra 𝔤 or its label a. If the index a of X does not occur among the indices of S, the subtraction is zero, so that in the resulting diagram, X is not connected to S. From this observation, we see (again) that connected Feynman diagrams have a single index a.

A second elementary step in the expansion defines a contraction between two existing vertices, decrementing k by two while leaving j unchanged. This step is

Sj,k-2=Sj,k-〈δ2Sj,kδXδY〉.(15)

One internal line is added by (15), by the joining of two external lines, which are then removed by the joining.

4.4.2 Gauge dependent RPT

The formulas (Equations 14, 15) extend to define RPT acting on gauge dependent Schwinger functions, defining the tadpole diagrams that couple the Schwinger functions to the gauge independent RPT.

4.4.3 Inductive steps

The integrations in Equations 14, 15, at fixed time, do not depend on the definition of the Hamiltonian.

For gauge dependent Schwinger function and tadpole RPT expansion steps, the integrations in Equations 14, 15 are at fixed time.

4.5 Power counting

4.5.1 The use of power counting

Power counting has an important role in establishing convergence as cutoffs are removed.

For the gauge independent Yang-Mills field, a goal of RPT is to define the S-matrix scattering amplitudes.

The power counting indices define the asymptotic properties of vertex factors. They do not specify the (gauge dependent) propagator of the Feynman diagrams.

Separate UV and IR tables summarize the key power counting properties of the fixed time RPT. UV and IR have distinct power counting Sobolev indices.

The power counting tables are SRI. Corrections to include the rest of the RPT series (called fluctuating terms or NSRI) are considered separately.

4.5.2 Vertex factors and propagators in RPT

The integrand for the bare vacuum diagram is the product of the Lagrangian L₁ with the product of two propagators P(p) and P(p′).

The second order vacuum diagram is the result of two inductive steps applied to the bare Lagrangian. The inductive steps add a new vertex L₂ and two new propagators P(p^‴) and P(p^{′*′′′*}), resulting in the integrand

L1L2P(p)P(p′)P(p‴)P(p′′′′).(16)

The higher order RPT entry is a product, called the vertex vector, of the L factors times the product of the P(p_*i*), called the propagator. Write E = FF. Consider the j covariant derivative of E, D^*j*E. Its propagator is the multiple p^*j*P(p) of the propagator P(p) of E. Any temporal p in this expression is non-anticipating in the fixed space RPT expansion. j = 1 is the Gribov ambiguity Hamiltonian extension, with the physical solution being the entropy production maximizing choice.

Let P_{ℓ, *k*} be the propagator for the Feynman diagram from RPT order 2ℓ − k, with 2k external legs, ℓ first inductive Steps, and ℓ − k second inductive steps. The integrand for this diagram is a combinatorial factor multiplied by

L1⋯LℓPℓ,k.(17)

There are possible multiple distinct topologies for this diagram. The entropy principle choice is made, both for the vertex functions L_*i* and the propagators P_{ℓ, *k*}.

The external legs are not dressed and do not enter in the UV and IR formulas.

4.5.3 Tensorial cancellations

Note the two step improvement in power counting in going from the UV and IR vacuum energy to the mass and from the mass to the charge.

To see the vanishing of the single order term, consider the correlation that results from the first order subtraction alone:

FμνFμνFμγ+FμνFμνFγμ=FμνFμν(Fμγ+Fγμ).(18)

The factor (F^μγ + F^γμ) traces two adjacent edges of the same plaquette twice, but with opposite orientations, so that the sum is zero.

This cancellation is SRI and does not occur with NSRI terms.

4.5.4 Power counting RPT

Power counting is explained in [7].

The power counting values for all fixed time k-particle insertions are given in Tables 1 and 2.

4.5.5 The Feynman gauge

Weinberg [7] recommends the Feynman and Landau gauges, defined in [7], eq. (15.5.25). For a blending of the two gauges, with a blending parameter ξ, the propagator P(p) is given by

P(p)=(2π)-1∫d4p[ημν+(ξ-1)pμpνp2]eip(x-y)p2.(19)

The choice ξ = 0 defines the Landau gauge, and the choice ξ = 1 defines the Feynman gauge.

The formula (Equation 19) from [7] has been Wick rotated into a Euclidean metric, whereby η_μν = δ_μν defines the Euclidean metric and the time integrals in Equation 19 are non-anticipating.

• Proposition 4.3.1 The SRI and the NSRI Euclidean Feynman gauge propagator is strictly positive in Fourier space and strictly positive definite in space-time variables.

Proof. This is an elementary consequence of Equation 19. □

4.5.6 Entropy principle Feynman diagrams

The details of the definition can be given in terms of the spherical harmonics expansion of the Feynman diagram. Consider the Feynman diagram expressed as a sum of Y_ℓ*m* basis elements. Entropy production is maximized separately in the choice of the m, ℓ, r coefficients.

The ℓ, m spherical harmonics describe the fixed time Feynman diagrams.

Consider first the m coefficients. The maximum entropy production state is equal to an equally weighted combination of the individual m states divided by the number of terms in the sum.

This state is the direct opposite of the mean, defined at fixed ℓ as the average of the individual m values in a single state. The entropy of the m maximum entropy production state is logm, and the entropy of the m mean is log1 = 0.

Continuing, the maximization of the ℓ entropy is similarly defined as the equally weighted combination of all ℓ states divided by the number of terms in the sum.

r is set to give regularity at the origin or regularity at ∞. The entropy maximized choice is the average of the two.

5 Convergence

5.1 Convergence to all finite orders

The conventional RPT vacuum energy is UV, with a cubic (0, −3) fixed time divergence. The same integrals, evaluated in the IR limit, are log divergent, (−1, 0). Here, the pair of numbers (a, b) indicates a temporal value a and a spatial value b. The separation of spatial and temporal power counting is not standard.

Any energy integral which is positive is not renormalized. The fixed time state norm integrals are finite and not renormalized.

Conventional (UV or IR) power counting is modified with a separate UV and IR power counting laws for fixed time.

The power is defined by the order of a Sobolev index, defined for a connected moment, that is, for the bare vacuum expectation value of a connected Feynman diagram.

Each order of RPT leaves the UV and IR asymptotics unchanged. Each breaking of an internal line to form two external legs improves the UV power counting by 2 and decreases the IR power counting by 2.

The time and space indices are listed separately, so that in the fixed time UV Table 1, the total index -3 is written as (0,-3) indicating a log temporal divergence and a cubic spatial divergence. To derive the mass entry from the vacuum entry, the Sobolev indices are increased by 2. This increase can occur in either of the temporal of spatial components, resulting in two possibilities, shown as (2, −3)_*t* + (0, −1)_*s*, with (2, −3)_*t* the temporal mass and (0, −1)_*s* the spatial mass.

The number of external legs k is a sum of its temporal and spatial components, k = (k_*t*, k_*s*). A term ±2(k_*t*, k_*s*) is set to zero by the entropy principle.

In the case of the charge, which renormalizes two particle states, the particles could be either t or s particles or one t and one s particle.

5.1.1 Convergence to all finite orders

The convergence is to all finite orders and ^*o*S′ convergence.

5.1.2 IR power counting

The mathematical description of IR power counting raises issues discussed in [32].

Table 2 is derived from Table 1 by inversion of space relative to the solid harmonics expansion. With this inversion, the temporal index increases by 1 with the change in time limit from small t (large 1/t) to large t.

The spatial index increases by three, reflecting the area of the unit sphere at ∞.

Energy entries from Tables 1 and 2 are related by the identity

UV Index+IR Index=-2.(20)

A single index as in Equation 20 fixes the sum of the space and time indices but not the individual indices.

5.1.3 Gauge dependent power counting

Schwinger functions are defined by the dot product of the Yang-Mills field operator FaμνFμνa with a test function f ∈ S. Due to the regularity of f, the Schwinger functions are UV and IR convergent to all orders. Tadpole diagrams are formed by the contraction of one or multiple Schwinger functions to a backbone formed by a gauge independent diagram. Due to the regularity of the Schwinger functions, the tadpole is more regular than its backbone.

5.1.4 ^*o*S′ convergence for power counting

• Proposition 5.1.2 Each entry in Tables 1 and 2 is convergent to all finite orders and in ^*o*S′. This statement applies to the SRI entries of the tables and the NSRI entries excluded from the tables.

Proof. An RPT power counting expansion term with j point vertices is bounded by derivatives up to an order which is a polynomial in j by the Sobolev imbedding theorem. The derivative bound implies a growth at most a polynomial in j also. By the definition of the ^*o*S′ topology, the power counting expansion term is in ^*o*S′.

In the fixed time RPT expansions, a generic spherical harmonics term has a mean value. This mean is SRI.

The number of spherical harmonics in the basis labeled ℓ, m is bounded by a polynomial in j, the number of vertices in the RPT expansion. This can be seen explicitly in the connected j rows of the tables. In particular, the number of NSRI terms is bounded by a polynomial in j times the number of SRI terms. A polynomial in j does not modify the ^*o*S′ convergence.

5.1.5 Power counting for fixed time RPT

• Proposition 5.1.3 The fixed time entropy principle vacuum energy diagram is UV cubically divergent. It is IR linearly divergent. The fixed time charge diagram j = 2 is finite (UV and IR).

• All two-particle insertions (UV and IR) with j ≥ 2 are power counting finite. The series of entropy maximizing k-particle insertions, k ≥ 0, as a function of j is convergent to all finite orders with ^*o*S′ convergence.

Proof. The proof of the power counting statements is direct as is convergence to all finite orders. □

5.2 The convergence proof

• Definition 5.2.1. A fluctuating term in a connected Feynman diagram is a term resulting from a choice of indices which do not occur in the SRI sum over indices.

This definition is illustrated by the next result.

• Example 5.2.1. The fluctuating vacuum diagrams are given by the formula

FaμνFμνa.(21)

The proof is by direct computation. It is the absence of the sum over repeated μν indices which makes (Equation 21) a fluctuating term. Additionally, the absence of a sum over a means that all formulas are interpreted with a fixed choice of a.

5.2.1 Renormalization and power counting

Assume removal of all overlapping divergences. Due to the removal of overlapping divergences, each modification as discussed below is treated individually. No interaction between these individual modifications occurs.

The vertices which enter into a single renormalization correction to power counting have a common T^*a* label.

Theorem 5.2.1

• The fixed time RPT series is defined by Sobolev space Tables 1 and 2 indices.
• The entropy RPT series is convergent to any finite order of RPT and in the ^*o*S′ topology, including NSRI terms.

Proof. (1) The expansion is defined by a negative Sobolev index norm in the power counting values of Tables 1 and 2 to prove the first assertion.

(2) The convergence in ^*o*S′ follows from the linear in j table entries for the generic connected j moments. This convergence justifies the termination of RPT at finite orders. □

5.3 Temperedness: S′ convergence

Temperedness is one of the Osterwalder-Schrader axioms [31]. It is the statement that the RPT functionals, defined with FaμνFμνa quadratic pairs evaluated at a single point, determine an S′ functional which satisfies S′ continuity.

• Proposition 5.3.1 By use of the temperedness axiom [31], axiom EO' is satisfied.

Proof. All proofs are given in the presence of cutoffs, with the result being uniform in cutoffs.

By use of the [2], Sobolev space ^*o*S′ convergence extends to convergence in S′ with the identical Sobolev space norms. A Feynman diagram defines an amplitude, which is a function of space and time, or more typically as a function of the Fourier transformed momenta p. Uniform estimates on the Feynman diagram integrand from [2] go beyond the power counting exponent which show the degree of exponential rate of convergence, in that the results are independent of the restriction to coinciding points.

The [2] estimates are uniform, in the sense of applying to all gauge invariant Feynman integrands which are UV finite. The estimates are very general, allowing Lagrangians which contain polynomial expressions of the field variables of any degree.

Theorem 5.1 of [2] asserts that for separated space time points in an n-tuple, with an arbitrary subset of a ν-tuple approaching one another, the difference variable function space S′ defined relative to S has a bound in terms of the ν-tuple and its approach to zero separation which is unrelated to (uniform in) the separation of the remainder of the n tuple, independently of n. This statement is given quantitatively as the existence of (non-negative) constants c_*n*, k_*n* and ℓ_*n* with the function S ∈ S′ satisfying

where

d(y1,⋯,ym)=maxi,j|yi-yj|r(y1,⋯,ym)=mini≠j|yi-yj|.(23)

If Equations 22, 23 are satisfied, then the function S has a unique extension to the associated Schwartz nuclear space.

5.3.1 Temperedness E0'

5.3.1.1 c_*n* Combinatorics

The combinatorics associated with the convergence have been subsumed within the constant c_*n* which is allowed an arbitrary dependence on n in Equations 22, 23.

5.3.1.2 r Convergence

The r property is a weak form of UV convergence.

To prove the r bound, recall from Theorem 5.2.1, the convergence with Sobolev norms is defined by Tables 1 and 2.

5.3.1.3 d Convergence

The d property is a weak form of IR convergence. It is a weak form of clustering. Clustering is an IR issue. The d bound is a power law cluster separation decay under large separation of the arguments x_ℓ. It is weaker than the clustering axiom itself, in that the cluster decay axiom does not depend on the number m of clusters. It is weaker in that the rate k of clustering decay is only assumed to be strictly positive.

Clustering is proven separately for each phase labeled by a basis element T^*a* of the gauge Lie algebra 𝔤.

The two cases of fixed time and fixed space are considered separately.

For the cluster property (Equation 23), L_∞ bounds are specified for the difference variables. For this purpose, Holder continuity of the difference variables is established.

5.3.1.4 Gauge independent clustering

Power counting estimates are given by Tables 1 and 2, for fixed time power counting.

Consider multiparticle clustering and a single multiparticle cluster X_*i*, i ≤ m. Each of the r_*i* single particle states in X_*i* has coordinates x_*i* + x_{r_*i*}, with the central position x_*i* held fixed in the analysis of clustering. For each x_*i*, define a 3-sphere S_*i* with a common (i independent) radius r. The distinct spheres S_*i* are nearly overlapping in the limit of large radius r with fixed centers x_*i*. For convenience, we treat the spheres as identical, equal to a single 3 sphere S. The number of external legs in each cluster is bounded by j_*i*/2, with j_*i* the number of vertices in the cluster i. Normally, this number of external legs would be called k_*i*, but this notation is avoided due to the occurrence of a different k in Equation 22.

A common cluster decay is to be obtained for each cluster, with a spherical angular independent constant, vanishing as a power of r as the consequence. In the limit, the fixed differences among the r_*i* do not contribute.

Taking the most divergent entries form Table 2, the IR Sobolev index is −3 − 2k with k ≤ m = n/2. The Imbedding Theorem gives Holder continuous derivatives of order −3 − m² ≤ −3 + n². (Equation 23) is satisfied.

5.3.1.5 Gauge dependent clustering

The Schwinger functions satisfy (Equation 23) due to the regularity of the test functions.

Recall that the tadpole is at least as regular as its backbone. Clustering for the backbone is as addressed above.

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Footnotes

Edited by: Snezhana I. Abarzhi, University of Western Australia, Australia

Reviewed by: Alex Klimenko, The University of Queensland, Australia

Sergey Chefranov, A. M. Obukhov Institute of Atmospheric Physics (RAS), Russia