Quantum Field Theory: A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields

. A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator (cid:10)((cid:12), (cid:13)) no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian (cid:15)((cid:10)) (cid:16) exists and that the corresponding (cid:17) ∗ - algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the (cid:20)((cid:10) (cid:21)(cid:22) ) (cid:16) , (cid:23) ≥ 2 quantum field theory models are Lorentz covariant.


Introduction
Extending the real numbers ℝ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution ( ) as a nonstandard point function.Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials.Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis.For further information on classical model theoretical nonstandard analysis namely , we refer to [5]- [8].Abbreviation 1.1In this paper we adopt the following canonical notations.For a standard set we often write ! .For a set ! let

Non-triviality
For every infinite set in the standard universe, the set % d|d ∈ * ( is a proper subset of * .Definition 1.12 A set is internal if and only if is an element of * for some ∈ V(ℝ).Let ; be a set and = % z ( z∈€ a family of subsets of ; .Then the collection has the infinite intersection property, if any infinite sub collection « ⊂ š has non-empty intersection.Nonstandard universe is isaturated if whenever % z ( z∈€ is a collection of internal sets with the infinite intersection property and the cardinality of š is less than or equal to i. Remark 1. 4 For each standard universe ¬ = ©(;) there exists canonical language / -and for each nonstandard universe • = ©(F) there exists corresponding canonical nonstandard language / = / ® any closed formula A whatsoever.
Abbreviation1. 2 In this paper we adopt the following notations [8].For a standard set we often write !, let != % * | ∈ ! ( # .We identify ) with ) # i.e., ) ≡ ) # for all ) ∈ ℂ. valued #-norm on a space T. Moreover 〈 , 〉 is #-continuous on Cartesian product T × T, where T is viewed as the #-normed space %T, ‖•‖ # (.Definition 1.14 A non-Archimedean Hilbert space T is a #-complete inner product space.Two elements and C of non-Archimedean Hilbert space T are called orthogonal if 〈 , C〉 = 0. Definition 1. 15 The graph of the linear transformation U: T → T is the set of pairs %〈Å, UÅ〉|(Å ∈ R(U))(.The graph of the operator U, denoted by Γ(Т), is thus a subset of T × T which is a non-Archimedean Hilbert space with the following inner product (〈Å I , [ I 〉, 〈Å , [ 〉).Operator U is called a #-closed operator if Γ(Т) is a #-closed subset of T × T. Definition 1. 16 Let U₁ and U be operators on H.If Γ(U₁) ⊃ Γ(Т), then U I is said to be an extension of U and we write U I ⊃ U. Equivalently: U I ⊃ U if and only if R(U₁) ⊃ R(U) and U I Å = UÅ for all Å ∈ R(U).
Thus for symmetric operators, we have U ⊂ U * * ⊂ U * , for #-closed symmetric operators we have U = U * * ⊂ U * and, for self-#-adjoint operators we have U = U * * = U * .Thus a #-closed symmetric operator U is self-#-adjoint if and only if U * is symmetric.
The main purpose of the present paper is to extend the result of [8] to ( ) , > 2. Our notation and definitions are the same as in [8].

Let
The formal expressions for the Hamiltonian and Lorentz transformation generators are given by [8] T where is the free energy density with hyperfinite cut-off q ∈ ℝ U and Ø ( , ). In and its direct consequence • , e = 1,2,3. (1.40) #b , where Ðd, ÃÑ is an #-closed interval in ℝ * À,345 # .A causal shadow of š b is defined to be the diamond In this section we consider the basic properties of T s,û and ë s,û jn , e = 1,2,3 in particular, the first order estimates they satisfy.Note that T s,û and ë s,û jn , e = 1,2,3 are well defined operators on a non-Archimedean Fock space ℱ # .We take the definition of ℱ # and the definition of the pointwise-defined time-zero field operators on ℱ # as in [8] (see [8,Section 9]).The spatially cut-off Hamiltonian is defined as self-#-adjoint operator on a non-Archimedean Fock space ℱ # [8].
0, e = 1,2,3.The spatially cut-off Hamiltonian reads where is the interaction energy density.The operator T s (•) has been studied in [8] and is known to be a self-#-adjoint semibounded operator on ℱ # .For the region Ô € u , defined above in section 1 we set now with ¶ > 0, and .
We assume now that and two additional technical conditions on the and We rewrite now the operator U j,s (Ò) as U j,s (Ò) = U j,s where (2.10) Where s n is given by (1.29) and Ò ∈ j Here s is the number operator with hyperfinite cut-off q and we have used the s -estimate [8]: Let • be a Wick monomial 12 with a kernel • ∈ / # ℝ * Â #bù , then where d + Ã ≥ .A similar decomposition holds for s n (Ò), e = 1,2,3.The result reads: . Then, That is convenient to approximate the operators ë s,û jn , e = 1,3,3 by the operators ë s,-,û jn , e = 1,3,3 with an additional momentum cut-off where U j,s,-and U €,s,-are defined by cutting off all the momentum integrals at |e| > .That is, U j,s and U €,s , are expressed as a sum of Wick monomials (2.12) each of which is replaced in the definition of U j,s,-and U €,s,-by Here ' -e I , … , e ù = 1 if |e z | ≤ ≤ q for all 1 ≤ • ≤ , and ' -e I , … , e ù = 0 otherwise.We abbreviate also Note that as a rule, estimates that hold for ë s,û jn also hold for ë s,-,û jn , uniformly in .For example, for all ∈ ℝ À0,6 # * , ≤ q: and for I + ≥ 2, where the constants are independent of .As a domain of admissible vectors in Proof Note that for e = 1,2,3 Moreover, by (2.14) we can choose d so that The second part of (2.21) follows by a similar consideration,

Quadratic estimates
In this section we prove the self-#-adjointness of the operators ë s,- jn , e = 1,2, by interpreting the operator U j,s,-as generalized Kato perturbation [8].Thus we need proving quadratic inequalities where d -and Ã are constants with d -depending on .Here is finite constant and ( ) satisfies conditions (2.5).

Theorem 3.1
The operators ë j,s,- zn , e = 1,2,3 are essentially self-#-adjoint on R # .There are constants d and Ã independent of , such that for < q and e = 1,2,3 Remark 3.1 For we use the "pull through formula" (3.5).Let U s = #-T j,s + © s ^^^^^^^^^^^^^^^ and We shall always be concerned with operators T that are essentially self-#-adjoint on domain OE 345 # defined in (2.17), and whose perturbation © is a finite sum of Wick monomials with #-smooth kernels.It follows that d( ) is defined on the #-dense domain and that (3.3) holds on this domain.
Lemma 3.2 Suppose that U s = #-T j,s + © s ^^^^^^^^^^^^^^^ satisfies the above conditions.Let [ ∈ OE 345 #g , where () − ) is in the resolvent set of U s for all ≥ 0. Then for ∈ ℕ a positive integer where š = %• I , … , • ( be a set of distinct ordered positive integers, (1, ) = %1, 2, . . ., (, d € = -∏ d HI e z ! for " > 0, d € = I for " = 0.The sum in (3.5) takes place over all partitions of %1, 2, . . ., ( into disjoint subsets š I , . . ., š ƒ0I (including permutations among the subsets) for ˜ = 0, 1, . . ., .The elements of each š z are taken in natural order.Let È != È($), È()) = (U s − )) yI , where … , cd e z % , ©h … h for " > 0 and © € = 0 for " = 0. Note that the sum (3.5) includes terms where « ƒ0I is empty but not š I , . . ., š ƒ ; this convention adjusts the sign (−1) ƒ correctly.The ˜ = 0 term is simply È I d (I,ù) [.Proof In order to apply (3.5) to the proof of (3.1) we must be able to estimate the commutators (3.8) Proof Let ℱ # , ∈ ℕ * be the -particle Fock space.Now ; s I e is defined on R for all e and since ; s I e maps ℱ ù # into ℱ ùyI # , it is sufficient to prove that (3.8) holds for [ ∈ R ∩ ℱ # with the constant independent of .We remark that by the methods of the previous lemma it is easy to show that the integrand in (3.8) is uniformly bounded in e, but different methods are necessary to prove it integrability.Now we define where I e, & is given by (2.9); therefore we obtain Replacing now e by & in (3.9) we get where d is a constant and We shall write this symbolically as ƒ & ƒ , suppressing the other variables.In obtaining (3.
For then the integral over p is a convolution between , and the integral over & ƒ is the square of the / # #-norm of this convolution.

Now we get
In order to justify the replacement of ƒ & ƒ by ƒ (&), we set and therefore we obtain to (3.12), we obviously get s on the left and s g from the first term on the right.To estimate the second term, we note that .Therefore the integral of the second term in (3.12) can be estimated by But, as before, this is the square of the / # -#-norm of the convolution of the function [ with a rapidly decreasing function and so it can be estimated by where the constant is independent of ∈ ℕ * .The third term resulting from (3.12) can then be estimated by the generalized Schwarz inequality applied to . Hence s is bounded as claimed.The single commutators (3.6) are all that we need estimate.For let š = %• I , … , • (; then oeU j,s for [ in the dense set R I,n = U s,- jn + Ã R as in (3.4).This choice of [ ensures that È s,-(−Ã)[ ∈ R I,n is in the domain of all the operators we wish to apply to it.Here Ã is chosen so large that ) and (3.17).However, by Proposition 3.7, the statement Ù ƒ,n implies ƒ0I,n , e = 1,2,3.

Higher order estimates
In this section we derive higher order estimates of the following form and where d -and -are constants depending on .The estimates (4.1) are used to prove that the powers ë j,s,- jn ƒ are essentially self-#-adjoint on OE 345 # and do not survive in the #-limit: → # q; on the other hand, the estimate (4.2) does transfer to the #-limit = q and, in fact, enables us to prove that this #-limit exists.For real ß ∈ ℝ * Â # we define the generalized number operator with hyperfinite momentum cut-off q ∈ ℝ À = U j,s,-Ò where me have converted all but one T j,s,-+ š I/ into an integral of products of annihilation operators.We apply the pull through formula  and so we must argue more carefully.Define now the operators A n ( )= T j,s,-+ š (ùyI)/ T j,s,-+ U j,s,- jn + U €,s,- jn + Ã , e = 1,2,3 on the domain R. It is sufficient to prove that R is a #-core for A n (1).For then (4.12) extends from R to R(A n (1)); by induction (4.13) holds on R å ë s,- jn + Ã (õˆ") w é ⊂ R(A n (1)), and the proof of the lemma is complete.As in the proof of Theorem 3.8, we consider a sequence of values ƒ = ˜/«, ˜ = 0, 1, . . ., «, and regard the operator as a perturbation of A n ( ƒ ).By (4.12) for any Å ∈ R, where the constants Ã and are seen to be independent of ƒ ∈ Ð0,1Ñ.But, as in the next lemma, T j,s,-+ š (ùyI)/ U j,s,-T j,s,-+ š Hence, by choosing hyperinteger « ∈ ℕ * 6 , « > -, we have for Å ∈ R, where d = « yI -< 1.That is, C is a Kato perturbation of A n ( ƒ ).Note that domain R is a #-core for A n (0).This follows from the facts that (4.8) holds when = 0, i.e. when ë s,- jn is replaced by T s,- jn = T j,s,-+ U €,s,- jn , and that powers T s,- jn ù are essentially self -#-adjoint on R. From 4.14 we see that R is also a #-core for A n (0) + = A n ( I ) and that R(A n ( I )) = R(A n (0)) Continuing in this way we reach the conclusion that R is a #-core for A n (1).To complete the estimate (4.1), we dominate powers of ë s,- jn , e = 1,2,3 by powers of T j,s,-.

Lemma 4.3
Let ˜∈ ℕ * be a positive hyperinteger.Then there are positive constants Ã and -, where -depends on such that Here 2 is the order of the interaction.
Proof Here 2 is the order of the interaction.Since T j,s,-+ Ã ƒ is essentially self-#-adjoint on R it is sufficient to prove (4.15) for [ ∈ R. Now because of the momentum cutoff, ë j,s,- jn has the form ë j,s,- jn = T j,s,-+ ∑ • z , where • z is a Wick monomial (2.12) whose kernel has #-compact support.where A is a bounded operator.For then it is clear by hyper infinite induction that T j,s,-+ Ã y ƒ is bounded.Take W of the form (2.12) with < 2 .Then By an extension of the basic estimate (2.13) to cover the case of operator-valued kernels, it follows that is a bounded operator.This completes the proof of the lemma.Note that by the generalized spectral theorem [8], the dependence of Ã can be incorporated into constant -.
, where Ã is a large positive number.By the previous two lemmas we have that Since R is a #-core for T j,s,-+ A ƒ , it follows from (4.15) that Therefore, by (4.17),
We insert this inequality into (4.19) and estimate the integral over the "variables" of š I .Say š I = %• I , … , • ð (.We must estimate where does not depend on the variables of š I , for which we recall that « = š ∪ š 0I ∪ … ∪ š z0I .Now In this way we integrate over the variables of š ∪ ... ∪ š z to obtain By a change of variables we can rewrite the sum over ˜ and š as a sum over subsets %1, 2 , . . ., "( of 1, 2 , . . ., .Using the estimates (3.19) and (4.7), we get where the " = 0 term is simply T j,s,-+ š w é, we obtain by the inductive hypothesis, ù0I,s,-= T j,s,- where the constant is independent of .Corollary 4.6 Let > 0 and be a positive integer.Then there are constants d and Ã independent of such that (2) For • + ˜≥ 2, (5.9) Proof Equation (5.7) is a consequence of estimates developed in [8] for Wick monomials with one creating and one annihilating leg.These estimates involve / I # -/ 6 * #≈ #-norms on the kernels such that As an example of (5.7), we consider the case • = 1 and ˜ = 2.As in (5.4), We see that for particle vector see [8].According to the definition (5.10) by (5.5) and (2.9) we obtain Notice that (6.1) is operator equality, since for ℝ * Ä,345 # valued function Ò, s (Ò) is a self-#-adjoint operator whose domain includes R } ë s zn + Ã I/ •.In addition, we prove on the domain Here the vectors ( , ) and Λ » ( , ) are in Ô € u , and the forms in (6.2) are #-continuous in and by the first-order estimate (5.16) and results of [8] sect.6.
Notice that the main part in the proof of (6.1) is to verify the commutation relation (1.15) for Ò ∈ j 6 * Ô € u , ℝ * À,345 # and • a cut-off function for the region Ô € u .For convenience, we assume that a function Ò with support contained in the region Ô B defined by and where > > 0 is some small enough number.This represents no loss of generality since any Ò in can be presented as a sum of such Ò.It follows from this assumption that if is related to a non-Archimedean von Neumann algebra ℜ(š b ) generated by the set The main parts of the proof are as follows: where ë s zn ( ) = Ð -exp(−• T s )Ñë s zn Ð -exp(• T s )Ñ.Note that a zn ( ) is well-defined and three times #-continuously #-differentiable by (5.19) and [8, Section 6]: for ˜= 0, 1, 2 , . ... Obviously one obtains, Part2.The commutators in (6.7)-(6.8)can be evaluated.On R # × R # one obtains, in the sense of bilinear forms, where s n , e = 1,2,3 is a locally correct momentum operators where Part5.The commutators on the right of (6.16) can be evaluated by passing to the sharp time fields, where the subscript " indicates the time dependence of a function Ò.The result for | | ≤ > reads Since supp Ò ⊂ Ô B , we can integrate (6.17) with respect to and thus on domain R T s 0b × R T s 0b we obtain In this section, we will discuss the generalized spectral theorem in its many aspects.This structure theorem is a concrete description of all self-#-adjoint operators.There are several apparently distinct formulations of the spectral theorem.In some sense they are all equivalent.The form we prefer in this section, says that every bounded in ℝ * À # self-#-adjoint operator is a multiplication operator.This means that given a bounded in ℝ * À # self-#-adjoint operator on a non-Archimedean Hilbert space T # , we can always find a #-measure l # on a #-measure space ë and a unitary operator ¬: T # → / # ë, m # l # so that (¬ ¬ yI Ò)( ) = a( )Ò( ) for some bounded ℝ * À # -valued #-measurable function a on ë.In practice, ë will be a union of copies of ℝ * À # and a will be so the core of the proof of the theorem will be the construction of certain #-measures.Our main goal in this section will be to make sense out of Ò( ), for Ò a #-continuous function.We will consider also the #-measure defined by the functional: Definition 7.1.The operator #-norm of a linear operator : T # → T # is the largest value by which stretches an element of T # , otherwise operator is called unbounded in ℝ * À # .We often write bounded operator instead bounded in ℝ * À # and unbounded operator correspondingly.
Proof of the Theorem 7.1.Let Å( ) = ( ).Then ‖ Å ( )‖ ℒ ó # = ‖ ‖ t # (#(n)) so Å has a unique linear extension to the #-closure of the polynomials in C # (i( ).Since the polynomials are an algebra containing š, containing complex conjugates, and separating points, this #-closure is all of C # (i( ).The first and simplest application of the l z # is to allow us to extend the #-continuous functional  In this section we will show how the spectral theorem for bounded in ℝ * À # self-#-adjoint operators which we developed in section 9 can be extended to unbounded in ℝ * À # self-#-adjoint operators.

10 )
(3.5)  to pull the d (I,ƒ) through the È n , and we dominate the factor T j,s,-+ -∑ l(e z ) e ƒ & ùƒ l I , … , l ƒ ×|n )Let us consider a typical factor È !© s € !È !ˆ" , regarded as a function of the variables e z " , … , e z @ , where • A ∈ š , ; = 1, … , .Because of the momentum cut-off, the estimates (3.16) and (3.23) hold: Each such monomial • z maps domain R into a set of vectors which have a finite number of particles and which are of #-compact support and 6 * ℝ * Â # #-almost everywhere in the momentum variables.It follows that ë j,s,- jn ƒ can be expanded on R into a sum of welldefined products of the form = -∏ T j,s,- z w'ˆ" ù •Hj • z w'ˆw where -∑ • HI = ˜, and • represents a typical Wick monomial in ë j,s,- jn .Each such product can be dominated by T j,s,-+ Ã ƒ provided that Ã is chosen sufficiently large, say Ã > 2 ˜l( ).It suffices to show that • T j,s,- e ù G e I , … , e ù × |n ) |o-× d * e I ••• d e ù , where G e I , … , e ù = }T j,s,-+ d − 2 l • z }T j,s,-+ d ± l e I ± ⋯ ± l e ù • yz , where the ± is chosen according to whether the corresponding d # e is an d or d * e .Since −2 l ≤ ±l e I ± ⋯ ± l e ù the operator #-norm ‖G e I , … , e ù ‖ # ≤ |• e I , … , e ù |.

3 .
ProofThe Corollary follows immediately from the Theorem by means of (4.5).Remark 4.1 The estimates(3.19)and (4.22) do not permit us to dominate the operator T j,s,-itself by − U j,s,-