JHEP05(2018)191 Springer May 20, 2018 May 27, 2018 May 30, 2018 oint energy, : : : March 16, 2018 : and Renormalons, Revised Accepted Published Received ntization. en applied to renormalizable this situation is built into con- ads to the desirable feature of ds, as well as general relativity. al classical theories, an alternative Published for SISSA by https://doi.org/10.1007/JHEP05(2018)191 . 3 1803.05823 The Authors. c Nonperturbative Eﬀects, Renormalization Regularization

Traditional quantum ﬁeld theory can lead to enormous zero-p , [email protected] Department of Physics, UniversityGainesville, of FL Florida, 32611-8440, U.S.A. Department of Mathematics, UniversityGainesville, of FL Florida, 32611-8440, U.S.A. E-mail: ArXiv ePrint: Models of Quantum Gravity and nonrenormalizable covariant scalarSimpler ﬁelds, models fermion are ﬁel oﬀered as an introduction to aﬃneKeywords: ﬁeld qua which markedly disagrees with experiment.ventional canonical Unfortunately, quantization procedures. Forquantization identic procedure, called aﬃnehaving a ﬁeld vanishing quantization, zero-point le energy. This procedure has be Open Access Article funded by SCOAP Abstract: John R. Klauder Quantum ﬁeld theory with no zero-point energy JHEP05(2018)191 ], ]. 1 2 3 5 2 dλ λ for ) is ) ) )] = i (1.1) λ y ( 0 ( | c † x, λ ˆ ), a for- π ), where ) ( , ∂/∂λ x λ ) ( N , ideally is ( x φ x i λB ( ) ( 0 † ˆ W | φ y , and Hilbert ) ) + ( ), and A 2 x / serves to main- − 1 s ). x ··· x, λ − x, λ − | ( ( † ∂/∂λ δ sN λ ed by ) ( | B 2 R λ , and its momentum R = [ ( x } )11, and [ 2 ( ~ mentum, we substitute } y i ) = 0 for all ) L ≡ ommutation relation [ A 2 1 λ y † − ,... (length) ( ) ( ) ∈ the quantum Hamiltonian, x 3 CQ) when a classical ﬁeld x , c s 1 h an annihilation operator. x s x ) , ∝ ( ( , κ i ≡ x 2 3 ily canceled by normal order- ) ˆ ( N ≡ − δ 0 φ b , = 0 for all ( | ways a term composed only of x c potential term, normal order- A 1 dλ d ) ) ( dλ d i ) x λ ) φ → 0 ( ] = ( | { † ) ∈ { τ ) A ) , . . . , x x, λ x x, λ y 1 ( ( ( ( , s x x, λ φ s B ( A ( ) ) ,A f R A (0). Local operator products are given ) x , ∈ ( , where R A x dλ ∈ – 1 – ) ∂/∂λ, λ ∂/∂λ, λ 2, [ ( ( λ h h √ † † x, λ / < ǫ < W ) ) ( ], which leads to renormalized (R) products such as )] 2 B x ) 2, where ( x, λ x, λ λ ( ( √ ) — a ﬁeld that is featured in the enhanced quantization ( A , etc.; here, the ﬁxed factor / A B x τ ( † − )] with 0 )11, where dλ ) Z Z φ † x ) λ ) ) ( ( x = ≡ ∞ c x A ( x, λ ( ( x, λ π H ( A + < [ B ) + ≡ † B ~ R ) i ) 2 ) λ x λ x ( x, λ ≡ † ( − ( ) ( A κ ) = ) ] — which leads to the Poisson bracket A x x 1 W ( ( x, λ ≡ ˆ ˆ κ π ( ) = [ ) B when smeared with suitable elements x → → ) = ( R )11. As usual, the ground state of the Hamiltonian, designat ˆ b λ φ ) y x, λ ∞ ) ( ( x x = ( − ( B → π κ 2 R This situation is inevitable with canonical quantization ( However, there is also a diﬀerent approach. In place of the mo x dilation ﬁeld ) ) ( N < < W x x δ ( ( ~ ˆ mal Lie algebra for the aﬃne group. As the analog of a current c the real ‘model function’; for the example oﬀered in section 0 by an operator product expansion [ φ Here the unique, normalized, ‘no particle’ state for which i an aﬃne quantization of these variables leads to ﬁeld and ing cannot eliminate thecreation zero-point operators. energy since there is al 1 Introduction The source of zero-pointing energy for of a the free HamiltonianUnfortunately, scalar operator, for ﬁeld a which is non-free leaves read scalar every ﬁeld, term say wit with a quarti 3 Ultralocal scalar model: a4 nonrenormalizable example Additional studies using aﬃne variables Contents 1 Introduction 2 Nonrenormalizable models exposed φ which is given by tain proper dimensions. Such local products become terms in space is given by suitable combinations of terms such as all the (EQ) program [ JHEP05(2018)191 , . 0, − ∞ =0 the has 2 and g ) (2.1) (2.2) (1.2) t D H g > ( q = 0 N < ⊂ 2 1 ) implies 0 q R free model − ( g> 2 2 , ) L D t ~ cross back and ( for all q ˙ 6∈ q T/ ). i 1 2 D ) =0 2) 0 solution is given , = 0, has a domain 0 [ N g | / λ dt ~ ( 0 T † g ] D c ) → +1 4 R +1) , which has a classical T/ never cross n s − N 3 g ( ( 2) . iven by i 0 1 . Clearly, / = g q R 2 , λ − is continuously connected − ( − g> e ) g . Speciﬁcally, a free model +1 2 N 2 rather than the 0 ) I λ ∞} n / ′ D x L ( ) ( le models. Consider the one- /∂λ =0 ( i q 2 y ) g ( q → − (because ∈ A on within λ n raint that its associated domain e − D ) ( i g 2 , while a h ) dt < c 0 q ′ ] ) N ··· with no spatial derivatives, then, | β 2 ( ˙ q 4 [ ∂/∂λc ′′ † ∂ ( q 2 0 T ) , λ − 1 n V ( ) an interacting model is continuously 2 0, as normalized expectations of the R q N − ) h n λ and ) ~ ) ( h , λ + 0, > λ ′′ i/ 2 ( ( 2 q B belong to ... x h c e ( q 7 ( continuously connected to the ‘free model’! lead with an annihilation operator). The → , n ′′ 5 , . . . , x + A , , and, to obtain a unique ground state, we q + h g ′ 0. The free model, with pseudofree model † 3 1 H ∂/∂λc , while the interacting theory, with any not , 2 q 2 ) 1 )= ∞ q 1 not ,... ≥ , λ – 2 – [ ˙ T − =1 3 ( 1 ) , ∞} n T 0 g q , λ (0)= /∂λ 2 )), for some x λ , q 1 2 R ( ( 1 x , ), for any Z x c ∂ ( )Σ f . In fact, solutions to the equations of motion when : ′ ( dλ < β g φ q − − =0 q dt < ( 2 A ′′ unique ground state n N terms in ] ) { + and does q 2 V = ) = 0. For the example of section λ 0 t ( q 0 ( ) = = λ θ → c + all ( + g 0 g> c )] 2 2 cos( ) 2 has a ) D q 0) = Σ g> = 2 [ ˙ λ x , = 0. In brief, as B ( ′ H D , which ensures that T 0 0) = lim q with normalizable eigenfunctions are positive), and (iii) | 6 π R ∂/∂λ, λ , ; ) 1 2 ∞ ′ ( h : β q ) = h (1 + t ,T ; (because ∂/∂λ, λ / = q 0 solutions belong to a ( ′′ ( + 2 { q 0, (ii) q ] is given by h are Hermite functions. On the other hand, the propagator for t ,T λ ( [ 3 → ′′ dλ = f ≥ q R 2 g ( =0 K ) cos( H =0 ∞ n ) would conﬁrm. λ pf g } ( B ) c = 0, solutions to the equations of motion within K D 1.2 q R ( q ±| , where the coupling constant n dt h . Thus, the ‘interacting models’ are ] = 1, { 0 ﬁnd that the limiting behavior of such solutions as 4 ) = t ~ − ( 0 Quantum mechanically, the propagator for the free model is g Hence: (i) g> ) q t D ( pseudofree model [ where A simple ‘toy’ example canvariable help problem understand with nonrenormalizab a classical action functional given b 2 Nonrenormalizable models exposed when smeared with suitable elements We say that the connected with the free actionis functional but with the const of functions solution is given by Hilbert space has combinations of to a solution that belongs to are always positive org always > negative has a domain of functions gq no zero-point energy that all eigenvalues of by forth over and, in particular, while solutions to the equations of moti Hamiltonian density of which requires that with We require that top line of ( insist that JHEP05(2018)191 2], ] + / and and 2 k = 0. s (3.3) (3.2) (3.1) (3.4) a k i Q Q ) k 2 0 Q 0 to ap- 0; , a m ψ ~ , which, in | and → ; s + ) k a k k . a > 2 P k φ φ k ( ∝ P . 0 is understood ,Q s Y a k iltonian prior to dφ 2 i ). k − P − s 2 k → and ( a a φ k ) [ ... a dφ 3.1 h k x , 2 1 ,a is large but ﬁnite and s s H P ~ − ) of step a d ; { s , ) s a K − and s i ,a 3.1 /a a ~ a 6 k ~ ; 4 φ ) k k ) = h φ φ k x → 4 k ( ( 6 on for a regularized classical k ( ry because if it is taken about sary that φ Y Y f 0. To ensure a reasonable limit → ∞ φ k ,Q 0 − − 0 1 π k , where 6! g e e g → } K P ( − Z Z a + has the form exp[ k ) + / / a ,... i k k 2 k i H 2 2 k 4 k ). This model has also been discussed φ ) ′ 0; φ dφ dφ 2 0 ± x x h This result implies that any attempt at a fulﬁlls the relation ( , s s ψ 4 | m k a a 1 φ i k − ) ) f 2 0 k ± , and the characteristic function of the regu- 0. Clearly, the interaction term has changed + φ in a power series, which leads to ,a ,a 1 x , 4! 0; h ~ ~ m 2 ( k ; ; 0 k ∞ ψ s k δ – 3 – π | f (since the distribution is ‘tall and narrow’ about + + φ ( /a ( u < < a 2 ~ Domains matter! = 1 2 ≡ { Y sj i ) k ) 2 k a − φ x } Z ( e ( k ) φ , a 0 if ′ h is the spatial volume implicit in ( Y ∝ ~ π ~ ∈ x 2 k / 0. Since there is no connection of the dynamics be- − ; ( ≡ a s f X j e k i a 1 2 ) k j φ 1 k k , π 2! -dimensional spatial lattice for ( → u 2 k , ) φ φ ≤ ∞ ( s − φ k k ) = x ( θ − h K a ( if if Z V Y 1 φ e e , where we assume the last term is polynomial in { π, φ s ). Moreover, at each site ( Z Z } ∈ a k ) = k s k k ) = K } 0 and ) H , a ,Q a π, φ k ~ , where ( ; = Π ; P ) = Π k u > V k ) = Π ( , . . . , k H f k 2 φ f ], and we will oﬀer just an overview of its quantization. ( 0. Here the ground state ( ( ,Q 6 → K k , k – 1 if K s 1 → P C 4 C k ; ≡ → H < Y ~ { ~ K a ) ( ) u k = O ( θ k −∞ , φ + k Let us ﬁrst introduce an Next we discuss a quantization based on CQ for the lattice Ham 4 k π ( represents a cell volume. Clearly, a suitable limit of Q k s 0 The ﬁnal issue involves taking the continuum limit as it is helpful to expand the term involving larized overall ground-state distribution is given by where vanishes if The normalized representation Schr¨odinger such that The classical Hamiltonian for this model is given by the paths that contributeperturbation to series the must path be integral. the taken free about theory the every term pseudofree theo will diverge! 3 Ultralocal scalar model: a nonrenormalizable example g before, e.g., [ where proximate the spatialHamiltonian integral, given which by leads to the expressi with the Poisson bracket taking a spatial limit with a In order to achieve a meaningful continuum limit, it is neces where the present case, means that tween spatial points, it follows that at each lattice site H JHEP05(2018)191 ] , ) ], ). ~ x x 7 s } 2 nal ( / d ˆ κ 2 (3.8) (3.9) (3.5) (3.7) (3.6) 2 m [ (3.10) ) ,... x → 3 bmλ ( , , ) − f s 2 k x e , a R ( 2 1 / κ dφ R 1 ] to yield the ribution when ) 4 k − ) = 0 and, for − 1 s | ∈ { x . φ φ s λ ba ( 0 | y, a (generalized) d 2 j 0 g . − o ψ ) k + (1 4 ) = ˆ is a ﬁxed constant so 2 φ − ) λ | ) = exp[ dφ 2 k λ x ( b k f 2 . κ, ( φ c ) . ( φ (ˆ for all s to operators φ 2 2 0 | ) m , s 0 C ′ K − ba s s 2 m g ) 2 a b an is chosen as a ba H ) x 2 − | , and + ( + ,a + − tion 4 3 (1 λ ˆ ∝ ~ φ 2 | ; 2 (1 ), and the classical Hamiltonian − 2 2 k − k | − a ) x − φ k | φ i ( ( λ x j 2 k dλ/ φ (0) ( κ Z 2 | 2 k ~ ) φ δ φ ~ − | s ~ φ 2 ; e ), when very tiny, provides the overall 2 0 a F h s 4 3 λ ~ ) s ) satisﬁes 0. In this case, s a ( i m 2 ) z φ ,a 3 4 ~ + ba ( ~ − ,a − / ) and ; → 0 2 ~ s k a e + ; x a ψ 0 ) + φ ( k 2 ( i k g φ x + φ ) ~ φ ( Z ( /∂λ undergo an operator product expansion renor- x k 2 k Z − κ 2 – 4 – ( λ/ if e 2 − ) ∂ π Z e x e ) − 2 ( s /∂φ ) ) ~ − 0 for this solution has been shown [ 2 s if x ba − e ) = ˆ ( ∂ 1 ( ba 2 x h φ ( → ~ ( − ~ ) 1 / k ˆ s κ s x 1 Z ~ − 2 ( 2 a ) h b − s k κ − ) is a regularized version of 1 2 φ a ) = Π Z s k ba x 1 an integral of the ground-state distribution is proportio ( 2 1 − 2 ( . Besides a Gaussian distribution found by CQ, the resultant if x s ) ba ˆ z e φ ) = d − φ ) 2 1 n ( − 1 x | ≪ 0 Z . This result is a standard result of the Central Limit Theore ( Z k 3 2 b ψ Z ˆ κ k φ ∞ k k − | )( s Π Π ) = ∂/∂λ, λ X 0 0 ( ba prevents divergence of an integral of the ground-state dist h → → ), and it formally follows that , and suitable factors in κ, φ a a s − (1), in which case the pre-factor ( x Z ( ) = < R < ′ a ( 2 1 ˆ O φ = exp = lim ˆ k φ ( H ) = lim φ , 0 + κ, f (ˆ → ≡ is dimensionless. The ground state ) is the only other result of the Central Limit Theorem, namel R 1 ( = ) ′ K − 1, but when s C ) F x 1, has a form such that λ 3.9 H s ( ba φ As an example, suppose we take the limit Next we introduce classical aﬃne ﬁelds ≪ ba | ≫ = 0), leading to the result that the continuum limit becomes s k k φ form ( ba for some The exponent malization which leads to Moreover, the classical limit as and | where to ( starting classical Hamiltonian. Poisson distribution (based on the new moments which implies a Gaussian (= free) ground state for CQ. A quantum study begins by promoting the classical aﬃne ﬁelds normalization. Once again we study the characteristic func where and thus φ that On the same spatial lattice as before, the quantum Hamiltoni becomes JHEP05(2018)191 , ], j er ) 17 – m ], and (3.11) -point ~ 14 13 ommons , – 1 10 n leads to a ingular ), which may 2 ~ ( , O | λ | ] has led to gravitational 9 oreover, the spectrum of , World Scientﬁc, Singapore dλ/ , ~ the ground state implicitly that are lied to other problems. The 8 cal free action functional is / redited. 2 ariant scalar ﬁelds [ Further use of aﬃne variables lpful suggestions. seveal papers, e.g., [ els [ bmλ g action functional and which is ) has a uniform spacing (2 − 0. Just as with the ‘toy’ example e i 3.11 , McGraw-Hill, New York (1980). → ~ 0 λ/ , Cambridge University Press, Cambridge g ) x ( if . e − – 5 – 1 h Z (1973) 295 x s d 14 ), which permits any use, distribution and reproduction in Quantum Field Theory Z b ], for which conﬁrmation requires multiple selected comput − 20 – , and which, as promised, correctly reﬂects a vanishing zero 18 , 1 CC-BY 4.0 ···} ) = exp , J. Funct. Anal. 3 f This article is distributed under the terms of the Creative C Enhaced Quantization: Particles, Fields & Gravity Beyond Conventional Quantization ( , , 0 2 , C Quadratic Forms and Klauder’s Phenomenon: A Remark on Very S 1 , , and from a path integral viewpoint, this domain modiﬁcatio 0 2 ]. ∈ { 4 j Perturbations (2015). (2000). This last reference also discusses diﬀerent singular terms [3] B. Simon, [2] C. Itzykson and J.-B. Zuber, [1] J.R. Klauder, [4] J.R. Klauder, where in section Enhanced quantization and aﬃne variablesuse have of also aﬃne been variables app to discuss idealized cosmological mod be relevant in alternative applications. 4 Additional studies using aﬃne variables energy [ References The referee is thanked for several usefulOpen corrections and Access. he any medium, provided the original author(s) and source are c references therein. Moreand quantum complex gravity [ applications includesimulations. cov Acknowledgments bounces rather than anapplied initial to idealized singularity gravitational of models the has universe. been given in diﬀerent set of pathsthe relevant than quantum in Hamiltonian the associated with free-model ( path integral. M Attribution License ( The resultant characteristic function is given by strictly larger than theunchanged domain in of the the limit classical of interactin the coupling constant which does not representdescribed a here traditional reﬂects free the model. fact that However, the domain of the classi JHEP05(2018)191 , , ]. ] , y SPIRE ]. IN [ (1971) 227 (2015) 83 SPIRE , (2015) 124018 8 IN time Dimensions Singularity Smooth quantum (2016) 124053 ][ ne quantum gravity , he physical Hilbert 182 (1970) 307 arXiv:1501.02174 D 92 [ 18 D 93 ]. (2012) 285304 ]. , ]. SPIRE A 45 Nonadiabatic bounce and an Vibronic framework for quantum ]. Phys. Rev. SPIRE IN ]. , , z, IN SPIRE ][ nd W. Piechocki, nd W. Piechocki, Phys. Rev. (2015) 061302 IN , ][ arXiv:1512.00304 [ . ]. SPIRE ][ Theor. Math. Phys. SPIRE IN J. Phys. IN 2, , D 92 ][ ][ ≥ Acta Phys. Austriaca Suppl. SPIRE n , theorem IN Commun. Math. Phys. , , 4 n ][ – 6 – φ limit gr-qc/0102041 (2016) 064080 [ arXiv:1206.4017 Phys. Rev. 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