Basics of scalar and vector Finite Quantum Field Theories are recalled, stressing the importance of the quantization of classical physical fields as Operator-Valued-Distributions with specific fast decreasing test functions of the coordinates. The procedure respects full Lorentz and symmetry invariances and, due to the presence of test functions, leads to finite Feynman diagrams directly at the physical dimension D = 2. 4. In dimension 2 it is only with such test function that the canonical quantization of the massless scalar field is found to be fully consistent with the most successfull Conformal Field Theoretic approach, pioneered by Belavin, Polyakov and Zamolodchikov in the early 1980's. The question is then raised how Poliakov's wordline path integral representation of the relativistic string could possibly lead tofinite Feynmann diagrams. The natural way of inquiries is through the extension of the string formalism with classical convoluted coordinates leading then to Operator-Valued-Distributions and thereby to Finite Quantum Field Theories. It is shown that in the process some age-old certitudes about quantized strings are somewhat jostled.

Basics of scalar and vector Finite Quantum Field Theories are recalled, stressing the importance of the quantization of classical physical fields as Operator-Valued-Distributions with specific fast decreasing test functions of the coordinates. The procedure respects full Lorentz and symmetry invariances and, due to the presence of test functions, leads to finite Feynman diagrams directly at the physical dimension D = 2.. 4. In dimension 2 it is only with such test function that the canonical quantization of the massless scalar field is found to be fully consistent with the most successfull Conformal Field Theoretic approach, pioneered by Belavin, Polyakov and Zamolodchikov in the early 1980's. The question is then raised how Poliakov's wordline path integral representation of the relativistic string could possibly lead tofinite Feynmann diagrams. The natural way of inquiries is through the extension of the string formalism with classical convoluted coordinates leading then to Operator-Valued-Distributions and thereby to Finite Quantum Field Theories. It is shown that in the process some age-old certitudes about quantized strings are somewhat jostled.

prior to the construction of the Lagragian density. The use of DR does not however address directly to the origin of these divergencies but just avoids them in going to an hypothetical space in D = 4-? dimensions. T LRS was developped in Ref. [11,12]. Since the early applications of this scheme [13,14] the calculation of radiative corrections to the Higgs mass [15] and the treatment of the axial anomaly [16,17] are relevant illustrations of the practical use of the T LRS procedure in the D = 4 context. It was shown recently how T LRS solves the long-standing consistency problem [18] encountered between EqualTime (EQT) and Light-Front-Time (LFT) quantizations of bosonic twodimensional massless elds. Our purpose here is to confront the ndings of [18] with the standard bosonic string theory approach of [19,20] and elaborate on the values of the critical dimension for the cancelation of the conformal anomaly.


# II .

THE MATHEMATICAL SETTING


# Classical wave equations

To the original classical eld-distribution ?(x 0 , x 1 ) is associatted a translationconvolution product ?(?) built on a rapidly decreasing test functions ?(x 0 , x 1 ), symmetric under reexion in the variables x 0 and x 1 . In Fourier-space variables this linear functional can be written as an integral with the proper bilinear form ? p, x ?= p a g a,? x ? (g a,? = diag{1, -1})
(? * ?)(x 0 , x 1 ) = dp 0 dp 1 (2?) 2 e -??p,x? ?(p 0 , p 1 )f (p 2 0 , p2 1 )
, where ?(p 0 , p 1 ) (resp. f (p 2 0 , p 2 1 )) is the Fourier-space transform of ?(x 0 , x 1 ) (resp. of ?(x 0 , x 1 )). Hereafter ?(x 0 , x 1 ) will stand for (? * ?)(x 0 , x 1 ).

The wave-equation for the classical convoluted distribution in space-time variables is obtained from the hyperbolic partial dierential equation (HPDE)
??(x 0 , x 1 ) = ? 2 x 0 -? 2 x 1 ?(x 0 , x 1 ) = 0.(2.1)
A solution of the Cauchy problem in the sense of convolution of tempered distributions is nothing else than D'Alembert's (1717 -1783) solution. It can be written as
?(x 0 , x 1 ) = 1 2? d 2 p?(p 2 0 -p 2 1 )?(p 0 , p 1 )e -??p,x? f (p 2 0 , p 2 1 ),(2.2)
with ?(±|p 1 |, p 1 ) = ? ± (p 1 ) . Canonical quantization of the zero mass scalar quantum operator valued-distribution (OPVD) eld ?(x 0 , x 1 ) proceeds from Eq.(2.2) via the correspondance, in terms of creation and annihilation operators, {? -(p) ? a ? (p), ? + (p) ? a(p)}, with commutator algebra [a(p), a + (q)] = 4?p?(p-q) and a vacuum | 0 > such that a(p) | 0 >= 0 ?p. That is London Journal of Research in Science: Natural and Formal
?(x 0 , x 1 ) = 1 4? ? 0 dp p
[a(p)e -?p(x 0 -x 1 ) + a ? (p)e ?p(x 0 +x 1 ) ]f (p 2 ).

(2.3)

Then, one easily evaluates the commutator of two free scalar OPVD to
?(x), ?(0) ? ??(x) = - ? ? ? 0 dp p sin(px 0 ) cos(px 1 )f 2 (p 2 ). (2.4)
This integral is nite without the test function and the limiting procedure where f 2 (p 2 ) ? f (p 2 ) = 1 refers to important mathematical properties of metric spaces (whether Minskowskian or Euclidean) [18].

Going to light-cone (LC) variables x 0 ±x 1 = x ± is motivated by Dirac's early observation that the LC-stability group is maximal: LC-dynamics has much to share with gallilean dynamics (e.g.relative motion of LC-interacting particles decouples from global center of mass motion...). However in the LC-variables the nature of the initial Klein-Gordon equation in Eq.(2.1) is changed to a characteristic initial value problem (CIVP) relative to the partial-dierential equation
? + ? -?(x + , x -) = 0(2.5)
with initial data on characteristic surfaces
?(x + , x - 0 ) = f(x + ), ?(x + 0 , x -) = g(x -),(2.6)
and the continuity condition
?(x + 0 , x - 0 ) = f(x + 0 ) = g(x - 0 ). (2.7)
At rst sight the LC-Lagrangian is singular
1 : W (x, y) = ? 2 L ?[? -?(x)]?[? -?(y)] = 0
, but the appearence of a primary contrainst is known to be of no physical signicance [21]. 1 The Hessian is indentically null


# The ET-LFT consistency problem

Nevertheless the consistency of the solutions in the two reference frames cannot be established without further insight. This is just the content of Ref. [18], with two main conclusions:

-On the one hand, full consistency of EQT and LFT quantizations can only be achieved when elds are considered as OPVD with partition of unity test-functions f (p + 2 ) such that, for the light-cone momentum p + , lim
p + ?0 + f (p + 2 ) p + = 0.
-On the other hand operator series in the Discretized-LC-Quantization (DLCQ) nd their natural handling of divergences in the substraction scheme embedded in the OPVD formulation. The net eect of the PU-test function is the appearence of its inherent RGscale parameter (?).


# London Journal of Research in Science: Natural and Formal

Then the LF-formulation and CFT analysis of 2d-massless models are in complete agreement in their representation of the energy-impulsion tensor in term of innite dimensional Virasoro Lie-algebras.

The motion under consideration here is taking place on a 2d-worksheet embedded in a D-dimensionnal space. The initial eld variables are then x a (?, ? ), p a (?, ? ) elevated to OPVD. A well-dened Lagrangian is then obtained in terms these regular eld variables X a (?, ? ), P a (?, ? ). After dealing with the LC-gauge conditions the equation of motion for X a (?, ? ) is just that of Eq.(2.1) with appropriate position and time variables. Accordingly the sum of the zero-point energies of the rst quantized string is just
(D-2) 2 ? n =0
n. The well-known conventional evaluation of this sum is given by the Zeta-
function ?(s) = ? n=0 1 n s with ?(-1) = -1 12 .
The critical dimension for the absence of the overall conformal anomaly must then be such as to suppress that one with the cental charge c = 1 coming from the 2d worksheet analysis and thus obeys (D-2) 2 ?(-1) = -1, that is D = 26! However, even though at the same time this reasoning based on Zetafunction was already under scruteny [24], this critical value survived the long haul! In the advocated 2d QFT treatment the key role is in the pseudo-function distribution extension Pf ( 1 p 2 ) of 1 p 2 at the origin. It is dened by the integral
I N = ? 0 d(p 2 )Pf ( 1 p 2 )f (p 2 ) = def lim ??0 [ ? ?? d(p 2 ) p 2 + 1 ? ? ? 2 d(p 2 ) p 2 + 2 ln(?)] = ln( ? 2 ? ) (3.1)
where ? is the dilatation-scale inherent to the construction of the test function f (p 2 ) [7,14]. The term in ln(?) corresponds to the general Hadamard substraction procedure to generate a Finite part (F.p.).


# III. THE QUANTUM BOSONIC STRING [19, 23_27]

3.1. Equations of motion of the scalar bosonic string in the LC-gauge


# TLRS and the Renormalization Group

London Journal of Research in Science: Natural and Formal

The factor ? is arbitrary 2 with no physical meaning unless explicit symmetry violations need enforcement. Consider now the identity
IP f (?) = d 2 (p) (2?) 2 ) f (p 2 ) p 2 ? d 2 (p) (2?) 2 (p + q) 2 p 2 (p + q) 2 f (p 2 ), = 1 0 dx d 2 (p) (2?) 2 (p 2 + q 2 (1 -x) 2 ) [p 2 + q 2 x(1 -x)] 2 f (p 2 ), = 1 4? (ln( ? 2 ? ) -1). (3.2)
This is easy to understand due to the identity in the UV limit of the p-integration where f
[(p + q) 2 ]f (p 2 ) ? f 2 (p 2 ) ? f (p 2 )
. Moreover the overall O(2) p-invariance implies that terms linear in p do not contribute to the integral. Consider then the one loop Feynman diagram in relation to the energy-momentum tensor of the X-eld and in the same UV limit 3
? ab|cd (q) = D8d 2 p (2?) 2 t a,b (p, q)t c,d (p, q) p 2 (p + q) 2 f [p 2 ]f [(p + q) 2 ], = D 8 1 0 dx d 2 p (2?) 2 t a,b (p, q, x)t c,d (p, q, x) [p 2 + q 2 x(1 -x)] 2 f [p 2 ],(3.3)
with
t a,b (p, q) = p a (p + q) b + p b (p + q) a -? a,b (p.(p + q)), t a,b (p, q, x) = (p -q(1 -x)) a (p + qx) b + (p + qx) a (p -q(1 -x)) b -? a,b [p 2 -pq(1 -2x) -q 2 x(1 -x)].
The presence of the test-function f [p 2 ] ensures the existence of this phase-space integral, which otherwise would exibit divergences when p ? ? . The common pratice in the far past was to consider their cancelations by appropriate counter terms. In that case the only surviving regular contribution to ? ab|cd (q) is 4
? reg ab|cd (q) = D 8 (2q a q b -q 2 ? a,b )(2q c q d -q 2 ? c,d ) 1 0 dxx 2 (1 -x) 2 d 2 p (2?) 2 [p 2 + q 2 x(1 -x)] 2 = - Dq 2 M 192? (? a,b -2 q a q b q 2 )(? c,d -2 q c q d q 2 ) (3.4)
2 For Gauge Theories ? is related to the gauge xing parameter [12].

3 This is the 2-points-function, eq.( 9.158), of Poliakov's monograph. A coupling vertex factor would be ? g 2 2 f acd f bcd = ? g 2 2 C A ? ad . 4 Here q M is with Minkowski's signature opposite to Euclid's one. ) what is at sake is the sum (e.g. Trace) of the eigen-modes of this matrix. It can be diagonalized by a unitary transformation with a preserved Trace equal to 4. The result 5 is then just the same critical dimension for the absence of the conformal anomaly 5 In the perpective of the analytic continuation of sect. (3.1) it is instructive to note how here this decomposes as


# London
-q 2 M 4? (D-2) 2 * 84
6 ,4 from the trace itself and 1 6 from the nal x-integration
1 0 dxx(1 -x) = 1 6 cf Appendix B
obtained in the rst quantization framework, that is D cr = 26. It is clear then that the elimination of diverging contributions by counter-terms just leaves the evaluation of (3.4) in keeping with the ndings of [19].

However our TLRS formalism shows that this is not the end of the story.Indeed from examples (3.1,3.2) we observe that diverging integrals in p 2 and p 4 carry essential dependencies on the RG-parameter ?. Then the complete ?-dependence governing the RG-analysis of the critical equation is concerned with the behaviour of the central charge under the ow of the renormalization group (RG). Zamolodchikov realized this as early as 1986 with his c-theorem [29]:

"There is a function C on the space of unitary 2d-eld theories that monotonically decreaes along the RG-ows and which coincides with the Virasoro central charge c at xed points."

It takes the form
µ d dµ C(µ, ?) ? µ ? d d( µ ? ) C( µ ? , 1) = ? d d? C(?, 1) = -?(i, ?)g(i, j)?(j, ?)
where the Calan-Symanzik ?-function at xed point is independent of ? and takes the primitive value [30] 6

LambertW (6) .

With the stress energy-tensors ?(z) ? T z,z and ?(z) ? T z,z the C-function and the metric write [31,33]
C = - 1 2? real surface dz ? dz < ?(z) ?(z) > c | IR(T LRS limit) (3.5)
and
g (z,z) = 6? 2 µ 4 < ?(z) ?(z) > c | IR(T LRS limit) ,
London Journal of Research in Science: Natural and Formal

where the subscript c at the bracket indicates connected collerator contributions. µ is an arbitrary inverse distance inherent to the construction of the TLRS test function as a partition of unity with a dimensionless argument (cf footnote 5). The elds ? i (x) originate from local coupling sources ? i (x).

Let us consider the correlator of two stress tensors on the plane in the TLRS context [31] < T ?,? (x)T ?,? (0
) >= ? 3 ? 0 dµC(µ) d 2 pf (p 2 ) (2?) 2 exp(?px) (g ?? p 2 -p ? p ? )(g ?? p 2 -p ? p ? ) p 2 + µ 2 .
We are only left with the unknown scalar function of the mass scale µ, the spectral density [32] C(µ). Its properties have to comply to the following requirements:

(i) Reexion positivity of the euclidean eld theory, i.e. unitarity of the Hibert space, implies C(µ) ? 0, (ii) Due to dim(T ?? ) = 2 the spectral density is a dimensionless measure of degrees of freedoom, (iii)The form of C(µ) in a scale invariant eld theory is completely xed by it dimensionality. Since dµC(µ) is dimensionless one may not exclude C(µ) ? c µ . This IR divergence at µ = 0 is fully understood in the TLRS context [7,12] as long as the scaling limit to 1 of the test fuctions is not taken too early.
Indeed the correlator is 6 < ?(x)?(0) > = c? 3 ? 4 |x| ? 0 dµ µ f (µ 2 ) d 2 pf (p 2 ) (2?) 2 exp(?p.x) p 2 + µ 2 , = - c 12? ln(? 2 )? 4 |x| [? E + ln( ?|x| 2 )], = 1 4? ln(? 2 ) 2c |x| 4
(iv) Conformity with conformal invariance is exibited through the 1 |x| 4 dependence in agreement with the results of [18](Eq.( 56)) for < 0|T (z)T (w)|0 >. The study of the central charge C from Eq.(3.5) on a 2d-curved manifold [34] has established the general validity of Zamolodchikov c-theorem. It is sucent, for our purpose, to consider only a at real surface with coordinate parametrization {z, z} = ? exp(±??) which leads to 7 , 8   6 It is always possible to write the initial PU-test function regulating the p-integral as
f 2 (p 2 ) ? f (p 2 )f (p 2 + µ 2 ) ? f (p 2 )f (µ 2 ), for, in the UV-limit, f (p 2 )f (p 2 + µ 2 ) ? f 2 (p 2 ) ? f (p 2 )
, whereas in the IR-limit the remaining f (µ 2 ) function just validates the corresponding integral. 7 Note that in the initial {z, z}-integrals the factor is 1 |z-z| 4 so that the ?-integral is on the variable v = ? 2 sin 2 (?), hence the independent factorization of the remaining ?-integrals with the appearance the ubiquitous 1  12 factor [18](eq.56). 8 The TLRS analytic evaluation of g(v 2 ) is proportional to the dierence of step-functions [16,32].The nal v-integration is then trivial, after Hadamard substractions of diverging contributions in ln(?), leaving the ln(? 2 ) factor.  
[?(v -x11) -?(v -x12)], with x11 = (? 2 ) ( 1 ? ) , x12 = (2? 2 ) ( 1 ? )

# London
(?) = - 1 32 2? 0 d(?) sin 2 (?) ? 0 d(v) f (v 2 ) v 2 = 1 32 2? 0 d(?) sin 2 (?) ? 0 d(v) d dv ( 1 v )f (v 2 ) = - 1 32 2? 0 d(?) sin 2 (?) ? 0 dv v g(v 2 ) with g(v 2 ) = d dv f (v 2 ) = - 1 32 ln(? 2 )lim ??0 { 1 ? ? 2 -? ? d(?)[ 1 sin 2 (?) + 1 cos 2 (?) ]} = 1 12 ln(? 2 ) (3.6)
It is plain to see that this result is in agreement with the observation about the unicity of the solution, up to to an arbitrary constant (here ln(? 2 )), of "Cayley's identity" known as the "Schwarz derivative" [18].

Recently J.F. Mathiot established that, within general arguments valid in the TLRS framework, the trace of the energy-momentum tensor in 4-dimensions does not show any anomalous contribution even though quantum corrections are considered [35]. It is then our concern to turn now to the determination of the critical dimension D cr for the absence of the overall conformal anomaly with p 2 and p 4 divergences of the Poliakovtensor treated in the TLRS formalism(cf Appendix A). As mentioned after Eq.(3.4) the elimination of diverging contributions by counter-terms just leads to the evaluation in keeping with the ndings of [19], that is D cr = 26 . However with TLRS the situation is dierent as shown in Appendix A. The surviving initial Poliakov-term comes with extra TLRS ?-independent components. The immediate issue is then the fate of the D cr = 26 value under these additional TLRS terms 9 .Following Poliakov's analysis [19] a direct calculation of ?

--|--(q, ?) shows explicitly the critical value D cr = 4, as detailed in Appendix B. Consider now the diagonalization of the normalized matrix ? ab|cd (q) with a Lagrange parameter ? in relation to the stress-energy constraint T ab = 0. At the value D cr = 4 ? is completely xed, indicating that reparametrizations of the world-sheet and conformal rescaling allow to fully x g ab to anything wanted.

As a nal additional observation it is instructive to consider the string description for the VVA-anomaly [22] versus its direct calculation with TLRS [16,17]. In the string treatment of the massless case (cf Eq.(6.44) of [22]) "explicit divergences are made of a dierence of two tadepoles type and hence do not contribute in dimensional regularization, whereas for the remaining terms integrations are elementary, and the result is, using Î?"-function identities, easily identied to the standard result for the massless QED vacuum polarization". In TLRS the calculation is directly in dimension D = 4 with the IV. FINAL REMARKS usual ? 5 and all contributions are either null or nite: a simple bookeeping leads then to the standard VVA-anomaly without further ado. The TLRS procedure does provide a very clear and coherent picture. All known invariance properties, besides those of the VVA-anomaly, are preserved [1315]. It is a direct consequence of the fundamental properties of TLRS. As an "a-priori" regularization procedure, it provides a well dened mathematical meaning to the local Lagrangian we start from in terms of products of OPVD at the same space-time point. It also yields a well dened unambiguous strategy for the calculation of elementary amplitudes, which are all nite in strictly 4-dimensional space-time and with no new non-physical degrees of freedom nor any cut-o in momentum space.

In summary the strategy developped here was based on the passage from rstquantization to second quantization of the bososnic string. It is characterized by the introduction of the notion of L.Schwartz's Pseudo-Functions [8](cf Eq.(3.1)) with their dilatation scale dependences. This result is at variance wih the usual dilatation-scale independant Zeta-fuction evaluation of the discrete sum on inverse quantum n of rstquantized space-time objects. Actually it is easy to see that the standard evaluation of the Zeta-function through normal Eulers'integral in the integration interval (0, ?) should be considered as the limit ? ? 0 of the same integral in the interval (??, ? 2 ? ), thereby collecting rst from the logarithmic term the contribution ln( ? 2 ? ) and not the value ?(-1) = - 1  12 .

The main conclusion is then that String Theory in the OPVD picture reduces to Finite Quantum Field Theory, directly in 4-dimensions with no trace anomaly of the energymomentum tensor , and in the limit where the tension along the string becomes innite.
31![Journal of Research in Science: Natural and Formal 15 Volume 23 | Issue 8 | Compilation 1.0 2023 Great Britain Journal Press Finite Quantum-Field Theory and the Bosonic String Formalism: A Critical Point of View Here, from the embedding of the 2 -d worksheet, D does stand for D -2. Follwing sect(3.1](image-2.png "31 ©")
33![Journal of Research in Science: Natural and Formal 15 Volume 23 | Issue 8 | Compilation 1.0 2023 Great Britain Journal Press Finite Quantum-Field Theory and the Bosonic String Formalism: A Critical Point of View](image-3.png "33 ©")
![](image-4.png "C")
			
Finite Quantum-Field Theory and the Bosonic String Formalism: A Critical Point of View Volume 23 | Issue 8 | Compilation 1.0 © 2023 Great Britain Journal Press

			
given by Eq.(A.9) of Appendix A.London Journal of Research in Science: Natural and Formal

			
Volume 23 | Issue 8 | Compilation 1.0 © 2023 Great Britain Journal Press

			
©

		
		

			

## ACKNOWLEDGEMENTS

This study nds its origin through numerous discussions with Professor Ernst Werner from the Theoretical Physics section of Regensburg University. The rst outcomes were publications [17,18]. In the early 2021 we undertook the present work. Up to April 2021 Ernst Werner contributed activly and continuously to its developments. He departed unexpectedly on May 12th 2021. This publication is then dedicated to his memory. Our collaborations and constant friendship lasted ever since the Thesis of A. Lacroix-Borderie submitted on September 9,1994 at the Université de Strasbourg.

We are grateful to André Neveu for sharing his past experience on the subject and his quest for clarifying comments along this presentation.

We acknowlege constant support from Denis Puy, Head of the "Laboratoire Univers et Particules" UMR-5299 of IN2P3-CNRS and Université de Montpellier and from Dominique Pallin, Head of the "Laboratoire de Physique Corpusculaire" UMR-6533 of IN2P3-CNRS and Université de Clermont Auvergne.

			

			

				


* 
	
		
		
			NNBogoliubov
		
		
			OSParasiuk
		
	
	
		Acta. Math
		
			97
			227
			1957
		
	




* 
	
		Introduction to the Theory of Quantized Fields
		
			NNBogoliubov
		
		
			DVShirkov
		
		
			1980. 1990
			J. Wiley & Sons, Publishers, Inc
		
	
	3rd edition




* 
	
		
		
			HEpstein
		
		
			VGlaser
		
	
	
		Ann. Inst. Henri Poincaré XIXA
		
			211
			1973
		
	




* 
	
		
			GScharf
		
		Finite Quantum Electrodynamics: the Causal Approach
		
			Springer Verlag
			1995
		
	




* 
	
		Lagrangian Field Theory
		
			RStora
		
	
	
		Proceedings of Les Houches
		
			CDewitt-Morette
		
		
			CItzykson Eds
		
		
			Gordon
		
		
			Breach
		
		Les Houches
		
			1973
		
	




* 
	
		
			ANKuznetsov
		
		
			AVTkachov
		
		
			VVVlasov
		
		hep-th/9612037
		Techniques of Distributions in Perturbative Quantum Field Theory
		
			1996
		
	




* 
	
		
		
			JMGracia-Bondia
		
	
	
		Math. Phys. Annal. Geom
		
			6
			59
			2003
		
	




* 
	
		
		
			JMGarcia-Bondia
		
		
			SLazzarini
		
	
	
		J. Math. Phys
		
			44
			3863
			2003
		
	




* 
	
		
		
			PGrangé
		
		
			EWerner
		
	
	
		Nucl. Phys. (Proc. Suppl.) B
		
			161
			75
			2006
		
	




* 
	
		
			LSchwartz
		
		Théorie des Distributions
		Paris
		
			Hermann
			1966
		
	




* 
	
		
			SSSchweber
		
		An Introduction to Relativistic Quantum Field Theory
		New-York
		
			Harper and Row
			1964
		
	




* 
	
		
		
			GHooft
		
		
			MVeltman
		
	
	
		Nucl. Phys. B
		
			44
			318
			1972
		
	




* 
	
		Renormalization
		
			JCollins
		
		
			1987
			Gambridge University Press
		
	




* 
	
		
		
			PGrangé
		
		
			EWerner
		
	
	
		J. Phys.A
		
			44
			385402
			2011
		
	




* 
	
		
		
			BMutet
		
		
			PGrangé
		
		
			EWerner
		
	
	
		J. Phys.A
		
			45
			315401
			2012
		
	




* 
	
		
		
			PGrangé
		
		
			J.-FMathiot
		
		
			BMutet
		
		
			EWerner
		
	
	
		Phys. Rev. D
		
			80
			105012
			2009
		
	




* 
	
		
		
			PGrangé
		
		
			J.-FMathiot
		
		
			BMutet
		
		
			EWerner
		
	
	
		Phys. Rev. D
		
			82
			25012
			2010
		
	




* 
	
		
		
			PGrangé
		
		
			JFMathiot
		
		
			BMutet
		
		
			EWerner
		
	
	
		Phys. Rev. D
		
			88
			125015
			2013
		
	




* 
	
		
			PGrangé
		
		
			EWerner
		
		math-ph/0310052v2
		Proceedings of "Light Cone meeting:Hadrons and beyond
		
			SDalleyEditor
		
		"Light Cone meeting:Hadrons and beyondDurhan (UK
		
			5th-9th August 2003. 2003
		
	
	Fields on Paracompact Manifolds and Anomalies




* 
	
		
		
			PGrangé
		
		
			J.-FMathiot
		
		
			EWerner
		
	
	
		Int. J. Mod. Phys. A
		
			35
			2050025
			2020
		
	




* 
	
		
		
			PGrangé
		
		
			EWerner
		
	
	
		Mod. Phys. Lett. A
		
			33
			22
			1850119
			2018
		
	




* 
	
		
		
			AMPolyakov
		
	
	
		Phys.Lett. B
		
			103
			207
			1981
		
	




* 
	
		Gauge Fields and Strings
	
	
		Contemporary Concepts in Physics
		London-Paris-New-York
		
			1987
			3
		
	




* 
	
		
			BHat_Eld
		
	
	
		Quantum Field Theory of Particules and Strings
		
			Addison-Wesley Publishing Company
			1992
			75
		
	




* 
	
		
		
			LFaddeev
		
		
			RJackiw
		
	
	
		Phys. Rev. Lett
		
			60
			1692
			1988
		
	




* 
	
		Quantization Without Tears
		
			RJackiw
		
		arXiv:hep-th/9306075
		
			1993
		
	
	MIT preprint CTP 2215




* 
	
		Perturbative Quantum Field Theory in the String-Inspired Formalism
		
			CSchubert
		
	
	
		Phys.Rept
		
			355
			
			2001
		
	




* 
	
		
		
			OAlvarez
		
	
	
		Nucl.Phys. B
		
			216
			125
			1983
		
	




* 
	
		
		
			TSBunch
		
	
	
		General Relativity and Gravitation
		
			15
			27
			1983
		
	




* 
	
		
			BZwiebach
		
		A _rst course in string theory
		
			Cambridge University Press
			2004
		
	




* 
	
		
			JPolchinski
		
		String theory
		
			Cambridge University Press
			2001
			1
		
	




* 
	
		
			KBecker
		
		
			MBecker
		
		
			JHSchwartz
		
		String Theory and M-theory
		
			Cambridge University Press
			2007
		
	




* 
	
		String Theory in a nutshell
		
			EKiritsis
		
		
			2007
			Princetown University Press
		
	




* 
	
		
			PGinsparg
		
	
	
		Applied Conformal Field Theory
		
			1988
		
	




* 
	
		
		
	
	
		MIT Spring Lecture
		
			19
			2007
		
	




* 
	
		
			CItzykson
		
		
			JMDrou_E
		
		Savoirs Actuels, Inter Editions du CNRS
		
			1989
			2
		
	
	Théorie Statistique des Champs




* 
	
		Éléments d'introduction á l'invariance conforme
		
			YGrandati
		
		
			Ph
		
		
			PDi Francesco
		
		
			DMathieu
		
		
			Sénéchal
		
		String. 12
	
	
		Conformal Field Theory
		New-York
		
			Springer-Verlag
			1992. 1997
			17
			159
		
	
	Finite QFT




* 
	
		
		
			ABZamolodchikov
		
	
	
		Pisma ZH. Eksp. Teor. Fiz
		
			43
			730
			1986. 1986
		
	
	JETP Lett.




* 
	
		
		
			Polchinski
		
	
	
		Nucl.Phys. B
		
			303
			226
			1988
		
	




* 
	
		
		
			SSalmons
		
		
			PGrangé
		
		
			EWerner
		
	
	
		Phys.Rev.D
		
			65
			125014
			2002
		
	




* 
	
		
		
			ACappelli
		
		
			DFriedan
		
		
			JLLatorre
		
	
	
		Nucl.Phys.B
		
			352
			616
			1991
		
	




* 
	
		
		
			EKneur
		
		
			ANeveu
		
	
	
		Phys.Rev.D
		
			101
			74009
			2020
		
	




* 
	
		
		
			DFreidman
		
		
			AKonechny
		
		e-Print:hep-th/0910.3109
	
	
		J. Phys.A
		
			43
			215401
			2010
		
	




* 
	
		
		
			HOsborn
		
		
			GMShore
		
		Print:hep-th/9909043
	
	
		Nucl.Phys.B
		
			571
			287
			2000
		
	




* 
	
		Finite QFT, Bosonic String
		
			J.-FMathiot
		
	
	
		London Journal of Research in Science: Natural and Formal
		
			36
			2150265
			2021
		
	
	Int. J. Mod. Phys. A