Abstract
We reconstruct causal perturbation theory in a way compatible with light-front dynamics on the basis of Rohrlich’s invariant null-plane coordinatization. We show that, when the causality axiom is referred to the \(x^+\) time coordinate, the retarded and advanced distributions are possibly non-null on the entire \(x^-\)-axis, despite which Bogoliubov–Medvedev–Polivanov’s axioms still have a perturbative solution whose practical implementation is exposed in detail. Our results are then applied to the construction of the free fields Feynman’s propagators, for which instantaneous terms arise in the fermion and massless vector cases, and then to Yukawa’s model for the interaction of a neutral pseudo-scalar field with nucleons.
Similar content being viewed by others
Notes
This is to say, that every point of the surface can be mapped into any other point of it by some element of the stability group.
There is also a difference in the method of quantization of fields: In Ref. [98] it is replaced by the analytic representation of the propagators, which in the case of the massless vector field uses the Lagrangian approach by the introduction of Lagrange’s multipliers. In our work we quantize the fields by directly constructing Fock’s space; a résumé of this quantization method is exposed in Appendix A. Particularly, the photons space is constructed with the gauge-fixed wave-functions, which will lead to a difference in the radiation field commutation distribution.
So-called Cauchy–Kovalevskaya’s problem for initial data on non-characteristic surfaces, and Goursat’s problem on characteristic ones [101].
Certainly, the initial data must be given in the two characteristic surfaces, but, as is explained in Ref. [6], one of them can be replaced by an asymptotic condition of vanishing at infinity.
In the conception of Epstein and Glaser [58],in fact, the switching function “is” the coupling constant of the theory.
Strictly speaking, this axiom concerns unitarity over Fock’s space of physical states; when it is expanded in order to contain non-physical states, then it must be replaced by a pseudo-unitarity axiom—this occurs in the construction of gauge theories, as will be explained in a future work.
The notion of a Schwartz’s multi-index is defined, for example, in Ref. [106]: A multi-index \(k\in {\mathbb {R}}^N\) is a sequence of non-negative numbers, \(k=(k_1;\cdots ;k_N)\), \(k_j\ge 0\), for which the following notations are established:
$$\begin{aligned} |k|\equiv \sum \limits _{j=1}^N k_j,\quad x^k\equiv \prod \limits _{j=1}^Nx_j^{k_j},\quad k!\equiv \prod \limits _{j=1}^Nk_j!,\quad D^kf(x)\equiv \prod \limits _{j=1}^N\partial _{x_j}^{k_j}f(x). \end{aligned}$$It must be phenomenologically constructed in the same way as a Lagrangian density: By imposing the symmetries of the theory, but it must be restricted to the first order in the coupling constant terms, only. For example, the construction based on quantum gauge invariance can be found in Ref. [75].
Our convention for Fourier’s transform of the function \(f\in {\mathscr {S}}\left( {\mathbb {R}}^4\right) \) is the following:
$$\begin{aligned} {{\hat{f}}}(p)=(2\pi )^{-2}\int d^4x f(x)e^{ipx},\quad px=p_+x^++p_\perp x^\perp +p_-x^-\,. \end{aligned}$$We conventionally use lower indices for momentum variables; according to the metrics given in Eq. (3) we have the equivalence: \(p_+=p^-\), \(p_\perp =-p^\perp \) and \(p_-=p^+\). Additionally, we will denote: \(d^3\varvec{p}:=d^2p_\perp dp_-\).
The definition of the singular order at the origin is similar to that at the \(x^-\)-axis, except that all the four variables are scaled. In momentum space [64]: Let \({{\hat{d}}}\in {\mathscr {S}}'({\mathbb {R}}^m)\) be a distribution, and let \(\rho \) be a continuous positive function. If the limit
$$\begin{aligned} \lim _{s\rightarrow 0^+}\rho (s)\left\langle {{\hat{d}}}\left( \frac{p}{s}\right) ;{\check{\varphi }}(p)\right\rangle =\left\langle {{\hat{d}}}_0;{\check{\varphi }}\right\rangle \end{aligned}$$exists for all test function \(\check{\varphi }\in {\mathscr {S}}({\mathbb {R}}^m)\), then \({{\hat{d}}}_0\) is the quasi-asymptotics of the distribution \({{\hat{d}}}\) at \(p\rightarrow +\infty \), with regard to the function \(\rho \). \(\square \)
This means that \({{\widehat{D}}}^F_{ab}(p)\) is transverse both to \(p^a\) and \(\eta ^a\).
This model was also studied by Chang and Yan in Ref. [20] in Schwinger’s functional-derivative method. However, in light-front dynamics literature it is customary to find the analysis of the neutral model with the \(\gamma ^5\) matrix replaced by the unit matrix 1, under the same name of “Yukawa’s model”; clearly, in it the meson field is not pseudo-scalar, but scalar. That model is studied, for example, in Refs. [7, 43, 44].
Do not confuse with the positive and negative frequency parts, which are denoted by the sub-indices “\({\pm }\)” without parenthesis: \(\psi _{\pm }(x)\). For example, the negative frequency part of \(\Lambda _{(+)}\psi \) is denoted by \(\psi _{(+)-}\), and so on.
References
P.A.M. Dirac, Forms of relativistic dynamics. Rev. Mod. Phys. 21(3), 392–399 (1949)
G.E. Ramos, Teoria de Campos no Plano Nulo: Um estudo, Masters dissertation, IFT-UNESP (2005)
G.E. Ramos, Formulação Canônica no Plano Nulo, PhD. thesis, IFT-UNESP (2009)
H. Leutwyler, J. Stern, Relativistic dynamics on a null plane. Ann. Phys. 112, 94–164 (1978)
F. Rohrlich, Theory of photons and electrons on null planes. Acta Phys. Austriaca 32, 87–106 (1970)
F. Rohrlich, Null plane field theory. Acta Phys. Austriaca 8(Suppl), 277–322 (1971)
S.J. Brodsky, H.-C. Pauli, S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone. Phys. Rep. 301, 299–486 (1998)
S. Fubini, G. Furlan, Renormalization effects for partially conserved currents. Phys. Phys. Fizika 1, 247–299 (1965)
S. Weinberg, Dynamics at Infinite Momentum, Phys. Rev. 150, 1313-1318 (1966). Erratum: Phys. Rev. 158, 1638 (1967)
H. Bebié, H. Leutwyler, Relativistically invariant solutions of current algebras at infinite momentum. Phys. Rev. Lett. 19, 618–621 (1967)
R.A. Neville, Quantum Electrodynamics in a Laser Pulse, Ph.D. Thesis. Syracuse University (1968)
R.A. Neville, F. Rohrlich, Quantum electrodynamics on null planes and applications to lasers. Phys. Rev. D 3(8), 1692–1707 (1971)
S.-J. Chang, S. Ma, Feynman rules and quantum electrodynamics at infinite momentum. Phys. Rev. 180, 1506–1513 (1969)
J.B. Kogut, D.E. Soper, Quantum electrodynamics in the infinite-momentum frame. Phys. Rev. D 1(10), 2901–2914 (1970)
R.A. Neville, F. Rohrlich, Quantum field theory off null planes. Nuovo Cimento 1(4), 625–644 (1971)
H. Leutwyler, J.R. Klauder, L. Streit, Quantum Field Theory on Lightlike Slabs, Nuovo Cimento, Vol. LXVI A, N. 3, 536–554 (1970)
J.H. Ten Eyck, Problems in Null-Plane Quantum-Electrodynamics, Ph.D. Thesis. Syracuse University (1973)
J.H. Ten Eyck, F. Rohrlich, Equivalence of null-plane and conventional quantum electrodynamics. Phys. Rev. D 9(8), 2237–2245 (1974)
S.-J. Chang, R.G. Root, T.-M. Yan, Quantum field theories in the infinite-momentum frame. I. Quantization of scalar and dirac fields. Phys. Rev. D 7, 1133–1146 (1973)
S.-J. Chang, T.-M. Yan, Quantum field theories in the infinite-momentum frame. II. Scattering matrices of scalar and dirac fields. Phys. Rev. D 7, 1147–1161 (1973)
T.-M. Yan, Quantum field theories in the infinite-momentum frame. III. Quantization of coupled spin-one fields. Phys. Rev. D 7, 1760–1780 (1973)
T.-M. Yan, Quantum field theories in the infinite-momentum frame. IV. Scattering matrix of vector and dirac fields and perturbation theory. Phys. Rev. D 7, 1780–1800 (1973)
S.J. Brodsky, R. Roskies, R. Suaya, Quantum electrodynamics and renormalization theory in the infinite-momentum frame. Phys. Rev. D 8, 4574–4594 (1973)
S.D. Drell, D. Levy, T.-M. Yan, Theory of deep-inelastic lepton-nucleon scattering and lepton-pair annihilation processes. I. Phys. Rev. 187, 2159–2171 (1969)
S.D. Drell, D. Levy, T.-M. Yan, Theory of deep-inelastic Lepton–Nucleon scattering and lepton pair annihilation processes. II. Deep-inelastic electron scattering. Phys. Rev. D 1, 1035–1068 (1970)
S.D. Drell, D. Levy, T.-M. Yan, Theory of deep-inelastic Lepton–Nucleon scattering and lepton-pair annihilation processes. III. Deep-inelastic electron-positron annihilation. Phys. Rev. D 1, 1617–1639 (1970)
S.D. Drell, T.-M. Yan, Connection of elastic electromagnetic nucleon form factors at large \(Q^2\) and deep inelastic structure functions near threshold. Phys. Rev. Lett. 24, 181–186 (1970)
T. Maskawa, K. Yamawaki, The Problem of \(P^+=0\) mode in the null-plane field theory and Dirac’s method of quantization. Progr. Theor. Phys. 56, 270–283 (1976)
H.-C. Pauli, S.J. Brodsky, Solving field theory in one space and one time dimension. Phys. Rev. D 32, 1993–2000 (1985)
H.-C. Pauli, S.J. Brodsky, Discretized light-cone quantization: solution to a field theory in one space and one time dimension. Phys. Rev. D 32, 2001–2013 (1985)
J.P. Vary, X. Zhao, A. Ilderton, H. Honkanen, P. Maris, S.J. Brodsky, Applications of basis light-front quantization to QED, Nuc. Phys. B (Proc. Suppl.) 251-252, 10-15 (2014)
X. Zhao, H. Honkanen, P. Maris, J.P. Vary, S.J. Brodsky, Electron g-2 in light-front quantization. Phys. Lett. B 737, 65–69 (2014)
R.J. Perry, A. Harindranath, K.G. Wilson, Light-front Tamm–Dancoff field theory. Phys. Rev. Lett. 65(24), 2959–2962 (1990)
B.L.G. Bakker, A. Bassetto, S.J. Brodsky, W. Broniowski, S. Dalley, T. Frederico et al., Light-front quantum chromodynamics. A framework for the analysis of hadron physics. Nucl. Phys. B (Proc. Suppl.) 251–252, 165–174 (2014)
E. Tomboulis, Quantization of the Yang–Mills field in the null-plane frame. Phys. Rev. D 8, 2736–2740 (1973)
A. Casher, Gauge fields on the null plane. Phys. Rev. D 14, 452–464 (1976)
B.M. Pimentel, A.T. Suzuki, Light-cone gauge and the (causal) principal value prescription. Phys. Rev. D 42, 2115–2119 (1990)
B.M. Pimentel, A.T. Suzuki, Causal prescription for the light-cone gauge. Modern Phys. Lett. A 6, 2649–2653 (1991)
R. Casana, B.M. Pimentel, G.E.R. Zambrano, SQED\(_4\) and QED\(_4\) on the null-plane. Braz. J. Phys. 44, 398–409 (2014)
B.M. Pimentel, A.T. Suzuki, G.E.R. Zambrano, Functional analysis for gauge fields on the front-form and the light-cone gauge. Few-Body Syst. 52, 437–442 (2012)
D. Bhamre, A. Misra, V.K. Singh, Equivalence of one loop diagrams in covariant and light front QED revisited. Few Body Syst. 59, 107 (2018)
A.T. Suzuki, T.H.O. Sales, The light-front gauge propagator: the status quo. ar**v:hep-th/0408135 (2004)
V.A. Karmanov, J.-F. Mathiot, A.V. Smirnov, Regularization of the fermion self-energy and the electromagnetic vertex in the Yukawa model within light-front dynamics. Phys. Rev. D 75, 045012 (2007)
B.L.G. Bakker, C.-R. Ji, Light-front singularities in the Yukawa model. Nucl. Phys. B (Proc. Suppl.) 161, 15–20 (2006)
D. Mustaki, S. Pinsky, J. Shigemitsu, K. Wilson, Perturbative renormalization of null-plane QED. Phys. Rev. D 43(10), 3411–3427 (1991)
W. Heisenberg, Die ’Beobachtbaren Grössen’ in der Theorie der Elementarteilchen, Z. Phys. 120 (1943) 513–538. This paper can be found in: W. Blum, H.-P. Dürr and H. Rechenberg (Eds.), Werner Heisenberg. Gesammelte Werke / Collected Works. Series A / Part II. Original Scientific Papers (Springer, 1989), in which excelent comments on Heisenberg’s motivation for the definition of the \(S\) matrix are given by R. Oehme
R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That (The Benjamin/Cummings Publishing Company, New York, 1964)
J. Glimm, A. Jaffe, Quantum Physics (Springer, Berlin, 1981)
H. Lehmann, K. Symanzik, W. Zimmermann, Zur Furmulierung quantisierter Feldtheorien, Nuovo Cim. 1 (1955) 205–225. Translation into English by C.F. Barger: On the formulation of quantized field theories (CBF Scientific Translations)
H. Araki, Von Neumann algebras of local observables for free scalar field. J. Math. Phys. 5, 1–13 (1964)
R. Haag, D. Kastler, An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)
J. Lacki, H. Ruegg, G. Wanders (Eds.), E.C.G. Stueckelberg, An Unconventional Figure of Twentieth Century Physics. Selected Scientific Papers with Commentaries (Birkhauser, New York, 2009)
E. Stueckelberg, D. Rivier, Causalité et structure de la matrice S, Helv. Phys. Acta 23 (1950) 215–222. This paper was reprinted in Ref. [52]
N.N. Bogoliubov, B.V. Medvedev, M.K. Polivanov, Voprossy Teorii Dispersionnykh Sootnoshenii (Moscow, Fizmatgiz, 1958). Translation into English: N.N. Bogoliubov, B.V. Medvedev and M.K. Polivanov, Problems in the Theory of Dispersion Relations (Institute for Advanced Study, Princeton, 1958)
N.N. Bogoliubov, D. Shirkov, Introduction to the Theory of Quantized Fields, 3rd edn. (Wiley, New York, 1979)
N.N. Bogolubov, A.A. Logunov, A.I. Oksak, I.T. Todorov, General Principles of Quantum Field Theory (Kluwer Academic Publishers, London, 1990)
B.M. Stepanov, On the construction of an \(S\)-matrix in accordance with the theory of perturbations (in russian), Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 1037–1054. Translation into English in: Eighteen papers on analysis and quantum mechanics, American Mathematical Society Translations, Series 2, Volume 91 (American Mathematical Society, 1970)
H. Epstein, V. Glaser, The role of locality in perturbation theory. Ann. Inst. H. Poincaré A 19, 211–295 (1973)
N.N. Bogoliubow, O.S. Parasiuk, Über die multiplikation der kausalfunktionen in der quantentheorie der felder. Acta Math. 97, 227–266 (1957)
S. Łojasiewicz, Sur le problème de la division. Studia Mathematica T. XVII I, 87–136 (1959)
B. Malgrange, Division des distributions. I: Distributions prolongeables, Séminaire Schwartz T. 4, exp. No. 21, 1–5 (1959–1960)
B. Malgrange, Division des distributions. II: L’Inégalité de Łojasiewicz, Séminaire Schwartz T. 4, exp. No. 22, 1–8 (1959–1960)
V.S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables (Dover, New York, 2007)
G. Scharf, Finite Quantum Electrodynamics. The Causal Approach, 3rd edn. (Dover, New York, 2014)
S. Fassari, G. Scharf, Borel Summability in Finite QED I: Outline and Results, in Stochastic Processes, Physics and Geometry II. ed. by S. Albeverio, U. Cattaneo, D. Merlini (World Scientific, Singapore, 1991), pp. 278–288
V.S. Vladimirov, Y.N. Drozzinov, B.I. Zavialov, Tauberian Theorems for Generalized Functions (Kluwer Acadamic Publication, New York, 1988)
M. Dütsch, F. Krahe, G. Scharf, Scalar QED Revisited, Il Nuovo Cimento 106 A, 3, 277–307 (1993)
M. Dütsch, T. Hurth, F. Krahe, G. Scharf, Causal construction of Yang–Mills theories. I. Nuovo Cim. A 106, 1029–1041 (1993)
M. Dütsch, T. Hurth, F. Krahe, G. Scharf, Causal construction of Yang–Mills theories. II. Nuovo Cim. A 107, 375–406 (1994)
M. Dütsch, T. Hurth, G. Scharf, Causal construction of Yang–Mills theories. III. Nuovo Cim. A 108, 679–708 (1995)
M. Dütsch, T. Hurth, G. Scharf, Causal construction of Yang–Mills theories. IV. Unitarity. Nuovo Cim. A 108, 737–773 (1995)
M. Dütsch, G. Scharf, Electroweak theory without spontaneous symmetry breaking, ar**v:hep-th/9612091 (1996)
A.W. Aste, G. Scharf, NonAbelian gauge theories as a consequence of perturbative quantum gauge invariance. Int. J. Mod. Phys. A 14, 3421–3434 (1998)
A.W. Aste, G. Scharf, M. Dütsch, Perturbative gauge invariance: electroweak theory II. Ann. Phys. 8, 389–404 (1999)
G. Scharf, Gauge Field Theories: Spin One and Spin Two. 100 Years After General Relativity (Dover, New York, 2016)
M. Dütsch, Slavnov–Taylor identities from the causal point of view. Int. J. Mod. Phys. A 12, 3205–3248 (1997)
M. Dütsch, Non-uniqueness of quantized Yang–Mills theories. J. Phys. A: Math. Gen. 29, 7597–7617 (1996)
D.R. Grigore, Trivial Lagrangians in the causal approach. Rom. J. Phys. 61, 320–332 (2016)
D.R. Grigore, Cohomological aspects of gauge invariance in the causal approach. Rom. J. Phys. 55, 386–438 (2010)
D.R. Grigore, The structure of the anomalies of gauge theories in the causal approach. J. Phys. A 35, 1665–1689 (2002)
D.R. Grigore, G. Scharf, No-go result for supersymmetric gauge theories in the causal approach. Ann. Phys. 17, 864–880 (1997)
G. Scharf, W.F. Wreszinski, B.M. Pimentel, J.L. Tomazelli, Causal approach to (2+1)-dimensional QED. Ann. Phys. 231, 185–208 (1994)
L.A. Manzoni, B.M. Pimentel, J.L. Tomazelli, Causal theory for the gauged Thirring model. Eur. Phys. J. C 8, 353–361 (1999)
L.A. Manzoni, B.M. Pimentel, J.L. Tomazelli, Radiative corrections to the causal Thirring model. Eur. Phys. J. C 12, 701–705 (2000)
J.T. Lunardi, B.M. Pimentel, J.S. Valverde, L.A. Manzoni, Duffin–Kemmer–Petiau theory in the causal approach. Int. J. Mod. Phys. A 17(2), 205–227 (2002)
J. Beltrán, B.M. Pimentel, D.E. Soto, The causal approach proof for the equivalence of SDKP\(_4\) and SQED\(_4\) at tree-level. Int. J. Mod. Phys. A 35(9), 2050042 (2020)
R. Bufalo, B.M. Pimentel, D.E. Soto, Causal approach for the electron-positron scattering in generalized quantum electrodynamics. Phys. Rev. D 90, 085012 (2014)
R. Bufalo, B.M. Pimentel, D.E. Soto, Normalizability analysis of the generalized quantum electrodynamics from the causal point of view. Int. J. Mod. Phys. A 32, 1750165 (2017)
O.A. Acevedo, J. Beltrán, B.M. Pimentel, D.E. Soto, Causal perturbation theory approach to Yukawa’s model. Eur. Phys. J. Plus 136, 895 (2021)
M. Dütsch, K. Fredenhagen, A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203, 71–105 (1999)
R. Brunetti, K. Fredenhagen, Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000)
M. Dütsch, K. Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity. Rev. Math. Phys. 16, 1291–1348 (2004)
R. Brunetti, M. Dütsch, K. Fredenhagen, Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13, 1541–1599 (2009)
K. Fredenhagen, K. Rejzner, Batalin–Vilkoviski formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697–725 (2013)
M. Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory (Birkkhäuser, London, 2019)
S. Hollands, R. Wald, Local wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289–326 (2001)
S. Hollands, R.M. Wald, Existence of local covariant time-ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002)
R. Bufalo, B.M. Pimentel, D.E. Soto, The Epstein–Glaser causal approach to the light-front QED\(_4\). I: free theory. Ann. Phys. 351, 1034–1061 (2014)
R. Bufalo, B.M. Pimentel, D.E. Soto, The Epstein–Glaser causal approach to the light-front QED\(_4.\) II: vacuum polarization tensor. Ann. Phys. 351, 1062–1084 (2014)
O.A. Acevedo, B.M. Pimentel, D.E. Soto, Epstein–Glaser’s causal light-front field theory, Proc. Sci. LC2019, 021 (2020)
A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Reprint ed. (Dover, 2011)
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale University Press, New York, 1923)
E. Goursat, A Course in Mathematical Analysis, vol. I (Ginn and Company, London, 1904)
S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford Classic Texts in the Physical Sciences, Clarendon Press, Oxford, 2006)
P.P. Srivastava, Perspectives of Light-Front Quantized Field Theory: Some New Results, ar**v:hep-ph/9908492v1 (1999)
L. Schwartz, Méthodes mathématiques pour les sciences physiques, 2 edn. (Hermann, Pairs, 1979)
F.G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd edn. (Cambridge University Press, Cambridge, 1998)
J. Hilgevoord, Dispersion Relations and Causal Description (North Holland, Amsterdam, 1960)
D. Prange, Lorentz covariance in Epstein-Glaser renormalization. ar**v preprint: hep-th/9904136 (1999)
T. Hurth, K. Skenderis, Quantum Noether method. Nucl. Phys. B 541, 566–614 (1999)
A.G. Sveshnikov, A.N. Tikhonov, The Theory of Functions of a Complex Variable (Mir Publishers, New York, 1971)
P.P. Srivastava, S.J. Brodsky, Light-front quantized QCD in the light-cone gauge: the doubly transverse gauge propagator. Phys. Rev. D 64, 045006 (2001)
K. Nishijima, Fundamental Particles (Benjamin, W.A, 1963)
V. Mukherji, A history of the meson theory of nuclear forces from 1935 to 1952. Arch. Hist. Exact Sci. 13(1), 27–102 (1974)
L.M. Brown, H. Rechenberg, The Origin of the Concept of Nuclear Forces (CRC Press, London, 1996)
L.M. Brown, H. Rechenberg, Quantum field theories, nuclear forces, and the cosmic rays (1934–1938). Am. J. Phys. 59(7), 595–605 (1991)
W. Heisenberg, Über den Bau der Atomkerne. I, Z. Phys. 77 (1932) 1–11. Translation into English in: D.M. Brink, Nuclear Forces (Oxford, 1965)
W. Heisenberg, Über den Bau der Atomkerne. II, Z. Phys. 78, 156–164 (1932)
W. Heisenberg, Über den Bau der Atomkerne. III, Z. Phys. 80 (1933) 587–596. Partial translation into English in: D.M. Brink, Nuclear Forces (Oxford, 1965)
A.I. Miller, Werner Heisenberg and the beginning of nuclear physics. Phys. Today 38(11), 60–68 (1985)
H. Yukawa, On the interaction of elementary particles I. Proc. Physico-Math. Soc. Jpn. 17, 48–57 (1935)
N. Kemmer, The charge-dependence of nuclear forces. Proc. Camb. Philos. Soc. 34, 354–364 (1938)
G.C. Wick, Range of nuclear forces in Yukawa’s theory. Nature 142, 993–994 (1938)
N. Kemmer, Quantum theory of Einstein-Bose particles and nuclear interaction. Proc. Roy. Soc. (Lond.) A 166, 127–153 (1938)
S. Glazek, A. Harindranath, S. Pinsky, J. Shigemitsu, K. Wilson, Relativistic bound-state problem in the light-front Yukawa model. Phys. Rev. D 47, 1599–1619 (1993)
O.A. Acevedo, B.M. Pimentel, Radiative corrections in the Yukawa model within the null-plane causal perturbation theory framework. Phys. Rev. D 103, 076022 (2021)
R. Casana, B.M. Pimentel, G.E.R. Zambrano, SQED\(_2\) on the null-plane. Revista Colombiana de Física 41(1), 220–222 (2009)
Acknowledgements
O.A.A. thanks CAPES-Brazil and CNPq-Brazil for total financial support, and B.M.P. thanks CNPq-Brazil for partial financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Prof. Günter Robert Scharf, in memoriam.
Quantization of fields on the null-plane
Quantization of fields on the null-plane
In quantum mechanics a particle is described by a wave-function. If the description is relativistic, such wave-functions are the positive frequency part of the so-called relativistic classical fields, this is to say, solutions of some relativistic equation of motion. The restriction to consider only the positive frequency part of the solution is done because in the real world the energy of the particle must be positive; the frequency and the energy are related by Schrödinger’s equation, which is an axiom in the quantum theory. In instant dynamics this could be confusing: We know that every component of every field satisfies Klein–Gordon–Fock’s equation, but it is impossible to put that equation in Schrödinger’s form. In light-front dynamics this issue is clear: Klein–Gordon–Fock’s equation can be put in Schrödinger’s form,
with the Hamiltonian:
All the solutions \(\varphi _1\) of this equation constitute the one-particle Hilbert space \({\mathcal {H}}_1\). This is sufficient if we want to treat free particles only, but when the particles are subjected to interaction we need to consider the possibility that they change their identity and their number. Of course, such changes cannot be described in detail: the very concept of particle is then lost; but we can describe the asymptotically free incoming and outgoing particles of a scattering process. And that could be done only if one works on a space containing states describing different number of particles.
Definition
Fock’s space \({\mathcal {F}}^{\pm }\) of bosons or fermions, respectively, is the direct sum of Hilbert’s spaces of symmetrized or anti-symmetrized wave-functions of n particles, \({\mathcal {H}}_n^{\pm }\), extended to all non-negative values of n:
Their elements are:
Particularly, the vacuum state of zero particles is:
Fock’s space \({\mathcal {F}}^{\pm }\) is equipped with the interior product \(\left( \bullet ;\bullet \right) \) defined as:
and norm induced by it:
with the restriction—over the states—that its value to be finite.
The passage from states of a given number of particles to another is done by the emission and absorption operators.
Definition
The emission operator of a particle with wave-function f, with \(f\in {\mathcal {H}}_1\), is the operator \(a^*(f):\text {Dom}(a^*(f))\subseteq {\mathcal {F}}^{\pm }\rightarrow \mathcal F^{\pm }\), defined as:
The absorption operator of a particle with wave-function f, with \(f\in {\mathcal {H}}_1\), is the operator \(a(f):{\mathcal {F}}^{\pm }\rightarrow {\mathcal {F}}^{\pm }\), defined such that:
The Hermiticity of the (anti-)symmetrization operators \(S^{\pm }_n\) implies the following relation between the emission and absorption operators:
The (anti-)commutation relations of the emission and absorption operators are found by using their definitions given in Eqs. (251) and (252). Let \(f,g\in {\mathcal {H}}_1\), \(\Phi \in \text {Dom}\left( [a(f);a^\dagger (g)]_{{\mp }}\right) \subset {\mathcal {F}}^{\pm }\). By direct calculus:
As we see, these relations consider the commutator in \({\mathcal {F}}^+\) (bosons space) and the anti-commutator in \({\mathcal {F}}^-\) (fermions space). Under the assumption that \({\mathcal {H}}_1\) is separable, it is possible to introduce a countable and orthonormal basis of functions \(f_j\):
in function of which every \(f\in {\mathcal {H}}_1\) is expanded as:
Therefore, defining the emission and absorption operators on those basis functions:
we obtain that \(a_j\) and \(a^\dagger _k\) satisfy the following relations [see Eqs. (254), (255) and (256)]:
Now we must specify the interior product in Eq. (254) in such a way that it is relativistic invariant.
Definition
Let \({\mathcal {H}}_1\) be the one-particle Hilbert’s space. Consider \(f,g\in {\mathcal {H}}_1\), and let \({{\hat{f}}}\) and \({{\hat{g}}}\) be their Fourier’s transforms. The interior product of them is defined by:
with \({\mathrm{d}}\mu _m(p)\) the relativistic invariant measure:
This measure \({\mathrm{d}}\mu _m(p)\) is defined on the “upper mass-shell”
so we identify the space \({\mathcal {H}}_1\) with the space of square-integrable functions on \({\mathcal {M}}^+\) with respect to the measure \({\mathrm{d}}\mu _m\):
We present the following results without proof—the proof in instant dynamics is given in Ref. [64]; in light-front dynamics the proof follows with simple modifications.
Theorem
Every irreducible representation of the (anti-)commutation relations of Eqs. (254) and (255), with vacuum state \(\Omega \), are equivalent to Fock’s representation.
Corollary
Every bounded operator acting on Fock’s space \(\mathcal F^{\pm }\) can be expressed in function of the emission and absorption operators \(a^\dagger (f)\) and a(f).
Also, Eq. (261) implies that, when one restricts the wave-functions to the null-plane, the measure of the interior product becomes independent of the mass. This leads to the following theorem [16]:
Theorem
Let \({\mathcal {F}}_\nu ^{\pm }\) (\(\nu =1,2\)) be two Fock’s spaces corresponding to particles with different masses \(m_\nu \), with \(m_1\ne m_2\). The Fock’s spaces \({\mathcal {F}}_1^{\pm }\) and \(\mathcal F_2^{\pm }\) are unitarily equivalents.
The next step in the construction of the quantum theory is the adequate election of the basis functions in \({\mathcal {H}}_1\). If the basis is chosen in a such a way to satisfy the equations and constraints of the classical field theory, then we will have no problem in recognizing that their evolution according to the Hamiltonian \(P_+\) is relativistic. Our strategy will be the following: We will write a basis for the physical solutions of Klein–Gordon–Fock’s equation; with them we will construct the complete multi-component fields by using the constraint equations and the adequate statistics. In order to do that, we remember that the initial value problem in the null-plane has the solution for the positive frequency part [5, 6]:
with \(\left\langle \bullet ;\bullet \right\rangle \) meaning that the distribution is applied to the function according to the formal rule given by the interior product \((\bullet ;\bullet )_1\), differentiating the notation in order to clarify that \(\left\langle \bullet ;\bullet \right\rangle \) is not an interior product, but the application of a distribution to a function. Also, \(D_+\) is the positive frequency part of Jordan-Pauli’s distribution. The expansion of f in the basis \(\left\{ f_j\right\} \) implies the following completeness relation:
The map** from the function f to the operators a(f) and \(a^\dagger (f)\) is making by:
Definition
Let \(f\in {\mathcal {H}}_1\) be a wave-function. The emission and absorption field operators in real space, \(a^\dagger (\varvec{x})\) and \(a(\varvec{x})\), respectively, are the operator-valued distributions which lead f to the operators \(a^\dagger (f)\) and a(f), respectively, by means of the rules:
Additionally, if \(f\in {\mathscr {S}}({\mathbb {R}}^3)\), we define the emission and absorption field operators in momentum space, \(a^\dagger (\varvec{p})\) and \(a(\varvec{p})\), respectively, as the distributional Fourier’s transforms of the corresponding field operators in real space.
Eqs. (257) and (266) imply that we can write the emission and absorption field operators as:
and in momentum space:
From these relations, jointly with the expansions in Eq. (259) and the completeness relation given in Eq. (265):
Definition
The quantized field operator is the operator-valued distribution u(x) defined as:
Its part corresponding to the absorption field operator is called “negative frequency part”, denoted \(u_-\); the corresponding to the emission field operator, “positive frequency part”, denoted \(u_+\).
To that definition many forms can be given. Using the expansion showed in Eq. (267):
Here we see that the dependence with the coordinates is contained in the functions \(f_j(x)\). Since \(\left\{ f_k\right\} \) is basis of \({\mathcal {H}}_1\), Eq. (271) implies that the quantized field operator satisfies the equation of motion of the classical field. Also, writing the function \(f_j(x)\) in Eq. (271) in function of their Fourier’s transform,
we obtain the following wave-packet expansion for the quantized field operator:
Finally, by applying Fourier’s transformation directly to the emission and absorption field operators a(x) and \(a^\dagger (x)\) in Eq. (270):
Once again the field operators \(a^\dagger (\varvec{p})\) and \(a(\varvec{p})\) can be expanded as in Eq. (268), with the same result as in Eq. (273); this shows that, in general: \({{\hat{a}}}_j=a_j\) and \({{\hat{a}}}_j^\dagger =a_j^\dagger \); this is not surprising, because \(a_j\) and \(a_j^\dagger \) are independent of the coordinates and momenta.
The most important property of the quantized field operators is the (anti-)commutation relation between their negative and positive frequency parts. Using the expression in Eq. (271) and Eqs. (259) and (265):
This commutator appears in a central rôle in CPT and was called “Wick’s contraction”: It comes from the solution of the classical field theory Goursat’s problem. If the same were calculated from Eq. (274), then we would obtain:
The comparison between Eqs. (275) and (276) allows the obtention of \(\left[ a(\varvec{p});a^\dagger (\varvec{q})\right] _{\mp }\).
Now we turn to the study of the scalar, fermion and massless vector fields.
1.1 Neutral scalar field
In this case we need to introduce one set of emission and absorption operators, only, \(a^\dagger (f)\) and a(f). The quantized field operator is:
Their commutation distribution will be Jordan-Pauli’s distribution, because it solves Goursat’s problem in the classical theory. Then:
and:
Jordan-Pauli’s distribution is:
their positive and negative frequency parts are, respectively, given by:
From Eqs. (278), (281) and (276) we obtain:
which implies:
Particularly, from Eq. (279) we can obtain the commutation relation at equal times. From Eq. (280), by integrating the variables \(p^+\) and \(p^\perp \), we find:
hence:
Here we observe the non-commutativity of the field in the \(x^-\)-axis, which does not violate causality. Additionally, taking the derivative with respect to \(y^-\) in Eq. (285):
In a Hamiltonian analysis, the conjugate momenta to the field \(\varphi \) is \(\pi =\partial _-\varphi \), so Eq. (286) is the canonical commutation relation at equal times. Equation (286), however, is unusual in the sense that it contains a factor of 1/2, which is due to the fact that \(\varphi \) and \(\pi \) are not dynamically independent in light-front dynamics [6]. It is usual that systems which are not constrained in instant dynamics turn to be constrained in the light-front one; we remember to the reader, in this sense, that the stability group of the null-plane is bigger than that of the spatial plane and the number of dynamical generators is less [2, 3]. Equation (286) is the same relation found by the Dirac–Bergmann algorithm [127]; we have found it without using this algorithm because we have constructed Fock’s space with an appropriate basis of one-particle wave-functions, which already satisfy all the constraints of the theory. Finally, we see that the commutation relations at equal times require only the knowledge of the field on the null-plane.
1.2 Fermion field
Consider now Dirac’s fermion field, so we will use anti-commutation relations. Since it describes two particles, each one with two polarization states, as it has spin 1/2, we will use four sets of emission and absorption operators: each set for each polarization of each particle. We start by remembering the reader that the classical Dirac’s field is decomposed into two parts by the projection operators \(\Lambda _{({\pm })}\):Footnote 14
From them, only \(\psi _{(+)}\) is dynamical and follows Klein–Gordon–Fock’s equation. The projection \(\psi _{(-)}\), on the other hand, is obtained from \(\psi _{(+)}\) by means of the constraint equation:
If we decompose \(\psi _{(+)}(x)\) in the basis of the space projected by \(\Lambda _{(+)}\):
then we can consider \(\alpha (x)\) and \(\beta (x)\) as fermion charged scalar fields, so their corresponding quantized field operators are:
They are subjected to the following anti-commutation relations:
and for the emission and absorption field operators (\(s={\pm }1\)):
Although the quantization is complete, it is convenient to use Eq. (288) to reconstruct the entire \(\psi (x)\) field and write the anti-commutation relation in function of it. Substituting Eqs. (290) and (291) into Eq. (289), we obtain \(\psi _{(+)}\). Then, using Eq. (288) we find \(\psi _{(-)}\); their sum—as in Eq. (287)—is the quantized field operator \(\psi (x)\). In this procedure, no degree of freedom is increased, and the complete field is implicitly written in function of the dynamical degrees of freedom of the theory, \(\alpha (x)\) and \(\beta (x)\). We obtain:
And for the adjoint field:
with the four-component functions \(u, {{\overline{u}}}\) and \(v,\overline{v}\) normalized so as to satisfy the sum rules:
With the expressions given in Eqs. (294) and (295) and using the anti-commutation rules of Eq. (293) we find:
and:
Therefore, since \(\Theta (p_-)-\Theta (-p_-)=\text {sgn}(p_-)\), the anti-commutator of Dirac’s field with its adjoint is:
We identify here the distribution S(x) which in the classical case solves Goursat’s problem for Dirac’s equation:
with:
1.3 Neutral massless vector field
As it is established at the classical level, the neutral massless vector field \(A^a\) has only two degrees of freedom, corresponding to the transversal components \(A^\alpha \) (\(\alpha =1,2\)), obeying Klein–Gordon–Fock’s equation:
The component \(A^+\) is zero—this is the null-plane gauge condition, imposed in order to have a well-defined initial value problem [6], and the component \(A^-\) can be obtained from Lorenz’s gauge condition, implied by the null-plane one in the free case:
By construction, only physical states are in Fock’s space, so that only the potentials \(A^a\) satisfying Eqs. (303) and (304) can be one-particle wave-functions. The covariance of Fock’s space is assured by the fact that the null-plane gauge condition completely determines the gauge.
Accordingly with our quantization program, the field operators associated to the components \(A_1\) and \(A_2\) are:
They satisfy the commutation relations:
From Eq. (307) we obtain:
and:
As in the fermion field case, it will be useful to write an expression for the complete vector field, for which we use the second of Eqs. (304), from that the component \(A^-(x)\) can be found:
where we have identified the “−” component of the polarization vectors \(\varepsilon _{1,2}(\varvec{p})^-\). For completeness we give the expression of the polarization vectors, which can be found in a classical analysis:
Being that way, we can write the quantized field operator:
With the aid of the commutation relations of Eq. (309) we can already calculate the commutation distribution of the quantized field operators in Eq. (312):
In these expressions, the light-like vector \(\eta ^a\) is defined as:
Therefore, defining the distribution \(D^{ab}(x)\) by:
we found that it is:
Rights and permissions
About this article
Cite this article
Acevedo, O.A., Pimentel, B.M. Null-plane causal perturbation theory. Eur. Phys. J. Plus 137, 287 (2022). https://doi.org/10.1140/epjp/s13360-022-02459-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-022-02459-3