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A Classical Approach to the Modeling of Quantum Mass

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A Classical Approach to the Modeling of Quantum Mass See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/262916117 A Classical Approach to the Modeling of Quantum Mass ARTICLE in JOURNAL OF MODERN PHYSICS · NOVEMBER 2013 DOI: 10.4236/jmp.2013.411A1004 READS 3 1 AUTHOR: Donald C Chang The Hong Kong University of Science and T… 135 PUBLICATIONS 3,747 CITATIONS SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate, Available from: Donald C Chang letting you access and read them immediately. Retrieved on: 07 March 2016 Journal of Modern Physics, 2013, 4, 21-30 Published Online November 2013 (http://www.scirp.org/journal/jmp) http://dx.doi.org/10.4236/jmp.2013.411A1004 A Classical Approach to the Modeling of Quantum Mass Donald C. Chang Div of LIFS, Hong Kong University of Science and Technology, Hong Kong, China Email: [email protected] Received September 1, 2013; revised September 29, 2013; accepted October 27, 2013 Copyright © 2013 Donald C. Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In modern physics, a particle is regarded as the quantum excitation of a field. Then, where does the mass of a particle come from? According to the Standard Model, a particle acquires mass through its interaction with the Higgs field. The rest mass of a free particle is essentially identified from the Klein-Gordon equation (through its associated Lagrangian density). Recently it was reported that a key feature of this theory (i.e., prediction of Higgs boson) is supported by ex- periments conducted at LHC. Nevertheless, there are still many questions about the Higgs model. In this paper, we would like to explore a different approach based on more classical concepts. We think mass should be treated on the same footing as momentum and energy, and the definition of mass should be strictly based on its association with the momentum. By postulating that all particles in nature (including fermions and bosons) are excitation waves of the vac- uum medium, we propose a simple wave equation for a free particle. We find that the rest mass of the particle is associ- ated with a “transverse wave number”, and the Klein-Gordon equation can be derived from the general wave equation if one considers only the longitudinal component of the excitation wave. Implications of this model and its comparison with the Higgs model are discussed in this work. Keywords: Mass; Vacuum; Quantum Excitation; Matter Wave; Particle-Wave Duality; Higgs Mechanism 1. Introduction Model! Yet, there are still many questions remaining to be We know the universe is made up of particles; most of answered. It is well aware in the physics community that which have a rest mass (except for photons and possibly the Standard Model (in the present form) has certain li- neutrinos). In the quantum field theory, a particle is re- garded as an excitation wave of a field. Then, how can a mitations. For example: particle acquire mass? According to the Standard Model It cannot explain gravitation. (SM), the mass of the particle is acquired through their Some consider it to be ad hoc in nature, requiring 19 interaction with the Higgs field, the excitation of which numerical constants whose values are arbitrary and is a scalar particle called “the Higgs boson” [1]. The can only be determined by experiments. Higgs particle has been theorized for a long time but was The specifics of neutrino mass are still unclear. It is not verified in experiment until very recently. This is due believed that explaining neutrino mass will require an to the very heavy mass of the Higgs particle and thus will additional 7 or 8 constants, which are also arbitrary require a very powerful accelerator to study it. parameters. Experiments to hunt for the Higgs boson was finally The Standard Model cannot explain the huge amount started at CERN about three years ago using the newly of dark matter and dark energy observed in cosmol- built Large Hadron Collider (LHC). On July 2012, two ogy. It also has difficulty to explain the observed experimental groups at LHC reported independently that predominance of matter over anti-matter. they found a new particle with a mass of about 125 GeV, Thus, there is still a long way to go for physicists to which could be the Higgs boson [2,3]. Earlier this year, develop a comprehensive theory that can explain our their report was confirmed by further experimental data physical world from the sub-atomic scale to the cosmos collected at LHC [4]. There was a lot of excitement in scale. There is no wonder why many physicists in recent the physics community about their findings. Their results years are exploring new theories beyond the SM, (e.g., are considered to be a major triumph of the Standard string theory, brane theory, super-symmetry, etc.). Open Access JMP 22 D. C. CHANG In this paper, we will explore a new approach to ex- Equation (6) is similar to Equation (4) if we interpret amine a specific problem in quantum physics. We will the parameter M in Equation (6) as the “mass” of the ask only one question, namely, what is the meaning of excitation wave. In another word, the coefficient associ- particle mass? We will first give a brief review on how ated with the quadratic term Φ†Φ in the Lagrangian den- particle acquires mass according to the Higgs model. We sity as shown in Equation (5) is identified as the square will explain why we need to consider alternative ap- of the rest mass (m2). Thus, in the quantum field theory, proaches. Then, we will present a new model which is the rest mass of a free particle is essentially identified based on more classical concepts. Finally, we will dis- from the Klein-Gordon equation (through its associ- cuss the implications of this model. ated Lagrangian density). Using this as a guide line, one can investigate how a 2. How Do Particles Acquire Mass in the particle may acquire mass in the quantum field theory. Higgs Model? Let us use the gauge boson as an example to demonstrate this mass-generating mechanism. In the SM, the gauge Let us first review briefly how mass is generated accord- field is introduced solely to allow the Lagrangian density ing to the Standard Model. Classically, mass is defined to be invariant under a local gauge transformation [5]. from its association with momentum. i.e., The excitations of this field are the gauge bosons. The p mv (1) gauge boson (such as photon) originally has no mass. This is not exactly how the quantum field theory The gauge boson acquires mass due to its interaction works. In the Standard Model of particle physics, mass is with a theoretical Higgs field, under the condition that a property associated with a specific field. The starting the local symmetry of this field is broken. This mecha- point is the energy-momentum relation attributed to the nism was proposed by Higgs, and others in the mid special theory of relativity (STR), 1960s [6-8]. The gauge field mediating the electro-weak interaction 22224 Epcmc (2) was introduced as a requirement of invariance of a La- To construct the corresponding quantum wave equa- grangian density under local SU(2) transformation as tion, one can use the correspondence principle by con- well as a local U(1) transformation. This approach was verting the energy (E) and momentum (p) of a particle first explored by Yang and Mills in 1954 [9], and was into operators of a field, such that Ei t and later developed into a theory to explain the unification of p i , i.e., the wave equation is electro-weak interaction in leptons by Weinberg and Sa- lem [10-11]. The following is a summary of essential 2 2 2 1 mc steps used in the Higgs mechanism to allow gauge bos- 22 0 . (3) ct ons to acquire mass: Step 1: First, one may construct a general Lagrangian (Here is Planck’s constant divided by 2π, is the density which is Lorentz-invariant and can be renormal- wave function of the field). This equation is called the ized, Klein-Gordon equation. If we use a unit system such that c = ħ =1, Equation (3) can be simply expressed as L V . (7) 2 Here Φ is chosen to be a two-component field, (which 22 m 0 . (4) 2 is sometimes referred to as the Higgs field), i.e. t A In the SM, the concept of “field” is being generalized. , (7a) Each type of particle is regarded as the excitation wave B of its own field Φ (which can be a real or complex scalar where both ΦA and ΦB are complex scalar fields, function, a spinor, or a vector field). If one constructs a ii,. Lagrangian density of this field in the form of AB12 34 2 Step 2: If one requires this field to have a local U(1) L M , (5) symmetry, one needs to introduce a vector gauge field (where μ is the usual summation index representing the Bμ(x), which obeys certain transformation law. This pro- 4-dimenional space), one can easily derive a generalized cedure is analogous to the introduction of the vector po- Klein-Gordon equation using Hamilton’s principle (i.e., tential Aμ into electro-magnetism. If one makes a further by applying the Euler-Lagrange equation), requirement that this field is also invariant under a local 2 SU(2) transformation, one needs to introduce a second 22 vector gauge field, Wμ(x).
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