Relativity in Physics for the 21St Century

The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

Relativity in Physics for the 21st Century

W-Y. Pauchy Hwang∗1

1Asia Paciﬁc Organization for Cosmology and Particle Astrophysics, Center for Theoretical Sciences, Department of Physics, and Institute of Astrophysics, National Taiwan University, Taipei, Taiwan 106

At the beginning of the 21st Century, we may declare that we are living in the quantum 4-dimensional Minkowski space-time with, via the gauge principle, the force-fields gauge-group structure, SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × SUf (3), built-in from the very beginning. From there, it emerges the Standard Model that describes the smallest units of matter such as electrons, neutrinos, and quarks. I would like to point out as well as to discuss a few major century errors dictated by the God, throughout the 20th Century. We, as the human being, are sometimes so helpless in the struggles for the true knowledge. 1 A Question to Dirac and thus we don’t need the infinite sea of electrons. P.A.M. Dirac discovers the Dirac equation, rather than inventing this equation [1]. In our Apparently, Dirac knew these also, since he World (or, our Universe), the basic units of mat- knows the language so well. Later on, he pub- ter, such as electrons, neutrinos, and quarks, lished “Quantum Mechanics” which the mean- are the smallest objects ever exist in nature. ing of quantization was elucidated in great de- These smallest units of matter obey Pauli’s ex- tail. There is no “second quantization” and, clusion principle, as fermions, satisfying the anti- also, there is no “electron sea” whatsoever. The commuting Dirac algebra. consistency and, perhaps, the completeness are At 1928, we already knew the electrons, but absolutely demanded for the language (and its not the positrons. This gave a few then-famous logic). “physicists” a real puzzle to tackle. In fact, Einstein’s 1905 revolution was not What if the positron still has not had discov- complete at E2 = ⃗p 2 + m2, which implies the ered till now? This would be refute to Einstein’s Klein-Gordon equation and which is not suitable relativity principle, rather than the “invention” in the description of the matter. The Einstein’s of the infinite “electron sea”, etc. Fortunately, revolution is thus completed by Dirac’s lineariza- the position is there and its company with the tion, the discovery of the Dirac equation. electron makes the four components that are In my personal opinion, the idea of the “elec- needed by Dirac’s linearization of Einstein’s re- tron sea” could not stand out since there is no lation E2 = ⃗p 2 + m2. This fact consolidates stopover at the bottom, at the negative infinity. the status of Einstein’s relativity principle and The quantization of the Dirac field begins with Dirac’s linearization, paving the way of describ- the field operator Ψ(x) that is the combination ing the motion using Newton’s doctrine. of the terms - the particle annihilation operators a(+)(⃗p,s) and the antiparticle creation operators a(−)∗(⃗p,s); there is no difference between these Thus, the God could fool all of us, the human two terms. In terms of the language, the anni- being, if so far there had not been positrons dis- hilation and the creation have the same status covered in our world. Then, Dirac had to figure and, if needed, they could be interchanged. The out why his equation has four components but mathematical framework is perfectly consistent the Nature only needed two (for the electron). It seems that the mathematics and the physics ac- ∗Email: [email protected] tually converge to only one thing.

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2 Another Century Error Dic- the existence of dark-matter lumps interpreted tated by the God as some simple adjustable outlet. Remember- ing that a star of five solar mass, in terms of Schwarzschild’s solution was given on 13 January the smallest units of matter (such as electrons, 60 1916 by Einstein (on behalf of Schwarzschild). quarks, etc.), has 10 units of matter - a gigantic The horizon idea was developed by Wheeler and number, the re-scaling of Newton’s gravitational others in 1960’s, leading to Hawking’s black constant has to be acceptable. holes. The mathematics is correct, so further pursuing is reasonable. However, at the turn of the 21st Century, it 3 Why Relativity for the 21st was realized that our Universe is composed by Century and Onward? 70% dark energy, 25% dark matter, and only 5% visual ordinary matter. According to the Stan- In the 20th Century, we have firmly established dard Model [2] (that describes our Universe), the the two pillars of modern physics, Einstein’s rel- 25% dark matter are neutrino halos, made up ativity principle and the quantum principle. In from cosmic background neutrinos (of three fla- the beginning of the 21st Century, we have real- vors, and antineutrinos) (CBν′s), just like cos- ized that there exist the smallest units of mat- mic microwave background (CMB), from the ter, including electrons, neutrinos, and quarks. Early Universe. To describe these smallest units of matter, we Neutrino halos, which are Fermi-Dirac gases, have the Standard Model [2] as the mathemati- obey the quantum principle, or Pauli’s exclusion cal framework. ν principle. For the Earth, the Fermi energy kF It is not outrageous to declare [4, 2] that of the Fermi-Dirac sphere would be 600 GeV , as we’re in fact living in the quantum 4-dimensional suggested by the Sam Ting’s Space-Station AMS Minkowski space-time with, via the gauge prin- experiments [3]. ciple, the force-fields gauge-group structure, The distance dictated by the quantum prin- SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × ciple, or by Pauli’s exclusion principle, is deter- SUf (3), built-in from the very beginning. In the mined by the Planck constant ~, which is many Standard Model [2], the language is “relativis- orders bigger than the Schwarzschild’s radius rh. tic quantum mechanics and quantum fields” [5]. So, if we consider a realistic physical sys- Since the left-handed and right-handed compo- tem that has five times in weight the neutrino nents of fermions enter the theory differently, the halo (the 25% dark matter) and the ordinary- Standard Model is a completely massless theory, matter object (such as the visual Earth), then, apart from the “ignition” term of the sponta- due to the universal gravitational force, the vi- neous symmetry breaking (SSB) [6]. sual ordinary-matter part, at the “last” stage, In fact, when we declare that we’re living would be dragged by the “incompressible” neu- in the quantum 4-dimensional Minkowski space- trino halo - thus, that a black hole cannot ex- time, we also need to know how the unit of mat- ist (i.e., that the final destiny of the neutrino ter know from the space-time point xµ to another halo determines the final destiny of the entire point xµ + δxµ; that is, the content of the gauge system). principle ∂µ → Dµ (the change in the derivatives, Therefore, this is the mistake (or error) dic- or in the velocities). This is the centerpiece of tated by the God. Our Universe is described, the Newton’s description of motions. This is also mathematically, by the Standard Model [2] and the centerpiece of the Standard Model [2]. this Universe has CBν′s and CMB upon its To begin our discussions about the nuclear and birth. The physicists in the generations of the particle physics for the 21st Century, we may be 1960’s, 1970’s and 1980’s didn’t know the con- overwhelmed by lots of “non-relativistic approxi- tents of our Universe, thus fooled by the God. mations” in our theoretical languages. Although Some might argue that Newton’s universal we don’t know how much revisions that might gravitational law may have trouble in light of be involved, the attitude that we actually ac- such dark-matter halos. But the re-scaling of knowledge to live in the quantum 4-dimensional Newton’s gravitational constant can always treat Minkowski space-time should be a positive first

14 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

step. It means the basic change [7] into our rel- the gauge principle, the gauge-group structure, ativistic quantum attitude from that based on SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × Newton’s classic era - in fact, we could just look SUf (3), built-in from the very beginning. into the textbooks (of the 20th Century) in gen- There are plenty of evidences for the family eral physics, into how we were taught in ordinary gauge symmetry - the existence of three “gener- quantum mechanics, and so on. We believe that ations” for leptons (if without the family gauge the recognition of where we are living (in the concept), the phenomenon of neutrino oscilla- quantum 4-dimensional Minkowski space-time) tions, etc. What if this family gauge symmetry should be the first revolution in our altitude - is not a gauge symmetry? It does not look like then the realization of electrons, neutrinos, and so - indeed, if yes, it solves a lot of puzzles; while quarks as the smallest units of matter will come if not, many questions remain [1]. as the next revolution in our altitude. In fact, the meaning of “generations” dis- appears completely, when the family SUf (3) gauge group is adopted for leptons while, for 4 We’re Living in the Quantum quarks, all the entries as the triplets of the quark 4-Dimensional Minkowski group SUQ(3) (not a gauge symmetry) are used. Space-Time. Therefore, the group meanings are rather clear and there is no need for “three generations” The If we try to look into the history of weak interac- Standard Model [2] does provide the proper nam- tions as a whole, we have lots of basic questions ings of the various concepts in the group theory. to ask but so far we are lacking most of exper- In other words, with the quark group SUQ(3), imental and theoretical answers. Theoretically, the entries for quarks are triplets under SUQ(3) Fermi’s current-current interactions allow us how while the entries for leptons are singlets under to write an interaction between two fermions (in SUQ(3) - everything has its own group mean- two Dirac particles). Renormalizability forbids ing. This is the group in addition to the gauge that, while the concept of gauge fields comes to groups SUc(3) × SUL(2) × U(1) (for quarks) and × × rescue that. The SUL(2) × U(1) theory of gauge SUL(2) U(1) SUf (3) (for leptons). “Gener- bosons with the Higgs mechanism becomes the ations” becomes a misnomer. partially-unified electroweak theory. The gauge principle serves as the thread to tie Further on, if we insist on “renormalizability”, everything together. For a basic unit of matter → we combine the SUc(3) strong interactions to at- (for leptons or for quarks), ∂µ Dµ brings in tain the SUc(3) × SUL(2) × U(1) gauge-group the gauge fields in the overall framework in a framework. But this ad-hoc add-up does not natural way. Experimentally, the Space-Station have smooth behaviors as Q2 → ∞ (or r → 0). AMS experiments [3] provide the powerful evi- ′ ′ That is why the SUL(2) × U(1) × SUf (3) gauge- dence of CBν s, so that both CMB and CBν s group structure sets in for leptons. The quark are there in our Universe. Our Universe is indeed world, with the SUQ(3) fields (SUQ(3) the quark described by the Standard Model [2]. SU(3), used to be called “flavor SU(3) symme- There is one more clarification: When we try”), sees the first (123) gauge symmetry while say that “the quantum 4-dimensional Minkowski the lepton world sees another (123) gauge sym- space-time”, it implies that the objects with the metry. It guarantees the smooth behaviors as Dirac algebra (i.e., the quantum objects) exist in Q2 → ∞ or as r → 0, in terms of asymptotic this space-time. In fact, the introduction of the freedom and free of Landau’s ghosts. Dirac equation with the fermion algebra should The existence of the almost-uniform 3◦ K cos- mark the first entry of the quantum principle. mic microwave background (CMB) gives the best Mathematics-wise, the Standard Model [2] is evidence that the force-fields gauge-group struc- basically a group theory, though fairly compli- ture and the quantum 4-dimensional Minkowski cated. The entry point is characterized by the ba- space-time are born simultaneously. The man- sic units of matter, which are expressed in terms ifold that describes our Universe should have of quarks, electrons, neutrinos, etc. The gauge this overall group structure - the quantum 4- principle, i.e., ∂µ → Dµ which varies with the dimensional Minkowski space-time with, via basic unit of matter, also brings in the various

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gauge fields (i.e., the force fields). The complex attempt to do the Lorentz parametrization [9] scalar fields enter for the massive gauge fields. also contained an error in the induced weak elec- As the beginning, there is a quark group SUQ(3) tricity form factor. for which the basic units of matter for quarks It is a relativistic system and the relativistic are triplets while the basic units of matter for method should be used. To iterate again, we leptons are singlets. The basic units of matter are in fact living in the quantum 4-dimensional are described by the Dirac equations of the anti- Minkowski space-time with, via the gauge prin- commuting and closed 16-elements Dirac alge- ciple, the force-fields gauge-group structure, bra. If we count everything carefully, all physics, SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × including the hadron CP -violating phase, neu- SUf (3), built-in at the very beginning. Thus, trino oscillations, etc., are taken into account. we should give up our past experiences with the The beauty of the Standard Model [2] is remark- non-relativistic methods, in our opinion. able. At the beginning of the 21st Century, we Because we live in the quantum 4-dimensional should use and develop the methods that respect Minkowski space-time, we should do everything Einstein’s relativity principle and the quantum in this space-time if the purpose is to understand principle. It is the first step to recognize where we’re living. The nuclear world, or the the we’re living in the quantum 4-dimensional quark world, should be described by Einstein’s Minkowski space-time with, via the gauge prin- relativity principle and the quantum principle. ciple, the force-fields gauge-group structure, Similarly, the atomic world, or the lepton world, SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × should be described by relativistic quantum me- SUf (3), built-in from the very beginning. chanics, rather than by the so-called “ordinary The Lee-Mo-Wu experiment [8] to test the quantum mechanics”. This observation puts a validity of Conserved Vector Currents (CVC) lot of burden on all the scientists in the entire marked the first quantitative weak-interaction communities of chemistry and of physics. This experiment, published in 1963, that tried to an- would be one important goal for the scientists of swer the basic questions regarding the weak in- the 21st Century. teractions. Physicists at the time realized that the A = 12 system is far from simple. Kim and Primakoff [9] invented, in 1965, the “elementary- 5 Another Forgotten Struggles particle treatment (EPT )” to interpret the Lee- in Relativity Mo-Wu experiment [8]. Whether the original Lee-Mo-Wu experiment Dirac’s basic discovery [1] of the Dirac equation [8] tests CVC became a big dispute, that is sum- for the matter implies that Lorentz invariance marized in the Commins’ textbook, published in and the complicated spin structures go hand in 1983 [10]. hand. That is why the Dirac equation and its Experimental-wise, the table, at that time, Dirac algebra is so unique, so nontrivial, should of Fermi functions contains an error, that tran- be 4 × 4, anti-commuting, and closed in the spired into the original Lee-Mo-Wu experiment. 16-elements. The “anti-commuting” means the Theoretical-wise, the Lee-Mo-Wu experiment operation (or, the invasion) of “the quantum uses A = 12 nuclei, which is too complicated principle” - noting that, classically, the anti- in interpreting. commuting nature does not belong. The story deviated from the right course, The first relativistic, though complicated, nu- mainly due to the complexity of the A = 12 clear system, the first such system that was sub- nuclei. In fact, in the original EPT treatment ject to experimentation, is the A = 12 triad [8]. [9], the introduction of the induced weak-electric 2 To interpret that, Kim and Primakoff [9] devel- FE(q ) form factor was in error and it turned out oped the framework called “Elementary-Particle to be numerically important [11, 10]. So, after Treatment (EPT )”. The A = 12 system is so more than twelve years, the story was finally set- complicated that the tabulated Fermi function tled with the paper of Hwang and Primakoff [11]. contained an error while experimentalists used The primary message, from the above A = 12 it without knowing. On the theory side, the first saga, is that we do know how to formulate the

16 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

problems in the relativistic quantum mechanics, interactions and in electromagnetic interactions but in detail we have to be careful with the com- - are they the same currents or not? “Strong” plexity of the Lorentz group and of other sym- CVC asserts that they are the same currents, metries. In short, all nuclei should be treated in made from quarks of only one kind and noth- a relativistic and quantum manner. ing more. This is the basic question to address, The internal quantum wave function of a nu- since after all the quarks in the constituent pic- cleus in the relativistic case is probably not very ture cannot be observed directly. In this arti- different from that in the non-relativistic approx- cle, CVC means “strong” CVC - that is, quarks, imation - judging from the fact that all nucleons apart from colors and flavors, are not duplicated. have to stick together to form a whole nucleus. “Strong” CVC and the absence of second-class But this should not be used as an excuse for not currents (SCC) would go hand in hand, using the developing the relativistic framework. In princi- standard definition of strong CVC and of SCC. ple, for the A = (Z,N) nucleus, we have to sepa- With the advancement of the electroweak the- rate the center-of-mass (CM) space-time coordi- ory, with the minimal Standard Model (SM) in nates (or something similar) and to write down particular, one would argue that it seems trivial the proper internal quantum wave functions. In to examine the question which CVC tries to ask - fact, we still don’t know how to do this exactly. “are they the same currents or not?”. In fact, the Let’s iterate: We actually live in the quan- Z0 boson in the minimal SM does couple to the tum 4-dimensional Minkowski space-time with, same iso-vector vector current but beyond the via the gauge principle, the force-fields gauge- SM there might be a source for a new iso-vector group structure, SUc(3) × SUL(2) × U(1) or vector current. So long as we examine the is- SUL(2) × U(1) × SUf (3), built-in at the very be- sue deeper, the question might remain. But it is ginning. The Standard Model [2] emerges from of interest to note that the question associated there [6, 5]. In this 21st Century, we should even- with strong CVC is partially answered; at least tually establish the suitable language for nuclear the advancement of the minimal SM does have physics and for particle physics. In fact, this is the impact on the issue. for all centuries, the 21st Century and onwards, In Chapters 4 and 5 of an advanced textbook as we believe that the thinking(s) of Newton’s by Commins and Bucksbaum [10], a nice exten- classic era would eventually cease to exist. sive review of Conserved Vector Currents (CVC), absence of Second-Class Currents (SCC), Par- 5.1 The Lee-Mo-Wu Experiment tially Conserved Axial Currents (PCAC), etc., is given in connection with muon capture and I think that we should give the due credit to the beta decays (in nucleons and nuclei). The sub- Lee-Mo-Wu experiment, in testing fundamental ject occupied the center stage of nuclear physics symmetries in this complicated world. and early particle physics since the early 1940’s Conserved vector currents (CVC), partially till the end of the 1960’s. conserved axil-vector currents (PCAC), and the In fact, after forty-five years or so of giving size of second-class currents (SCC) were among the platform entirely to the developments of the the top in the issues, when the weak interactions Standard Model (in particle physics) since the were first discovered. Now (in 2017), the Stan- early 1970’s, we should pay another review of dard Model [2], phrased in the so-called “con- this very broad subject. Hopefully, we would stituent” picture, takes up its final shape (on like to see which ways we are going. these currents) and, on the other hand, some So, what is important and indispensable out other basic questions in weak interactions still there, in terms of our future studies? It seems wait to be answered in a quantitative manner. that the descriptions in terms of the constituents When we begin in writing an article of this (quarks, and others) are so dominant that there kind (in the 21st Century), we need to remind is no need to get back to the language of old ourselves of the deep meanings of these funda- days. In fact, all great physicists in the sixties mental questions. For instance, what we mean or seventies wrote papers in terms of the matrix by a test of “strong” CVC, and of “weak” CVC, elements, using the language to express the basic etc. We have iso-vector vector currents in weak ideas. One purpose of this article is to examine

17 Regular Article The Universe, Vol. 5, No. 1 January-March 2017

this issue. constraints on some form factors and so to de- To some extent, the spirit of the Hwang-Kim- termine the size of the second-class currents. We Primakoff method or the “elementary-particle were lucky that we could do that. treatment” has been exercised to an extreme in Our current form differs sightly from the old 2 some sense. After the radiative muon capture Kim-Primakoff form (in which FE(q ) was ab- work, we turned our attention to the question sent) but I didn’t try to persuade much Henry of whether there is the second-class current; in and hence I already had the blessing from him. fact the experiments from University of Tokyo In the old form (1965-form), there is no weak- 2 and from Princeton University at that time indi- electricity FE(q ) term, which turns out to be of cated sizable second-class currents. Our analysis numerical importance. All my work so far had used the “parametrization” [11]: convinced Henry that I knows very well how to parameterize in a Lorentz covariant way. In 1981 12 12 < C(p1) | Vλ(0) | B(p2, ξ) > or 82, Val Telegdi used the term ”Hwang-Kim- √ qρ Qη 2 Primakoff methods” at the Symposium in cere- = 2ϵλκρηξκ FM (q ), (1) 2mp 2M bration of Henry’s 65th birthday, trying to explain our success in terms of the ”standard” picture - CVC, PCAC, and No Second-Class Cur- 12 12 < C(p1) | Aλ(0) | B(p2, ξ) > rent. √ · { 2 q ξ 2 The Hwang-Kim-Primakoff method could be = 2 ξλFA(q ) + qλ 2 FP (q ) mπ nothing but an application of Lorentz invari- · Qλ q ξ 2 ance to the physical processes of complicated − FE(q )}, (2) 2M 2mp systems, i.e. particles and nuclei, with sizes smaller than a few fermis. The typical nu- The spin-0 12C nucleus is described by a mo- clei include the famous A=12 nuclei, the isospin 3 3 mentum four vector p1. On the other hand, the and spin doublets such as ( He, H), and two- spin-1 12B nucleus is described by the four mo- nucleon systems, etc. while the processes in- mentum p2 and the polarization four vector ξ. clude all involving these nuclei so long as the We define qλ ≡ (p2 − p1)λ, Qλ ≡ (p2 + p1)λ, and same or related initial and final states. I think ≡ 1 M 2 (M1 + M2) with the symbol M or m for that application of the Hwang-Kim-Primakoff the mass of the object (such as the proton mass method has changed these (originally nuclear mp and the pion mass mπ). physics) problems into the elementary-particle- 2 2 2 The four form factors FM (q ), FA(q ), FP (q ), physics problems and makes the two disciplines 2 and FE(q ) are, respectively, the (nuclear) weak becoming one. In this sense, whether Lorentz in- magnetism, axial, pseudoscalar, and weak elec- variance applies to nuclear systems equally well tricity (pseudotensor) form factors. These four as elementary-particle systems and whether the Lorentz-invariant form factors constitute the ba- above Lorentz-covariant parameterizations could sis of the framework that respects Einstein’s rel- be applied is the gut of the issue at stake. ativity principle and the quantum principle - the To summarize what we have said so far, we framework that we are pursuing after. use the elementary-particle treatment, or the 12 12 − In other words, the beta decay B → Ce νê Hwang-Kim-Primakoff method, to describe the 12 12 and the muon capture µ + C → νµ + B, A = 12 nuclei to test symmetries including CVC, and the related reactions (via CVC, etc.), are the absence of Second-class Currents, and PCAC described basically by the four form factors - very successful so far. Of course, we should 2 FM,A,P,E(q ). It is this basic fact that the num- extend such tests more systematically and more ber of the observables in the various reactions is precisely. On the radiative processes (which were larger than that of the unknowns in the entire intended originally for the test of PCAC), it problem. turns out that we have to look into the theo- Well, using CVC and etc., we could describe retical framework first - the suggestion is to first four or five reactions near the thresholds simul- sort out how restore gauge invariance, CVC, and taneously in terms of four form factors near the PCAC, before we could use it as a genuine test thresholds - that is why we could put stringent of PCAC. Besides the tests of CVC, second-class

18 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

currents, PCAC, etc., we consider other tests four vector (which eventually is set to zero in such as absence of right-handed currents, etc., to view of Lorentz Invariance). broaden the scope of our studies. In all cases, we The transition amplitude for the beta decay 12 12 − assume the relativistic invariance to begin with. B → Ce ν¯e is given, in our normalization, Furthermore, how about CP or CPT invari- by ance? We know that CP is broken slightly if 12 12 − CPT invariance is sacred - the reason for this T ( B → Ce ν¯e) violation is so far mysterious. The weak interac- = √G <12 C(p ) | [V (0) + A (0)] |12 2 1 λ λ tions involve only the left-handed currents and B(p2, ξ) > ·iu¯e(pe)γλ(1 + γ5)vν (pν (3)). where are the right-handed currents? A ques- e e tion that already has the answer in the Standard For the definitions as in the above, consult the Model [4]. As we would like to understand the original reference [11]. For the basic notations baryon asymmetry in this Universe, we have to (such as γ-matrices, etc.), see the textbook - understand these symmetries or asymmetries in “Relativistic Quantum Mechanics and Quantum proposing the final theory of our Universe - Cos- Fields” by Ta-You Wu and W-Y. Pauchy Hwang mology or, in the mathematical term, the Stan- (World Scientific, Singapore, 1991) [7], or its dard Model [4]. In fact, tests of CVC, PCAC, 21st-Century Edition (2017). and the absence of second-class currents are the The e− decay energy and angular distributions first steps in our efforts to unravel these myster- and the corresponding asymmetry coefficient A− ies. are then

3 − 5.2 The Famous Example: The A=12 d Γ(e ) 2 √ ∼ G 2 − 2 Triads = [ 2F (0)] F−(Z,E )p E (∆ − E ) 8π4 A e e e e P − We are talking about the A = 12 (I,J ) = (1 + η + a−Ee)dEedΩe (1, 1+) and (0, 0+) nuclear states and the ×[1 − (h1 − h−1)(1 + α−Ee)cosθe transitions among them. The (1, 1+) states 3 2 1 are 12B(g.s.) (15.21 MeV), 12C(15.11), and +(1 − 3h )α−E ( cos θ − )], (4) 0 e 2 e 2 12N(g.s.) (14.98 MeV), relative to 12C(g.s.) (0.00 MeV). How do we treat the various reactions in a rela- A − − − tivistic manner - or in a Lorentz-invariant man- ≡ dΓ(e ; θe = 0) dΓ(e ; θ = π) − − ner? In general, the spin structures are fairly dΓ(e ; θ = 0) + dΓ(e ; θ = π) ∼ complicated but the spin structures are a nec- = −(h1 − h−1) essary consequence of Lorentz invariance - as {1 + α−[1 − (1 − 3h0)]Ee}. (5) a theorist, we often overlook this basic aspect. The problems of the A = 12 triads offer a “sim- Here we have ∆− ≡ M(12B) − M(12C). ple” exercise of dealing with the complicated spin F−(Z,Ee) is the well-known Fermi function for − structures. the β decays. h1, h−1, h0 are the populations 12 The electroweak reactions include of the Jz = 1, −1, 0 states of B (normalized so 12 12 − B(g.s.) → C(g.s.) + e +ν ¯, that h1 + h−1 + h0 = 1). cosθe ≡ pê · zˆ with θe 12N(g.s.) →12 C(g.s.) + e+ + ν, the angle between the electron momentum and 12C(15.11) →12 C(g.s.) + γ, the 12B polarization. The various coefficients are e− +12 C(g.s.) → e− +12 C(15.11), and given by µ +12 C(g.s.) → ν +12 B(g.s.). During the early 4 FM (0) 60’s, these are “frontiers” of the field. a− ≡ , We begin by writing down the matrix elements 3mp FA(0) of the polar vector and axial vector currents 1 FM (0) FE(0) α− ≡ ( − ) 12 12 Vλ(x) and Aλ(x) between B(p2, ξ) and C(p1) 3mp FA(0) FA(0) − as above (Eqs. (1) and (2)). Let’s try to explain ∆ FM (0) FE(0) η− ≡ (−2 + ). (6) the notations [11]: x, or xµ, is the space-time 3mp FA(0) FA(0)

19 Regular Article The Universe, Vol. 5, No. 1 January-March 2017

12 12 + Analogously, N(g.s.) → C(g.s.) + e + νe Indeed, Commins and Bucksbaum [10] in their is described exactly by the same formulas, ex- textbook quote g2 = −(0.4 0.9)g1/(2mp) (in 12 cept that the isospin rotation from B(g.s.) to part of the conclusion on p.197, g1 the nucleon 12 N(g.s.) is performed. We may assume that the axial coupling and g3 the second-class coupling) isospin symmetry is exact so that the four fac- [10]. (Maybe a better way to contrast with is tors would be identical - until the broken isospin to compare with the weak magnetism coupling symmetry is discussed. which is about four times bigger in size.) This The 12(15.110) →12 C(g.s.) transition or elec- reflects the status of the situation regarding the tron scattering is related to the above beta de- absence of the second-class currents - fairly large cays also by isospin rotations - the iso-vector po- errors thus leading to reasons for writing this lar vector parts (regarding the CVC story). review article. On the other hand, the transition amplitude Why are the right-handed currents missing, − 12 12 for the muon capture reaction µ + C → νµ + and to what precision levels? Is V-A exact? Why B is given by do only the left-handed currents couple to the weak bosons, W ± and Z0? These are very basic − 12 12 T (µ + C → νµ + B) questions to ask, after several decades of work- G 12 † † 12 ing with the constituent picture (of quarks) to = √ < B(p2, ξ) | [V (0) + A (0)] | 2 λ λ describe physics. Of course, it is worthy of work- · ing with the constituent picture for so long - for C(p1) > iu¯νµ (pνµ )γλ(1 + γ5)uµ(pµ). (7) example, the exactness of the V-A currents sim- Thus, the physical observables for the above ply implies the left-handed nature of the charged muon capture reaction can be expressed in term weak bosons. If the right-handed currents could 2 2 2 of the four factors FA(q ), FM (q ), FP (q ), and be found at a certain level, this may be taken as 2 2 2 the signature of the existence of the right-handed FE(q ) at a certain q (= 0.740mµ). By compar- ison, the q2 for the beta decays is approximately weak bosons. zero - the formulae in the above take care of this In other words, tests of the various symme- fact. tries could have very different fundamental im- The muon capture rate is specified in detail by plications. Test of strong CVC means [11] that [11]. So, if isospin symmetry is exact, we have there is only one conserved vector current and the case that the four factors (functions) describe that is the same iso-vector current entering the many reactions; near threshold, we have tests of electromagnetic and weak current. Existence of CVC and the absence of Second-class currents. the second-class currents on top of the ordinary In fact, the main conclusion which we were first-class currents means the duplication of the able to draw came from the fact that the early isospin quark doublet (i.e. additional (u, d)). In experiments in support of sizable second-class addition, tests of weak CVC and of weak PCAC currents is intrinsically contradictory to the ob- refer to the structure of the currents themselves. served rate for muon capture which, owing to Furthermore, different experimental tests of 2 2 q , determines very well the size of FM (q ). So, CPT or T keep going on. Presumably, these we cannot draw the conclusion simply from beta tests, if any violation be found, would bring us to decay reactions since the muon capture reaction another comprehension of the nature. So far, we should be in the game owing to its (better) sensi- have furthered our understanding of the small vi- tivity to certain form factors. The natural way is olation of CP but the understanding of its mean- to see all these is through the elementary-particle ing is still lacking. Clearly, the search along this treatment or the Hwang-Kim-Primakoff meth- line (along the line of Commins and Bucksbaum ods. [10]) is clearly fundamental and of utmost impor- I think that the questions these days would tance. We should not “shut” our eyes, though be, e.g., whether there still be some second- working more on “the constituent picture” for class currents, at the level of 10% or better; how decades. about the precision associated with the validity During the old days, we used the EPT method of CVC; how to get a convincing test of PCAC; to look at a few interesting cases. Apart from and so on. the isospin and spin doublet, such as (p, n) or

20 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

(3He, 3H) (to be described below) [12], we also the late Henry Primakoff, suggested to me to use considered the following cases. In muon capture radiative muon capture as a test of PCAC. The 12 − by C, there might be some interest [13] to mea- first thing to work on was µ + p → νµ + n + γ. sure the average polarization of the recoiling 12B. But the major problem is that when you add up Excitation of the 15.110 MeV of 12C by neutri- the Feymann diagrams for particles with sizes (p, nos and antineutrinos [14] would eventually be- n, etc.) gauge invariance (GI) is violated; but come of interest (and now it is). These belong to to test the validity of a general principle such the famous A=12 triad. Because of some inter- as PCAC, we need a quantitative prediction on est in nuclear parity violation, I also examined the amplitude. By looking at the algebraic con- parity mixing in 18F as seen by low-energy in- straints arising from GI, CVC, and PCAC, I pro- elastic electron scattering [15]. All of these are posed to Henry that we could use the linearity simply the beginning of the game, after we rec- hypothesis (LH) to fix the amplitude. That is, ognize that we are in fact living in the quan- we are able to solve the coupled equations de- tum 4-dimensional Minkowski space-time with, rived from CVC, PCAC, and GI, making use of via the gauge principle, the force-fields gauge- LH. The trouble is, as we found out later on, group structure SUc(3)×SUL(2)×U(1)×SUf (3) that our way to restore GI, subject to CVC and built-in at the very beginning [2]. PCAC in addition to GI, is in fact different from Nowadays we have a broader list of elec- the then standard method that takes care of GI troweak reactions that become available and are alone. In early 80’s, when I saw Henry and talked subject to studies for one reason or another. about our LH, Henry still thought that it is a Our capability to answer the basic questions is nice approach - I’m sorry I did not fight on. In thus strengthened and the EPT or HKP method 1983 Henry passed away because of the lymph should be sharpen to become the methodology cancer. Maybe we should be thinking of this for analysis of these basic questions. Those in- problem again, despite forty years or more later. − duced by neutrinos (solar neutrinos, accelerator After I “solved” µ + p → νµ + n + γ, it im- − 3 3 neutrinos, or reactor neutrinos), those induced mediately applied to µ + He → νµ + H + γ by the electron beams of different energies, and [16], the other well-known spin and isospin dou- those induced by the primary proton beams or blet via the so-called “elementary-particle treat- by the second pion or kaon beams - we are living ment” (EPT). The application to the A=12 tri- − 12 12 in an era that there are indeed so many possi- ads, µ + C(S = 0,I = 0) → νµ + B(S = bilities (or nothing impossible) in terms of the 1,I = 1) + γ, took the effort of another few experimental means. These days neutrinos or months [17]. Again, I used the LH to turn the antineutrinos, as now with oscillations, are sub- equations into a set of algebraic equations. ject to intensive studies. Theory-wise, the EPT In fact, one can always quantify the difference or HKP method provides a useful methodology between our LH approach and the “standard” at medium energies, particularly when relativis- way to restore GI. In other words, which method tic “corrections” becomes dominant at these en- in restoring GI is probably the second question ergies. Moreover, we in fact live in the quan- and in fact we could derive two predictions, and tum 4-dimensional Minkowski space-time, such then we could compare the two predictions with that we should adopt the framework that respects the experiment, if the data would be reliable. both Einstein’s relativity principle and the quan- This type of jobs have never been carried out - tum principle [2, 5]. Leptons and quarks are the maybe because the radiative muon capture reac- smallest units of matter. tions are already too complicated. The other alternative is that in our method 5.3 Radiative Processes we use LH to fix the amplitude that is consistent with GI, CVC, and PCAC simultaneously; not Let us begin with a debate (among medium- just with GI only. The old method treats only energy theorists) as a way to enter the center of with GI only; what happens to CVC and PCAC? the subject. We were interested in the validity So, we may answer the dispute “theoretically”. and, thus, the test of PCAC, which seemed to be I think that at that time I was too young to sit solved very soon. In late 1973, my thesis advisor, on the same problem - too bad to be so young.

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To explain it briefly, we invoke LH so that GI, So, in radiative muon capture (or radiative CVC, and PCAC can be taken care of simultane- processes of the extended objects), the em- ously (by solving the coupled algebraic equations phasis should be first focused on the test of coming out of GI, CVC, and PCAC) while in the “methodology”, before the tests of symme- the Low theorem approach (the “standard” ap- tries could be reliably addressed. Test of the proach) only the presence of gauge non-invariant “methodology” becomes an important issue if terms is secured by adding certain terms - in there are more than two methodologies for the LH, the argument of the various form factors is same problem. shifted, (q+k)2 → q2 while in Low theorem some On the symmetry front, the citing of our pa- k−dependent terms are added. The terms com- pers for CVC, second-class currents, and PCAC ing out of the (q + k)2 dependence (i.e., from the by Commins and Bucksbaum [10] have good rea- extended structure of the system) is the source sons. Our second lesson is that tests of funda- of GI violations - we need to fix this problem for mental symmetries are best phrased in terms of the amplitude before we can use it in making the matrix elements, not in terms of the language predictions. In our approach, CVC and PCAC in the constituent picture. Furthermore, we try in addition to GI all would be at stake and thus to write everything in a Lorentz (relativistically) we convert the problem into a set of algebraic invariant way. That means that Lorentz invari- equations. ance is more fundamental than other symmetries Linearity hypothesis (LH) that helps to solve (such as CVC, PCAC, the absence of the second- the constraints out of CVC, PCAC, and GI, al- class or right-handed currents, etc.) which we together, should not be the party to blame. The try to test. real reason is: Those people exercising the Low’s Consider the spin and isospin doublet such as theorem were so reluctant to give it up. In their (p, n), (3He,3 H), (15O,15 N), etc. and all the Low’s theorem, the term to restore GI is rather reactions involved by the same spin and isospin ad hoc; CVC and PCAC were left “violated”. I doublet [12]. For example, the triton β-decay, 3 3 − wonder whether Low would still keep his theorem H → He + e +ν ¯e, has been studied in untouched. quite details. This implies that the form fac- 2 2 ∼ Fortunately it remains the case that the test tors FV,M,A(q ) at q = 0 are well-known. These of PCAC could be best carried out in ordinary form factors are defined analogously by [12] muon capture such as µ + p → ν + n. The rates µ <3 H(p′, s′) | V (0) | 3H e(p, s) > from radiative muon capture are indeed very sen- λ ′ σληqη sitive to pseudoscalar form factors, but how to =u ¯(p )[F (q2)γ − F (q2) ]u(p),(8) V λ M 2m describe the radiative processes, the connection p of the radiative processes (with a photon) to the 3 ′ ′ 3 ordinary processes (without a photon) for ex- < H(p , s ) | Aλ(0) | He(p, s) > tended systems, is up in the air. In our approach, ′ 2 =u ¯(p )[FA(q )γλγ5 we elevate CVC and PCAC together with GI in 2Mq γ5 order to fix the radiative matrix element - unlike +F (q2)i λ ]u(p). (9) P m2 the Low theorem which takes care of GI only. π In fact, this is a general problem for radiative The other reaction well-studied is the muon cap- − 3 3 processes which are quite common, as most of ture reaction, µ + He → νµ + H, from which 2 2 time the hadrons involved are extended, i.e., not the form factors FV,M,A,P (q ) at some small q ∼ point-like. The q2-dependence together the ra- can be accessed (in this case we have q2 = 2 diative process now requires the treatment of the 0.96mµ). Of course, the elastic electron scatter- shift in (q + k)2. There was the so-called Low ing e−+3He → e−+3He and e−+3H → e−+3H theorem, but we think that it is wrong. Our first are much easier to study - from generalized CVC, 2 2 lesson of this article is that the relation between we have good knowledge of FV,M (q ) at these q . the radiative process and the related ordinary In other words, the various observables in process is expressed in terms of the matrix ele- these reactions can be correlated among them- ments, not in terms of the language in the con- selves; those observables which have been mea- stituent picture. sured experimentally could be correlated and

22 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

tested while those observables remain to be pre- ∫Dµ(k, q, Q) dicted for future experiments. 4 −ik·x In fact, as time goes by, some experiments be- = d xe < Nf (pf ) come feasible and, for one reason or the other, | T (Jµ(x)∂λAλ(0)) | Ni(pi) >, (16) are of great interest - such as neutrino or antineu- trino scattering of these targets, 3He and 3H, and and parity violation studies in electron scatter- | | ing off these nuclei. So, Hwang-Kim-Primakoff kµDµ(k, q, Q) = i < Nf (pf ) ∂λAλ(0) Ni(pi) > . methods are quite versatile, indeed. (17) We would like to discuss radiative muon cap- In the case of the nuclear spin and isospin dou- ture versus ordinary muon capture, in order to blet(s), our construction gives 34 form factors display one basic problem associated with the ex- for Vµλ(k, q, Q) and another 34 form factors for tended structure of the system. Of course, this Aµλ(k, q, Q). In this case, CEC gives us 16 con- is a general problem for any radiative process straints while CVC gives 8 constraints, and so (with an extra photon) with an extended system on. The important point is that it gives us the versus the ordinary process (without the extra ”hint” about the possible solution. photon). It is clear that the point-like particle That is where the complication comes from; it doesn’t have the same problem, that is, there sounds like that we invented the “linearity hy- is no displacement in the form factor such as pothesis (LH)” so that GI (or CEC), CVC, and (q + k)2 → q2 involved. PCAC all can be solved. It turns out that, if our When the HKP methods are applied to the solutions are not adopted, it would be difficult − 3 3 to another solutions that would be consistent si- radiative muon capture [16], µ + He → νµ + H + γ, one has to solve the tensors: multaneously with gauge invariance, CVC, and PCAC. Most of the approaches only take care of Vµλ(k,∫ q, Q) gauge invariance, but ignoring CVC and PCAC. 4 −ik·x In our view, linearity hypothesis should not be = −imp d xe < Nf (pf ) blamed as we may examine the CEC, CVC, and | T (Jµ(x)Vλ(0)) | Ni(pi) >, (10) PCAC constraints altogether. In fact, they call that “Low’s Theorem” when they take care of Aµλ(k,∫ q, Q) gauge invariance; it’s difficult to fight against some “theorems”, of course. − 4 −ik·x = imp d xe < Nf (pf ) The intention was the attempt to test PCAC but GI was violated when you sum up all the | T (Jµ(x)Aλ(0)) | Ni(pi) > . (11) “Feynman” diagrams with extensive vertices. In general, gauge invariance (GI) gives us the Maybe the appropriate question should be “how constraints which we call CEC: the GI, CVC, and PCAC get restored in radia- kµ tive muon capture” or something similar. This Vµλ(k, q, Q) =< Nf (pf ) | Vλ(0) | Ni(pi) >, mp is another 20th-Century error that we theorist (12) might have made, in relation to the tests of fun- kµ damental symmetries. Aµλ(k, q, Q) =< Nf (pf ) | Aλ(0) | Ni(pi) > . mp (13) 5.4 E.M. Henley and the Break-Up On the other hand, CVC and PCAC yield Channels (kλ + qλ) V (k, q, Q) My research works with Ernest M. Henley m µλ p started with the problems with the deuteron | | = < Nf (pf ) Vµ(0) Ni(pi) >; (14) and its break-up channels. Our focus is about parity violations in the electron-deuteron scat- (kλ + qλ) Aµλ(k, q, Q) tering, elastic and break-up channels. The mp real efforts are with the deuteron disintegra- | | = < Nf (pf ) Aµ(0) Ni(pi) > tion via break-up channels. The underlying the- +Dµ(k, q, Q), (15) ory is the Glashow-Salam-Weinberg (GSW) elec-

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troweak theory plus parity nuclear violations - wave functions. Similar expression was adopted nonleptonic parity violations represent basically 1 (f) | (A) | for the matrix element < S0(p ) Nλ (0) a new arena (than GSW). The research works D(p(i), ξ(i)) >. of this type, which are fairly complicated and On the other hand, we introduced in the pa- 3 tedious, provide good training for a young theo- per, with F = P1 (subchannel γ), rist, like me at that time. 3 (f) (f) | | (i) (i) When we treat break-up channels such as < P1(p , ξ ) Jλ(0) D(p , ξ ) > electron-deuteron scattering into the various ¯(f) (i){ Qκ γ 2 − qκ γ 2 } = iϵλρηκξρ ξη FV (q ) FR(q ) (np)-scattering states, each final state has its 2M 2mp Q own total angular momentum - so, by Lorentz ¯(f) (i) qκ ζ { qλ γ 2 +iϵρηκζ ξρ ξη FS (q ) invariance, each final state has to be treated sep- 2mp 2M 2mp arately. In other words, it seems that we don’t Q − λ F γ(q2)} gain too much. 2M T One comprehensive reference that can be re- (i) · qρ Qη {¯(f) ξ q ferred to is the paper by Hwang, Henley, Miller +iϵλρηκ ξκ 2mp 2M 2mp [18]. When the excitation energy is limited to a ¯(f) · certain (small) value so that the (np)-scattering (i) ξ q } γ 2 +ξκ F (q ), (20) 1 3 3 3 2m L states such as S0, P0, P1, and P2 dominate p the parity-conserving channels, the treatment is 3 (f) (f) (i) (i) (−) still systematic. < P1(p , ξ ) | Jλ(0) | D(p , ξ) > The fate of the paper is the same. The al- ξ(i) · q ξ¯(f) · q gebra looks very complicated; thus, no one else = i{ξ¯(f) − ξ(i) }Gγ (q2) λ 2m 2m M wanted to follow up (in the last forty years). But p p Q I think that the young experimentalists should ¯(f) · (i) λ γ 2 +iξ ξ GV (q ) eventually try while the young theorists should 2M (i) · ¯(f) · think how to do it relativistically. As we said, {¯(f) ξ q (i) ξ q } γ 2 +i ξλ + ξλ GT (q ) Einstein’s relativity principle and the quantum 2mp 2mp ¯(f) · (i) · principle will be the basic cornerstones of our ξ q ξ q Qλ γ 2 +i GS(q ) World - there is no escape from that. 2mp 2mp 2M The overall contribution on the cross section ¯(f) · (i) qλ γ 2 +iξ ξ GY (q ) would be the incoherent sum of the different 2mp channels, i.e. into the 1S , 3P , 3P , and 3P 0 0 1 2 ξ¯(f) · q ξ(i) · q q final states and others. Parity violation signals +i λ Gγ (q2). 2m 2m 2m Z would also be calculated channel by channel. We p p p 1 (21) define, for the S0 final state (subchannel α), 3 (f) (f) 1 (f) (i) (i) Similar expression for < P1(p , ξ ) | < S0(p ) | Jλ(0) | D(p , ξ ) > √ N (A)(0) | D(p(i), ξ(i)) >. (i) qρ Qη α 2 λ = 2ϵλκρηξκ FM (q ), (18) Well, there are at least four channels (which 2mp 2M we considered in the paper [18]) and in each channel there are the matrix elements of or- 1 (f) | | (i) (i) (−) < S0(p ) Jλ(0) D(p , ξ ) > dinary (parity-conserving) and parity-violating √ · (i) { α α 2 q ξ α 2 electromagnetic currents (such as above) and of = 2 ξλ GV (q ) + qλ 2 GS(q ) 0 mπ the neutral weak currents (for the Z -exchange); · (i) each matrix element is parameterized by a num- − Qλ q ξ α 2 } GT (q ) , (19) ber of Lorentz-covariant form factors. So, there 2M 2mp are too many unknowns - what’s the purpose of (i) (f) (i) (f) where qλ ≡ (p − p )λ and Qλ ≡ (p + p )λ. the Lorentz-covariant parametrization? In fact, Here the parity-conserving matrix element is a typical nuclear physics problem, if analyzed in the ordinary M1 transition while the parity- details, is often very complicated, but the im- violating matrix element refers to the contribu- portant aspect is that the problem can be dealt tions due to the odd-parity admixtures in the with in a systematic manner.

24 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

To proceed, we have to use the so-called EPT- tricks are there. NOIA/GI connections in order to make clear-cut predictions. The Nucleon-Only Impulse Approx- 5.5 Prospects of the Elementary- imation (NOIA) as constrained by Gauge Invari- Particle Treatment ance (GI) is a microscopic model that is used by us to be contrasted with the Elementary-Particle When we apply the elementary-particle treat- Treatment (EPT) framework - the contrast is ment (EPT) to particle or nuclear problems, i.e. carried out in the Breit frame. The above sen- to use EPT to start Lorentz parameterizations tence means a lot which we cannot elaborate here for the description of the problem, the language [18]. is at least correct and is suitable for the field of By invoking EPT-NOIA/GI, we basically en- nuclear physics or of particle physics. large the prediction power of the Hwang-Kim- On the radiative process versus the non- Primakoff methods. It is better than NOIA/GI radiative, the first prospect would be how to because the Lorentz-covariant form factors are modify the ordinary process to describe the ra- used. diative process (to add a photon in an extended In those years, the EPT-NOIA/GI exercises system). As said before, radiative muon capture have been too complicated, as it could be seen is just one example. There are so many radia- from appendices [18], such that the strength of tive processes which deserve our attention – it is the methodology seems to be obscure. Now, we a little strange as theoretical physicists that we, think that the Hwang-Kim-Primakoff methods among ourselves, still dispute over how to do it. should always be used when we want to test the On tests of the various symmetries, the sec- “rare” symmetries in particle or nuclear physics - ond prospect would be, in our opinion, further since the microscopic models are used with care. tests of the absence of the second-class currents For example, we don’t know why the Breit frame or more generally the right-handed currents, etc. has to be used - in fact, there are some differences These currents, e.g., the right-handed currents, in the other frames. the second-class currents, etc., define the bound- To summarize, the Hwang-Kim-Primakoff ary between our world and the other unknown method, or Elementary-Particle Treatment worlds - thus, understanding the boundary be- (EPT), will be the way to go in the 21st Century tween our world (i.e., the quantum 4-dimensional - a method that respects Einstein’s relativity Minkowski space-time) and others through the principle and the quantum principe. As for unusual symmetries should be one of the goals. the microscopic thinking such as NOIA/GI, Moreover, we would eventually have to come the further developments, consistent with the back to more refined tests of CVC, PCAC, CP, symmetries such as CVC, PCAC, and GI, are CPT, etc. This is the third prospect, the avenue needed. to study and to know everything. There is one story which I would like to tell. As a practitioner of the elementary-particle Roughly in 1987 or 1988 when I was at Carnegie- treatment, we try to put everything in the Mellon University, Brad Keister told me, on one Lorentz invariant form. But the microscopic day, that some of the form factors (which we in- models are often non-relativistic, or even rela- troduced in the paper) are not independent of tivistic but frame-dependent. The tests about 3 the others. He pointed to < P1 | Jλ(0) | D > relativistic invariance are often not easy. The (as above, reproduced for your reference) and in deuteron and the n-p scattering states offer us a the few days he gave me a page of notes prov- number of celebrated examples, so far as the rel- ing that. Later, some Russian paper (for which ativistic covariance is concerned. When we use the preprint was sent to my attention) also iter- the non-relativistic wave functions to make the ated this point. Basically, the constructions of connection, the “Breit-frame” is often used as ϵαβγδ would go back those without ϵ’s. In our the middle point for making connections (such case, using EPT-NOIA/GI connections, certain as the EPT-NOIA/GI connection) - but is it form factors are zero identically, in the case of re- necessary? NOIA represents the whole body of dundancy - but early on I didn’t know why but non-relativistic treatments, including those two- didn’t think about it further. But the hidden body potentials and meson-exchange currents. It

25 Regular Article The Universe, Vol. 5, No. 1 January-March 2017

would be very nice to further make these EPT- SUf (3), built-in at the very beginning - and NOIA/GI connections. we’re lucky to be able to see the lepton world, of There are a certain class of tests or prospects, atomic sizes, and also to see the “triple” quark such as radiative versus non-radiative, CVC, world, of nuclear sizes. (The triple quark world PCAC, no SCC, etc., which may be phrased in means that the entries would be written as mem- the model-independent fashion. However, there bers of the quark group SUQ(3) - used to be are a different class of problems, which would called “flavor SU(3) symmetry”). be inherently model-dependent, such as certain So, the Lorentz covariant formalism, such as tests of relativistic covariance. The two foun- the elementary-particle treatment (EPT), should dation pillars, i.e., Einstein’s relativity princi- serve as a major part of the language for (nuclear ple and the quantum principle, are in principle or atomic) physics in the quantum 4-dimensional frame-independent, but it seems to be that the Minkowski space-time (that is our World). We language of describing the various phenomena are confident that, in the 21st Century, the cannot be free of the frames. framework that respect Einstein’s relativity principle and the quantum principle, such as the Elementary-Particle Treatment (EPT), will stay 6 Closing Remarks there and will be used everywhere in practise. To close this article, we would like to ask ourselves - if the elementary-particle treatment or Appendix: A Mysterious Error the Hwang-Kim-Primakoff method is so simple and so versatile, why so few “nuclear physicists” In my experience as a medium-energy theorist, so far have ever tried to use the methods? Over the most vivid memory of making an error was the last forty years, I started to use the methods associated with the Annal Physics paper on the but eventually realized that I might be probably deuteron photodisintegration. the only one and thus I stopped at some point. I was known to be a physics theorist who (My efforts didn’t pay off as a young theorist.) published a lot of appendices containing many The real answer is that the real attempts are of- formulas, such as the Annal-Physics article ten quite complicated and most theorists would coauthored with E.M. Henley and G.A. Miller get scared away - with our Universe being the [18]. The EPT-NOIA (Elementary-Particle- quantum 4-dimensional Minkowski space-time, Treatment versus Nucleon-Only-Impulse- the first step is the discovery of Dirac’s lineariza- Approximation) that connects two language tion of Einstein’s basic relation, E2 = ⃗p 2 + m2, frameworks would be an example. As years went but the next steps have to go through these com- by, I would discover a couple of typo errors but plicated algebras. We in fact cannot hide our- very few cases would give numerical differences selves since we’re living in this strange space- that would be significant. time. In my 1983 Annal-Physics article with J.T. From the Kim-Primakoff attempt in trying to Londergan and G.E. Walker [19], I realized that understand the Lee-Mo-Wu experiment to the my forward-angle cross section was slightly lower formalism of Hwang-Kim-Primakoff and from than other people’s results. I tried to search the the low level of appreciation of the works of possible error between the formulas and the com- Hwang and Henley, the role of relativity is still puter code (of more than one thousand Fortran under-developed in the early 21st Century. We lines); for over half a year, I could not find an er- believe that the situation would gradually be ror and so decided to publish as such. (The pub- changed, for that we are in fact governed by the lication pressure for a young colleague started to two foundation pillars, i.e., Einstein’s relativity take a toll.) After a couple of years when I de- principle and the quantum principle. cided to move back to Taipei, I suddenly discov- Thus, after thinking of it over and over, ered one critical sign mistake in my complicated this world should be the quantum 4-dimensional formulas. I could have submitted an erratum Minkowski space-time with, through the gauge during my last few days at Indiana - I decided principle, the force-fields gauge-group structure, to check on it more. SUc(3) × SUL(2) × U(1) or SUL(2) × U(1) × However, in Taipei, it was very strange that

26 The Universe, Vol. 5, No. 1 January-March 2017 Regular Article

the identical computer code which I ran for the [2] W-Y. Pauchy Hwang, arXiv:1304.4705v2 problem could not reproduce the identical result [hep-ph] 25 August 2015; “The Standard (at Indiana, and later at Los Alamos). I strug- Model”. gled for a few years (in Taipei) and then gave [3] Samuel Ting, CERN Courier 56-10, 26-30 up. I had to conclude, in early 1990’s, that there (2016); and references therein. must be slight differences in the detailed ma- [4] W-Y. Pauchy Hwang, The Universe, 4-4, 7 chine languages between the one at Indiana or (2016); arXiv:1409.6077v2 [hep.ph] 24 Jun Los Alamos and the one in Taiwan - explaining 2016. why the identical computer code gives slightly [5] W-Y. Pauchy Hwang, The Universe, 3-1, different numerical results. (In fact, I never got 3 (2015); “The Origin of Fields (point-like my suspicion verified one way or the other. The Particles)”. [6] W-Y. Pauchy Hwang, The Universe, 2-2, 34 difference is about one percent or so, not very (2014); “The Origin of Mass”. big but fairly visible.) [7] W-Y. Pauchy Hwang and Ta-You Wu, “Rel- For many years, I kept asking why my late ativistic Quantum Mechanics and Quantum Professor Henry Primakoff was so patient [11], Fields”, the 21st-Century Edition (World in reading our first joint paper, asking me to do Scientific, Singapore, 2017). major revisions four times and minor revisions [8] Y.K. Lee, L. Mo, and C.S. Wu, Phys. Rev. for almost a hundred times. The human being is Lett. 10, 253 (1963). prong to make some errors (mistakes); that is, a [9] C.W. Kim and H. Primakoff, Phys. Rev. human being is originally made for, keeping try- B1447, 139 (1965); B566, 140 (1965). ing and making decisions, without guaranteeing [10] E. D. Commins and P.H. Bucksbaum, success. The nature of a human being is different “Weak Interactions of Leptons and Quarks” from that of a machine. In fact, I should have (Cambridge University Press, 1983), Ch. 4 understood all these long before. and Ch.5. Perhaps, we, human beings, are lucky to be [11] W-Y. P. Hwang and H. Primakoff, Phys. able to make mistakes occasionally and, later on, Rev. C16, 397 (1977), Beta Decay and to correct the mistakes - the characteristic for the Muon Capture in the A=12 Nuclei: Second- human beings but not for the machines. Class Current and Conserved Vector Cur- rent; W-Y. P. Hwang, Phys. Rev. C20, 814 (1979), Test of Fundamental Symmetries in Acknowledgements the A=12 Nuclei. [12] W-Y. P. Hwang, Phys. Rev. C17, 1799 I would like to acknowledge so many old friends (1978), Nuclear Muon Capture: Hyper- and sadly many of them had already passed fine Effects in Nucear Spin and Isospin away, including Henry Primakoff, Val Telegdi, [1/2±, 1/2] → [1/2±, 1/2] and [1+, 0] → C.S. Wu, Peter Rosen, and Ernest M. Hen- [0+, 1] Transitions. ley. The live friends include Chung W Kim, [13] W-Y. P. Hwang, C20, 805 (1979); (E) C20, Bernard Goulard, Stephen L. Mintz, John Ng, 2457 (1979), Average Polarization of the Re- and other “EPT” friends. In fact, Steve Mintz coil 12B from Muon Capture by 12C. was working with the break-up channel, with [14] W-Y. P. Hwang, Nucl. Phys. A329, 463 some useful results. Also, Bernard Goulard used (1979), Excitation of the 15.110 MeV State the EPT to treat the solar-neutrino reaction of 12C by Neutrinos and Antineutrinos. 15 15 + O → N + e + νe and related reactions. To [15] W-Y. P. Hwang, Phys. Rev. C20, 331 my knowledge, these people are a few royalists (1979), Parity Mixing in 18F and Low- of the “Elementary-Particle Treatment”. Energy Inelastic Electron Scattering. [16] W-Y. P. Hwang and H. Primakoff, Phys. Rev. C18, 414 (1978), Theory of Radiative References Muon Capture with Application to Nuclear Spin and Isospin Doublets. [1] W-Y. Pauchy Hwang, The Universe, 3-2, [17] W-Y. P. Hwang and H. Primakoff, Phys. 11 (2015); “Why is Our Space-Time 4- Rev. C18, 445 (1978), Theory of Radiative Dimensional?” Muon Capture by 12C.

27 Regular Article The Universe, Vol. 5, No. 1 January-March 2017

[18] W-Y. P. Hwang, E.M. Henley, and Ger- [19] W-Y. P. Hwang, J.T. Londergan, and G.E. ald A. Miller, Annals of Physics (New Walker, Ann. Phys. (N.Y.) 149, 335 (1983); York) 137, 378 (1981), Parity Violation in Deuteron Photodisintegration I: Theory Electron-Deuteron Scattering. II. Break-up and Results at Low Energies. Channels.