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Published on January 7, 2008

Author: Dixon


Slide1:  Breaking the Dirac Code Peter Rowlands Slide2:  The Dirac Code Slide3:  The Dirac Code The Dirac equation is the most fundamental in physics, the only equation which applies universally to the most fundamental particles or fermions. Somehow or other it must contain most of the information which makes up particle physics. But the Dirac equation is a cryptic one – ‘a riddle, wrapped in a mystery, inside an enigma’ – with a seemingly bizarre mathematical structure. Can we crack the code and find the physics that the structure of the equation obscures? Slide4:  The Dirac Code Q What does ‘cracking the code’ mean? Q How will we know when we have done it? A It means finding an expression of relativistic quantum mechanics which is transparent – which actually provides purely physical information. A We will know when we have done it when the mathematics is no longer arbitrary but an intrinsic part of the physical structure. Slide5:  The gamma matrices (gmm + im) y = 0 It is not difficult to see where the problems begin. The equation looks concise but a large part of it (98.4 %) is redundant. It is also scrambled and less symmmetric than it at first appears. (gmm + im) is a 4 × 4 matrix is a 4-component spinor y and each of the g terms is a 4 × 4 matrix. It is certainly the matrices that are the initial problem. Slide6:  The gamma matrices But what do these g terms actually do? There are five of them, but only four appear in the equation. Dirac introduced them to make quantum mechanics linear in space and time and needed algebraic square roots to do this. He needed a system of five operators in which (g0)2 = (g5)2 = 1 (g1)2 = (g2)2 = (g3)2 = – 1 and all terms anticommute with each other: g0 g1 = – g1 g0 , etc. He used matrices, but we needn’t do so. Quaternions Complex Quaternions:  Quaternions Complex Quaternions i2 = j2 = k2 = – 1 ij = – ji = k jk = – kj = i ki = – ik = j (ii)2 = (ij)2 = (ik)2 = 1 (ii)(ij) = – (ij)(ii) = i(ik) (ij)(ik) = – (ik)(ij) = i(ii) (ik)(ii) = – (ii)(ik) = i(ij) Anticommuting square roots of –1 and 1 have been around for a century and a half Pauli matrices Multivariate vectors:  Pauli matrices Multivariate vectors sx2 = sy2 = sz2 = I sxsy = – sysx = isz sysz = – szsy = isx szsx= – sxsz = isy i2 = j2 = k2 = 1 ij = – ji = ik jk = – kj = ii ki = – ik = ij There are two algebras isomorphic to complex quaternions Slide9:  Multivariate vectors and Clifford algebra Multivariate vectors are ones with a full product, combining the scalar and vector products: ab = a.b + i a × b So aa = a2 ii = 1 ij = ik etc. An algebra combining anticommuting square roots of 1 and –1 has also been around for more than a century. It is called Clifford algebra or geometrical algebra and is symbolized by Cl(m, n) of G(m, n). Slide10:  Pauli matrices create intrinsic problems There are only 2 degrees of freedom, defined by the complex plane. One of the three matrices is defined only as the product of the other 2, and so has complex coefficients where the others have real ones. The representation thus becomes asymmetric. Another problem is the skew-symmetry between sx and sz. Slide11:  The g matrices inherit these problems Slide12:  The g matrices inherit these problems Slide13:  as does the Dirac equation: Slide14:  and the free-particle amplitudes for y: Slide15:  The space-time signature becomes distorted The solutions have different phases, the first two being positive energy and the second two positive energy, and E and m are inextricably mixed. Also, we have terms like px + ipy and px – ipy, with no counterpart in nature. In the process we change the momentum-energy or space-time signature from the true + + + – to the distorted + + – –, that is we have two spacelike and two timelike components. Slide16:  Problems with baryonic structure Among other things, this makes it impossible to produce a satisfactory mathematical model for baryonic structure because the SU(3) strong interaction derives from the 3-D symmetry of the momentum operator which this distorted model denies. Slide17:  Problems with the g matrices They cause fragmentation of the equation, mixing up energy, momentum and mass terms. They take up too much logical space, requiring 16 pieces of information for one operation. They lack symmetry. There are 5 terms in the equation, but only 4 have a g matrix. Yet there is a fifth matrix (g5) in the algebra. The first thing to do is to defragment – that is, to separate energy, momentum and mass terms from each other. Slide18:  Defragmenting Dirac Slide19:  Defragmenting Dirac We need separate ‘bins’ for energy, momentum and mass. Slide20:  Why do we defragment? Because we have created a problem that need not exist. There is no necessity to interpret the g operators as matrices at all. We can do the same job using Clifford algebra. This will also solve the logical space problem by reducing the 16 operators of the 4 × 4 matrix to one. The problem is which? There are many Clifford algebras and many ways of applying it. Slide21:  Which Clifford algebra? The only point in applying a new algebra is to crack the code, and unlock the physical information in the equation. But it seems that we need the exact algebra to do this – experience shows that only one will work. Elegant reformulations of Dirac using quaternions and Clifford algebra already exist – but they don’t solve the fundamental problem. However, the correct algebra alone will not be enough. There will be no completely deductive mathematical path to unravelling the code, just as there wasn’t for Dirac in producing his equation. Quaternions Multivariate Vectors:  Quaternions Multivariate Vectors i j k quaternion units i j k vector units 1 scalar i pseudoscalar This is intriguingly close to twistor algebra (a complex 4-D space-time), now used in QCD. We have 4 real parts and 4 imaginary. Let’s try a combination of quaternions and vectors Slide23:  (±1, ± i) 4 units (±1, ± i) × (i, j, k) 12 units (±1, ± i) × (i, j, k) 12 units (±1, ± i) × (i, j, k ) × (i, j, k) 36 units This is the same as with the g matrices. We will soon effectively eliminate the vectors. There are 64 possible products of the 8 units Slide24:  There are 64 possible products of the 8 units Assuming that ± signs are determined by the order of multiplication, the 32 remaining units can be generated binomially from five composite ones, like the g matrices. There are numerous ways of doing this, but all basically require distributing the units of one 3-D structure among the rest, e.g., i i j k 1 i j k ik ii ji ki 1j Slide25:  go = ik g1 = ii g2 = ji g3 = ki g5 = ij (B) go = -ii g1 = ik g2 = jk g3 = kk g5 = ij (A) The g operators Such pentads can be easily mapped onto the g matrices, e.g. Slide26:  Choosing mapping (A), we obtain: The algebraic Dirac equation A key move now is to multiply from the left by j. This apparently trivial operation has profound consequences. Slide27:  This becomes: This switches the mapping to (B). The algebraic Dirac equation  Slide28:  The key result is that the equation is now fully symmetrical: And the quaternion operators provide the 3 ‘bins’ we require: k i j energy momentum mass The algebraic Dirac equation Slide29:  The real significance of the extra symmetry becomes apparent when we try inserting a plane wave solution for y: y = A e –i(Et – p.r) Then (kE + iiipx + iijpy + iikpx + ijm) A e –i(Et – p.r) = 0 (kE + iip + ijm) A e –i(Et – p.r) = 0 where p is multivariate. The algebraic Dirac equation Slide30:  (kE + iip + ijm) A e –i(Et – p.r) = 0 is only valid if A is a multiple of (kE + iip + ijm). A is a nilpotent or square root of zero. This must also be true of y, and must have been true even before we made the substitution of algebraic operators for matrices. The algebraic Dirac equation The Dirac 4-spinor:  The Dirac 4-spinor Of course, from the conventional Dirac equation we know that there are really four solutions within the 4-component spinor y. But with the new symmetrical form of the equation, and with the knowledge (from Hestenes) that a multivariate p or Ñ already incorporates fermionic spin, these are easy to identify fermion / antifermion  E spin up / down  p These options apply to both amplitude and phase. The Dirac 4-spinor:  The Dirac 4-spinor We note that, for a multivariate p, pp = (s.p) (s.p) = pp = p2 So we can also use s.p for p (or s.Ñ for Ñ) in the Dirac equation, where s.p is helicity, and s is a pseudovector of magnitude –1. We will only need to explicitly incorporate spin where the vectors are not multivariate (e.g. using polar coordinates). The Dirac 4-spinor:  The Dirac 4-spinor y1 = (kE + iip + ijm) e–i(Et – p.r) y2 = (kE - iip + ijm) e–i(Et + p.r) y3 = (-kE + iip + ijm) ei(Et – p.r) y4 = (-kE - iip + ijm) ei(Et + p.r) So we can represent the Dirac 4-spinor by a column vector with these 4 components: each of which is operated on by The Dirac 4-spinor:  The Dirac 4-spinor However, by removing the 4 × 4 matrices, we have effectively reduced the differential operator to a single term. This gives us the logical space to add an extra twist of symmetry to the equation, by making the operator a 4-component spinor, identical in representation to the amplitude. The Dirac 4-spinor:  The Dirac 4-spinor To do this, we transfer the variation in the signs of E and p from the phase term to the differential operator.       The Dirac 4-spinor:  The Dirac 4-spinor We see here that the Feynman representation of negative energy states being associated with negative time is automatically applied. Also, with only a single phase, the expression (± kE ± iip + ijm) or, more conveniently, (± ikE ± ip + jm) can be taken to refer either to the differential term in operator form, or to the amplitude produced by applying this to the phase.       The Dirac 4-spinor:  The Dirac 4-spinor In any case, the term (± ikE ± ip + jm) remains a nilpotent, and the expression (± ikE ± ip + jm) (± ikE ± ip + jm) = 0 (in which the first term is conveniently a row vector and the second a column vector) becomes a perfect expression of the Pauli exclusion principle.       Slide38:  Pauli exclusion, however, is not unique to free fermions, and this brings us to a step which, in retrospect, might appear revolutionary. We assume that all fermionic amplitudes in all states are nilpotent. We postulate that the most general form for a wavefunction is nilpotent, and that we should seek specifically nilpotent solutions for all problems. The algebraic Dirac equation Slide39:  We justify this on the grounds that: All fermionic states are Pauli exclusive. Calculations such as that for the hydrogen atom effectively assume this to be the case. It works. What it means is that the (± ikE ± ip + jm) operator must always produce nilpotent solutions even when E and p are covariant derivatives or contain field terms. The algebraic Dirac equation Slide40:  The Dirac nilpotent state vector The principle is revolutionary because it means we no longer need an equation of any kind. Relativistic quantum mechanics is completely specified by an operator of the form (± ikE ± ip + jm) which completely determines the phase that will provide a nilpotent amplitude. Finding the solution generally means finding the phase. Slide41:  The Dirac nilpotent state vector In fact, we only need the first term of the operator, say (ikE + ip + jm), as the other three terms are produced by automatic sign changes, and don’t represent independent information; and we will now use the convention that this really represents (± ikE ± ip + jm). It is difficult to see how fundamental physical information could be more compactified. (1/16 of equation) Slide42:  Using discrete calculus D F = y = ikE + iiP1 + ijP2 + ikP3 + jm Dy = 0 Significantly, D has no m term. Slide43:  The Dirac nilpotent state vector We can see that each complete fermion operator incorporates four individual creation (or annihilation) operators (ikE + ip + jm) fermion spin up (ikE – ip + jm) fermion spin down (–ikE + ip + jm) antifermion spin down (–ikE – ip + jm) antifermion spin up The nature of the state is determined by which of these is the lead term. The others can be regarded as vacuum states representing ones into which it could transform. Slide44:  The Dirac nilpotent state vector Because of the way they are defined, nilpotent operators are specified with respect to the entire quantum field. Formal second quantization is unnecessary. We can consider the nilpotency as defining the interaction between the localized fermionic state and the unlocalized vacuum, with which it is uniquely self-dual. The phase is the mechanism through which this is accomplished. Slide45:  The Dirac nilpotent state vector Defining a fermion implies simultaneous definition of vacuum as ‘the rest of the universe’ with which it interacts. The nilpotent structure then provides energy-momentum conservation without requiring the system to be closed. The nilpotent structure is thus naturally thermodynamic. Slide46:  Multiple meanings We can now see that the expression (ikE + ip + jm) (ikE + ip + jm)  0 has at least five independent meanings. classical special relativity operator  operator Klein-Gordon equation operator  wavefunction Dirac equation wavefunction  wavefunction Pauli exclusion fermion  vacuum thermodynamics Slide47:  Have we cracked the code? We now have an operator (ikE + ip + jm) that potentially incorporates all the physical information available to the fundamental physical state. If it is no longer ‘coded’, then this physical information should be transparent. Slide48:  Have we cracked the code? It is easy to show that we can use our operator to do conventional quantum mechanics, e.g. by defining a probability density by multiplying by its complex quaternion conjugate (ikE – ip – jm). We can also derive spin ½ in the usual way, and the one-handed helicity of massless fermionic states. And we can proceed to do QED, QFD, QCD, renormalization, etc. Slide49:  Particle states and interactions If the fermionic nilpotent is the most fundamental structure in physics – in effect, its fundamental unit, can it reproduce the fundamental particle states and their interactions? These two questions are not independent of each other. The first thing is to see if the structure of the nilpotent operator can give us any insight into the nature of fermionic interactions. Slide50:  Spherical symmetry: the point source In fact, this is precisely what it can do. But, first, assuming that the constraint of spherical symmetry exists for a point particle, then we can express the momentum term of the operator in polar coordinates, using the Dirac prescription, with an explicit spin term: We need the spin term because the multivariate nature of the p term cannot be expressed in polar coordinates. Slide51:  Spherical symmetry: the point source The nilpotent Dirac operator now becomes: Now, whatever phase we apply this to, we will find that we will not get a nilpotent solution unless the 1 / r term with coefficient i is matched by a similar 1 / r term with coefficient k. So, simply requiring spherical symmetry for a point particle, requires a term of the form A / r to be added to E. Slide52:  Spherical symmetry: the point source If all point particles are spherically symmetric sources, then the minimum nilpotent operator is of the form To establish that this is a nilpotent, we must now find the phase to which this must apply to create a nilpotent amplitude. This will quite quickly produce the characteristic solution for the Coulomb force (H atom, etc.) Slide53:  Spherical symmetry: the Coulomb potential The solution is straightforward. We apply this to the phase to find the amplitude, and equate the squared amplitude to zero. We have: Slide54:  Spherical symmetry: the Coulomb potential Equating constant terms, we find: Equating terms in 1/r2, with n = 0: Slide55:  Spherical symmetry: the Coulomb potential Assuming the power series terminates at n', and equating coefficients of 1/r for n = n': and When A = Ze2 we have the ‘hydrogen atom’ solution. Slide56:  Spherical symmetry: other solutions This solution does not require the usual two sets of interacting solutions for positive and negative, as the whole thing arrives as a package in the nilpotent formalism. It results purely from spherical symmetry. Only two other nilpotent solutions are possible assuming spherical symmetry: with additional potential ( r), we obtain the characteristic strong interaction infrared slavery / asymptotic freedom with any other additional potential (e.g. dipole / multipole), we obtain a harmonic oscillator (weak signature?) Slide57:  Spherical symmetry: the point source We have, of course, without mentioning anything about potentials or interactions, or anything physical at all, and only using the structure of the nilpotent operator, needed to maintain the spherical symmetry of a point-particle source, created the solution for the Coulomb or inverse linear potential. And we have shown that it is absolutely necessary to any fermionic state described as a point source, regardless of what other potentials may be present. We can now proceed to show that another fundamental potential can be derived from the structure of the nilpotent operator alone. Slide58:  What would a baryon state vector look like? A significant test of the validity of the nilpotent structure is the case of the baryon – a set of three interacting states which has no satisfactory representation within the conventional formalism. Conventionally, we consider a baryon to be made up of three fermionic components, to which we assign colour to overcome Pauli exclusion. Can we relate this concept of colour to the fundamental structure of nilpotents? Slide59:  What would a baryon state vector look like? Can we have a 3-component state vector? (ikE + ip + jm) (ikE + ip + jm) (ikE + ip + jm) = 0 Slide60:  What would a baryon state vector look like? Can we have a 3-component state vector? (ikE + ip + jm) (ikE + ip + jm) (ikE + ip + jm) = 0 But (ikE + ip + jm) (ikE + jm) (ikE + jm)  (ikE + ip + jm) (ikE + jm) (ikE + ip + jm) (ikE + jm)  (ikE – ip + jm) (ikE + jm) (ikE + jm) (ikE + ip + jm)  (ikE + ip + jm) So it is possible to have a nonzero state vector if we use the vector properties of p and the arbitrary nature of its sign (+ or –). Slide61:  What would a baryon state vector look like? A state vector of the form, privileging the p components: (ikE ± iipx + jm) (ikE ± ijpy + jm) (ikE ± ikpz + jm) has six independent allowed phases, i.e. when p = ± ipx , p = ± jpy , p = ± kpz But these must be gauge invariant, i.e. indistinguishable, or all present at once. Also, we must interpret the E, p, m symbols as belonging to a totally entangled state, rather than the subcomponents. Slide62:  The strong interaction in a baryon In principle, we can see the ‘quark’ structure as using the concept of spatial (rather than temporal) separation to represent the arbitrary nature of the direction of fermionic spin. One method of picturing the arbitrary nature of the phases (gauge invariance) is to imagine an automatic mechanism of transfer between them. Slide63:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE –i kpz + j m) (ikE + i ipx + j m) (ikE + i jpy + ij m) (ikE + i kpz + ij m) (ikE – i ipx + j m) (ikE – i jpy + ij m) (ikE – i kpz + ij m) (ikE + i ipx + j m) (ikE + i jpy + ij m) (ikE + i kpz + ij m) (ikE – i ipx + j m) (ikE – i jpy + ij m) (ikE – i kpz + ij m) Slide64:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) Slide65:  The strong interaction in a baryon (ikE – i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) Slide66:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) Slide67:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE – i jpy + j m) (ikE + i kpz + j m) Slide68:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) Slide69:  The strong interaction in a baryon (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) Slide70:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) Slide71:  The strong interaction in a baryon (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) +RGB (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) –RBG (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) +BRG (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) –GRB (ikE + i ipx + j m) (ikE + i jpy + j m) (ikE + i kpz + j m) +GBR (ikE – i ipx + j m) (ikE – i jpy + j m) (ikE – i kpz + j m) –BGR Slide72:  The strong interaction in a baryon This has exactly the same group structure as the standard ‘coloured’ baryon wavefunction made of R, G and B ‘quarks’,  ~ (RGB – RBG + BRG – GRB + GBR – BGR) That is, it has an SU(3) structure, with 8 generators. And, since the E and p terms in the state vector really represent time and space derivatives, we can replace these with the covariant derivatives needed for invariance under a local SU(3) gauge transformation. Slide73:  The strong interaction in a baryon A significant aspect of this SU(3) symmetry or strong interaction is that it is entirely nonlocal. That is, the exchange of momentum p involved is entirely independent of any spatial position of the 3 components of the baryon. We can suppose that the rate of change of momentum (or ‘force’) is constant with respect to spatial positioning or separation. A constant force is equivalent to a potential which is linear with distance, exactly as is required for the conventional strong interaction. Slide74:  The strong interaction in a baryon Very significantly, the full symmetry between the 3 momentum components can only apply if the momentum operators can be equally + or –. (P transformation; operator i) With all phases of the interaction present at the same time (perfect gauge invariance), this is equivalent to saying that left-handedness and right-handedness must be present simultaneously in the baryon state. In other words, the baryonic state must have non-zero mass via the Higgs mechanism (Tait Foundation problem). Slide75:  The strong interaction in a baryon It is significant that the baryon representation can only exist as a unified or entangled state. It is not really a representation of a combination of 3 independent fermions. It is equally significant that the representation is impossible in a conventional spinor formulation, with terms such as px + ipy, or in any representation in which the momentum operators cannot show the full affine nature of the vector concept. Slide76:  Have we cracked the code? The important thing is to see what physical information is identifiable within the structure we have created. The mathematics only becomes significant when it is associated with a physical application. We might expect, for instance, that the k, i, j operators are active physical elements (‘hypertext’) rather than passive mathematical objects. Slide77:  Multiple meanings for i, j, k The three quaternion operators i, j, k can be seen to have multiple meanings in the nilpotent formalism – as charge generators; as P, C, T transformation operators; and as vacuum generators. (1) The primary meaning is as charge generators. (2) Premultiplying the nilpotent gives vacuum, e.g. k(ikE + ip + jm) weak vacuum (3) Pre- and postmultiplying the nilpotent transforms via P, C or T, e.g.: k(ikE + ip + jm)k T transformation Slide78:  Vacuum Take (ikE + ip + jm) and postmultiply it by k(ikE + ip + jm). The result is (ikE + ip + jm), multiplied by a scalar, which can be normalized away. This can be done an indefinite number of times. k(ikE + ip + jm) behaves as a vacuum operator. So do i(ikE + ip + jm) and j(ikE + ip + jm). Slide79:  Vacuum operators We can see the three vacuum coefficients k, i, j as originating in (or being responsible for) the concept of discrete (point-like) charge. The operators act as a discrete partitioning of the continuous vacuum responsible for zero-point energy. In this sense, they are related to weak, strong and electric localized charges, though they are delocalized. Slide80:  Vacuum operators We can suggest specific identifications on the basis of the pseudoscalar, vector and scalar characteristics of the associated terms. k (ikE + ip + jm) weak vacuum fermion creation i (ikE + ip + jm) strong vacuum gluon plasma j (ikE + ip + jm) electric vacuum SU(2) The 3 additional terms in the Dirac spinor then become strong, weak and electric vacuum ‘reflections’ of the state defined by the lead term. Slide81:  CPT symmetry uses the same operators This is, of course, not a coincidence. P i (ikE + ip + jm) i = (ikE – ip + jm) T k (ikE + ip + jm) k = (–ikE + ip + jm) C -j (ikE + ip + jm) j = (–ikE – ip + jm) CPT - j (i (k (ikE + ip + jm) k) i) j = (ikE + ip + jm) CPT connects relativity with causality, only in nilpotent. Slide82:  Fermions and antifermion vertices Because the state vector always represents four terms with the complete variation of signs in E and p, an interaction vertex between any fermion / antifermion and any other (ikE1 + ip1 + jm1) (ikE2 + ip2 + jm2) will remove the quaternionic operators, leaving only scalars and vectors. When the E, p and m values become numerically equal, the vertex can be defined as a new combined bosonic state, with a single phase. Slide83:  Antisymmetric wavefunctions We may note that nilpotent wavefunctions or amplitudes are automatically antisymmetric: (± ikE1 ± ip1 + jm1) (± ikE2 ± ip2 + jm2) – (± ikE2 ± ip2 + jm2) (± ikE1 ± ip1 + jm1) = 4p1p2 – 4p2p1 = 8 i p1  p2 Slide84:  Interaction vertices Where there is an interaction vertex between two fermionic / antifermionic states, the signs of E and p of the second term, with respect to the first, will also determine the nature of the bosonic or combined state which may be created. Because there are three operators involved – i, j, k – there are also three possible bosonic states. Any transformation of a fermionic state can be represented as a bosonic state in which the old state is annihilated and the new one created. Slide85:  Bosonic states Spin 1 boson: (ikE + ip + jm) (– ikE + ip + jm) T Spin 0 boson: (ikE + ip + jm) (– ikE – ip + jm) C Bose-Einstein condensate / geometric phase, etc.: (ikE + ip + jm) (ikE – ip + jm) P Slide86:  Nonzero geometric phase The fermion-fermion state (ikE + ip + jm) (ikE – ip + jm) has many physical manifestations: Aharonov-Bohm effect Jahn-Teller effect quantum Hall effect Cooper pairs even-even nuclei Even spin 1 He3 can be accommodated (via s.p). Slide87:  Bosonic states Significantly, the spin 0 bosonic state cannot be massless, because, if it is nilpotent it automatically becomes zero. (ikE + ip) (– ikE – ip) = 0 This becomes a significant factor in the Higgs mechanism. It also implies that massless fermions cannot have the same handedness as massless antifermions. The conventional derivation of spin assigns left-handedness to fermions. Slide88:  The mediators of the strong force (gluons) The mediators of the strong force will be six bosons of the form: (ikE – iipx) (– ikE – ijpy) and two combinations of the three bosons of the form: (ikE – iipx) (– ikE – iipx) These structures are, of course, identical to an equivalent set in which both brackets undergo a complete sign reversal. The important thing here is that applying any of these mediators will produce a sign change in the p component that leads to mass. Slide89:  Bosonic states We can see how the 3 bosonic states are related to vacua produced by the 3 charge operators: weak spin 1 (ikE + ip + jm) k (ikE + ip + jm) k (ikE + ip + jm) k (ikE + ip + jm) … (ikE + ip + jm) (–ikE + ip + jm) (ikE + ip + jm) (–ikE + ip + jm) … electric spin 0 (ikE + ip + jm) j (ikE + ip + jm) j (ikE + ip + jm) j (ikE + ip + jm) … (ikE + ip + jm) (– ikE – ip + jm) (ikE + ip + jm) (– ikE – ip + jm) … strong B-E condensate (ikE + ip + jm) i (ikE + ip + jm) i (ikE + ip + jm) i (ikE + ip + jm) … (ikE + ip + jm) (ikE – ip + jm) (ikE + ip + jm) (ikE – ip + jm) … Slide90:  Bosonic states All these discrete vacuum states produce virtual boson states which have no effect on the fermion (ikE + ip + jm). So, each fermion becomes its own supersymmetric bosonic partner, and vice versa. This removes the need for renormalization in the case of free particles, while ‘renormalization’ of interacting particles becomes rescaling – charge values being determined by their interactions with all the others in the universe. Slide91:  We can show this nilpotent property relatively simply. The perturbation expansion for a first-order coupling of a virtual photon to an electron in a 4-potential (f, A) produces a wavefunction of the form: Renormalization Y1 = –eS [kE + ii s.(p + k) + ijm]–1(ikf – i s.A)(kE + iis.p + ijm)e–i(Et – (p + k).r) Observing in the rest frame of the electron and eliminating any external source of potential (by assuming only self-potential), then the external momentum k = 0 and i s.A = 0. So Y1 = –e S [kE + ii s.p + ijm]–1 ikf (kE + iis.p + ijm) e–i(Et – p .r) = 0 Slide92:  Renormalization of the charge and mass of free fermions in the conventional formalism is a result of creating a ‘redundancy barrier’ through an artificial singularity. Another example of this is the infrared divergence in the definition of the propagator. In the Feynman formalism, the fermion propagator has the form: Propagators with a singularity or ‘pole’ (p0) where p2 – m2 = 0. Slide93:  Propagators This singularity goes all the way back to having to separate positive and negative energy solutions in the conventional Dirac formalism. - Here, on either side of the singularity, we have positive energy states moving forwards in time, and negative energy states moving backwards in time, the terms. Using a contour interval method we arrive at: Slide94:  Propagators In the nilpotent formalism, the fermion propagator becomes: with no singularity or pole, because - is finite at all values. There is no infrared divergence because there is no separation of positive and negative energy states, or forward and backward times. Slide95:  Propagators The integral now becomes: where - are respective row and column vectors. The integral comes as a single package. and the adjoint Slide96:  Propagators 3 boson propagators can be defined by analogy with the fermion propagator. For example, spin 1: - For QED, where the boson is massless, we have: In the specific case of massless bosons, conventional theory states that ‘infared’ divergencies occur when such bosons are emitted from an initial or final stage which is on the mass shell. Such divergencies, however, will not occur where there is no pole. Slide97:  Higgs mechanism Imagine a virtual fermionic state with no mass in vacuum (ikE + ip) An ideal vacuum would maintain exact and absolute C, P and T symmetries. Under C transformation, (ikE + ip) would become (– ikE – ip) with which it would be indistinguishable under normalization. No bosonic state would be required for the transformation. Slide98:  Higgs mechanism If, however, the vacuum state is degenerate in some way under charge conjugation (as supposed in the weak interaction), then (ikE + ip) will be transformable into a state which can be distinguished from it, and the bosonic state (ikE + ip) (–ikE – ip) will necessarily exist. However, this can only be true if the state has nonzero mass and becomes the spin 0 ‘Higgs boson’: (ikE + ip + jm) (–ikE – ip + jm) Slide99:  Geometric phase: a prediction For nonzero geometric phase the spin 0 ‘bosonic’ state (ikE + ip + jm) (ikE – ip + jm) is such as would be required in a pure weak transition from –ikE to + ikE, or its inverse. Because the spin 0 state is necessarily massive, time reversal symmetry (the one applicable to the transition) must be broken in the weak formation or decay of states involving the nonzero geometric phase. Slide100:  Strings? The nilpotent operator (± ikE ± ip + jm) can be regarded as a 10-D object (embedded in Hilbert space): 5 for iE, p, m and 5 for k, i, j; and six of these (all but iE and p) are compactified. ‘Self-duality in phase space determines vacuum selection.’ It is a mass-shell system and incorporates the right groups. Slide101:  Conclusion This presentation gives only a few results that have been obtained by the nilpotent method. Further work shows that it can encompass a large amount of QM, QFT and the Standard Model, in a coherent structure, with some hints of what might lie beyond. The key to its success lies in the fact that it avoids the conventional distortion of the structure of 4-D space-time through the unique Clifford algebra which unlocks the Dirac code. Slide102:          The End Slide103:  SU(2)L  U(1) The acquisition of mass in the nilpotent formalism can be related to the capacity for change of sign in the p term with respect to that of the E term. In principle, a weak isospin transition can be seen as a change of the form (ikE + ip + jm)  a1 (ikE + ip + jm) + a2 (ikE – ip + jm) isospin up isospin down The down state introduces a degree of right-handedness which is not present in the up state, and which is not weak in origin. Slide104:  SU(2)L  U(1) Where the strong interaction is not involved, a partial p sign transition (involving vacuum operator i) can only be accomplished by involving the electric vacuum operator (j) as well as the weak one (k). The weak interaction preserves left-handedness in fermionic states and right-handedness in antifermionic states. So, in any pure weak transition, the anti-state to the state to be annihilated and the state which is to be created must exist as a spin 1 bosonic combination. Slide105:  SU(2)L  U(1) But fermion states with mass also carry a degree of right-handedness. A transition from left- to right-handedness, involving only fermionic states (not antifermionic), requires a vacuum which we can describe as ‘electric’. Only the electric vacuum carries a transition to right-handedness where the vector character (strong interaction) is absent. Slide106:  SU(2)L  U(1) And, to produce a pure transition from left- to right-handedness (and vice versa) without a change from fermion to antifermion requires an electroweak combination (jk, equivalent to i): (ikE + ip + jm) left-handed fermion (– ikE + ip + jm) weak transition to right-handed antifermion (ikE – ip + jm) electric transition to right-handed fermion Slide107:  SU(2)L  U(1) Using the concept of electric ‘charge’ as indicating the presence of right-handedness, we may identify four possible transitions (taking the ‘left-handed’ / ‘right-handed’ transition to mean ‘the acquisition of a greater degree of right-handedness’), and hence four possible intermediate bosonic states: LH to LH RH to RH LH to RH RH to LH The LH / RH transition clearly has the nature of an SU(2)L symmetry, with the requirement of 3 generators, which are necessarily massive, to carry the RH unrecognised by the interaction, and 2 of which carry electric ‘charge’ (+ and –), in addition to one which leaves the handedness unchanged. Slide108:  SU(2)L  U(1) This leaves the fourth transition state or equivalent as an extra generator with a U(1) symmetry. If we assume that massive generators are necessary for a ‘weak interaction’, and indicate its presence, we can assign the fourth generator to the pure electric interaction. Electric charge, however, is not the sole reason for the massiveness (and hence mixed handedness) of real fermionic states. So the absence of e does not indicate that a weak generator must be massless. So, the 2 generators without e are assumed mixed, the combination producing 2 new generators, one of which becomes massless and so carries the pure electric interaction. Slide109:  Representing interactions There are three fundamental ways of representing strong, weak and electric interactions: (1) Through the nilpotent formalism in terms of E and p, and their sign and component changes. (2) Through the conversion of E and p in the nilpotent formalism into covariant derivatives directly derived from the symmetry groups associated with the transformation mechanisms in (1). (3) Through the potential functions which, when added to E (and p), produce the same effect as in (2). Slide110:  Representation through symmetry groups The Dirac nilpotent has three terms of equal status: A pseudoscalar term (± iE) with a natural dipolarity connected with SU(2) weak interaction. A vector term (± p) related to strong SU(3). A scalar term (m) related to electric U(1). Slide111:  Representation through symmetry groups The three symmetry groups associated with the strong, weak and electric interactions come from the spherical symmetry ( conservation of angular momentum) necessarily implied when we define a point source. U(1) symmetry says that spherical symmetry is preserved whatever the length of the radius vector. SU(3) symmetry says that spherical symmetry is preserved whatever the choice of axes. SU(2) symmetry says that spherical symmetry is preserved irrespective of whether the rotation is left- or right-handed. Slide112:  The strong interaction in a baryon The covariant derivative under an SU(3) local gauge transformation is: or, in component form: In the strong interaction, the p or vector term may be considered as the ‘active’ component, and the E term as the ‘passive’. Slide113:  The strong interaction via covariant derivatives We now apply SU(3) generators to the baryon state vector to obtain: x active y active z active Slide114:  The electroweak interaction via covariant derivatives Applying these covariant derivatives to the nilpotent vertex which describes the weak interaction, we find that we can represent three components as ‘active’ and one as ‘passive’. Deriving covariant derivatives with Wm and Bm as the respective 4-vector generators for SU(2) and U(1), we have, for left-handed states: and, for right-handed: Slide115:  The electroweak interaction via covariant derivatives We note here that the electroweak interaction (or the weak component of it) is defined only in terms of a 2-component vertex, such as (ikE + ip + jm) (– ikE + ip + jm) Essentially, because of the pseudoscalar nature of the energy term, associated with k, i.e. the mathematical indistinguishability of +i and –i, the weak interaction is always defined as minimally dipolar, in the same way as the fermion always defines itself as a dipole with respect to vacuum (leading to half-integral spin). Slide116:  The electroweak interaction via covariant derivatives We now write a vertex for a standard electroweak transition in the form (ikE + ip + jm) (– ikE + ip + jm) = By choice of mass term, we can write this as: Slide117:  Representation by potentials Here, we see the origin of the scalar ‘passive’ components for the interactions, for this operator only produces nilpotent solutions if the potential term V(r) incorporates an inverse linear or Coulomb component (–A / r), equivalent to a U(1) symmetry. In the representation by potentials, using polar coordinates for iis., we write the nilpotent potent state vector under the action of a point source in the form: Slide118:  The Dirac nilpotent and symmetry breaking If we suppose that the strong, electromagnetic and weak interactions are determined by sources with respective vector, scalar and pseudoscalar properties, the ‘passive’ or Coulomb term that each interaction requires appears to be equivalent to the scalar values associated with these. These can be equated to the coupling constants associated with these interactions, and it is these that we may expect to be unified at Grand Unification. The ‘active’ parts of the strong and weak interactions represented by the non-Coulombic potentials can then be seen to result from their vector and pseudoscalar properties. Slide119:  Charge and spherical symmetry The scalar electric term can be expected to be equivalent to a pure magnitude (Coulomb term). The vector strong term requires an additional linear component (–Br). The pseudoscalar weak term requires an additional dipolar component (–Cr–3). Slide120:  Charge and spherical symmetry These three conditions give the only nilpotent solutions for the state vector, and they have the characteristics observed in the three interactions: inverse linear U(1) scalar phase + linear inverse linear SU(3) confinement + other polynomial SU(2) harmonic oscillator Slide121:  Acquisition of mass through the Higgs boson The coupling of a massless fermion, say (ikE1 + ip1), to a Higgs boson, say (ikE + ip + jm) (–ikE – ip + jm), to produce a massive fermion, say (ikE2 + ip2 + jm2), can be imagined as occurring at a vertex between the created fermion (ikE2 + ip2 + jm2) and the antistate (–ikE1 – ip1), to the annihilated massless fermion, with subsequent equalization of energy and momentum states. Slide122:  Acquisition of mass through the Higgs boson If we imagine a vertex involving a fermion superposing (ikE + ip + jm) and (ikE – ip + jm) with an antifermion superposing (–ikE + ip + jm) and (–ikE – ip + jm), then there will be a minimum of two spin 1 combinations and two spin 0 combinations, meaning that the vertex will be massive (with Higgs coupling) and carry a non-weak (i.e. electric) charge. Slide123:  Acquisition of mass through the Higgs boson So, a process such as (ikE + ip + jm)  a1 (ikE + ip + jm) + a2 (ikE – ip + jm) isospin up isospin down requires an additional Higgs boson vertex (spin 0) to accommodate the right-handed part of the isospin down state, when the left-handed part interacts weakly. This is, of course, what we mean when we say that the W and Z bosons have mass. The mass balance is done through separate vertices involving the Higgs boson. Slide124:  Majorana neutrino A Majorana neutrino might be considered as a superposition of state and antistate: a1 (ikE + ip + jm) + a2 (– ikE + ip + jm) It would be a result of the violation of weak charge-conjugation symmetry, and would be naturally CP violating. Slide125:  Berry phase: a prediction The mathematical dipolarity of the pseudoscalar weak charge appears to be the ultimate source of different phases of matter and phase transitions, when the indistinguishability of sign is allowed to effectively eliminate the weak component in fermion-fermion combinations, and so overcome aspects of Pauli exclusion. It is certainly the origin of the Berry phase and related effects (Aharonov-Bohm, Jahn-Teller, quantum Hall, Cooper pairs, etc.). Slide126:  Spherical symmetry: the Coulomb potential The derivation is, of course, perfectly standard, in many respects, but it has also some powerful additional features: It requires the manipulation of only one set of equations compared with the conventional two. This becomes a recurring feature of nilpotent methods, which greatly adds to their power. It derives not from any assumed potential, but purely from the spherical symmetry expected with respect to any point source. It will therefore be a feature of any fermion-fermion interaction at the quantum level, irrespective of type. Slide127:  The Dirac nilpotent in a strong potential We have already derived a 2-component potential purely from the baryon nilpotent structure. The scalar (or spherical symmetry) term A0 seems to require a Coulomb potential of the form A / r (or a potential energy qA / r), while The vector A term seems to require a linear potential of the form sr (or a potential energy qsr). In principle, this should apply to the interaction between quark and antiquark as much as that between three quarks. In the first case it is convenient to use a potential energy of the form –qs r. Slide128:  The Dirac nilpotent in a strong potential The Dirac nilpotent for a quark-antiquark pair now takes the form: Again, we look for a function that will make the eigenvalue nilpotent: Applying this, and expanding: Slide129:  The Dirac nilpotent in a strong potential Equate: coefficients of r2: coefficients of r: coefficients of 1 / r: coefficients of 1 / r2: constant terms: Slide130:  The Dirac nilpotent in a strong potential The function required where n = 0 is: The imaginary exponential terms in F can be interpreted as representing asymptotic freedom, the being typical for a free fermion. The imaginary part of the rg-1 term can be written as a phase, f(r) = exp ( iqA ln (r)) which varies less rapidly with r than the rest of F. Significantly, this is defined by the Coulomb term A. Slide131:  The Dirac nilpotent in a strong potential We can write down the functional term as: where Where r is small (at high energies), the first term dominates, approximating to a free fermion solution (asymptotic freedom). When r is large (at low energies) the second term dominates, bringing in the confining potential (s) (infrared slavery). Slide132:  The Dirac nilpotent in a strong potential Where the Coulomb part of the interaction dominates, we will have an energy-level series of the same form as that for the hydrogen atom except that here we write qA for A: Rather than signifying escape, as with the electron in the hydrogen atom, the condition resulting from E2 > m2 is that of asymptotic freedom, because of the continued presence (but reduced effect) of the confining linear potential. Slide133:  The nilpotent Dirac equation for the 3-quark case has an almost identical form to that for the quark-antiquark combination, though the radial distance this time is with respect to the centre of charge, and there will be variations in the values of the constants A and s. It is possible that there is a constant term in the potential energy expression for both cases (as presumed in lattice gauge QCD), but this will have no effect on the form of the solution, merely changing the effective value of E. The Dirac nilpotent in a strong potential Slide134:  The Dirac nilpotent in a strong potential We can use the full and Coulomb-like solutions to investigate the transition point at which infrared slavery becomes effective. From the full solution, let at zero effective energy (or infrared slavery). Then If, from the Coulomb-like solution, we take the ‘free-particle’ transition energy as the mass of the state m, and assume that this mass is mostly dynamic (gluonic) in origin, then we find qs r = 2E, suggesting a virial relationship, as would be expected with a linear potential. Slide135:  The Dirac nilpotent in a polynomial potential What happens if we try a spherically symmetric potential of the form crn, where n  2 or  – 2? Incorporating the Coulomb term required for spherically symmetry, we obtain, for the differential operator: As before, we need to find the function that will make the eigenvalue nilpotent. Slide136:  The Dirac nilpotent in a polynomial potential Applying the usual procedures, with a termination in the power series, we obtain: Equating constant terms: Equating terms in r2n, with n = 0: Equating coefficients of rn–1, where n = 0: Ac = – (n + 1) b (1 + g) (1 + g) =  iA Slide137:  The Dirac nilpotent in a polynomial potential Equating coefficients of 1 / r2 and coefficients of 1 / r, for a power series terminating in n = n', we obtain A2 = – (1 + g + n')2 + (j + ½) 2 and EA = a (1 + g + n') Combining these various expressions gives us or Slide138:  The Dirac nilpotent in a polynomial potential The formula provides the series of equally-spaced energy levels characteristic of the harmonic oscillator: If we took the minimum condition for A, the phase term required for spherical symmetry, to reproduce the random directionality of the fermion spin, as the half-unit value ( ½ i), then the formulae would coincide exactly. More significant is the fact that the solution allows the coefficient c in crn to be imaginary, making the total potential complex. Slide139:  The Dirac nilpotent in a polynomial potential One of the remarkable things about this result is that it is independent of the value of n in crn, as long as n  2 or  – 2, or how many terms of this kind appear in the polynomial function. In other words, there appear to be only three possible solutions to the Dirac nilpotent equation under the conditions of spherical symmetry which we would expect for point sources: inverse linear U(1) electromagnetism linear + inverse linear SU(3) strong force other polynomial + inverse linear ? harmonic oscillator Slide140:  The Dirac nilpotent and symmetry breaking Since two of the solutions are obviously associated with the electromagnetic and strong interactions, is it possible that the third is responsible for the weak force? The weak force is a creator / annihilator, like the harmonic oscillator. It is also dipolar (fermion / antifermion) or multipolar, and would be associated with an index n  – 2. This solution introduces a possible complexity into the potential energy / charge / vacuum coupling (as with CP violation); and a complexity of this kind would also suggest automatic dipolarity. Slide141:  Multivariate vectors A multivariate vector is one with a full algebraic product. ab = a.b + i a  b The full product of a vector with itself will then be of the form: aa = a.a = a2 Hestenes’ work showed that, using a multivariate p or , even in the Schrödinger equation, leads to the automatic generation of an extra term, representing spin. Slide142:  The creation of the Dirac state The Dirac state is the most efficient packaging of the 4 fundamental parameters: Time Space Mass Charge i i j k 1 i j k pseudoscalar vector (mult.) scalar quaternion Energy Momentum Rest mass ik ii ji ki 1j E p m Slide143:  The broken symmetry between the charges The combination of the algebras not only affects time, space and mass. It also breaks the symmetry between the charges. Weak charge Strong charge Electric charge ik ii ji ki 1j One charge (w) takes on pseudoscalar (timelike) characteristics; another (s) takes on vector (spacelike) properties; the third (e) remains scalar (masslike). Slide144:  The fermionic state The combined Dirac or fermionic state (± ikE ± ip + jm) is a charge state as well as an energy state. It is a nilpotent or square root of zero: (± ikE ± ip + jm) (± ikE ± ip + jm) = E2 + p2 + m2 = 0 Slide145:  The Dirac nilpotent state vector For a ‘free’ fermion, the phase (exp (–i(Et – p.r)) provides the complete range of space and time translations and rotations, but if the E and p terms represent covariant derivatives or incorporate field terms, then the phase term is determined by whatever expression is needed to make the amplitude nilpotent. In other words, we don’t require either the Dirac equation or a specification of 4 terms for quantum physics, only the operator: (ikE + ip + jm) Slide146:  Bosonic states The 3 vacua also help to explain the meaning of the 4 terms in the Dirac 4-spinor. There is 1 real state (the lead term) and 3 potential (vacuum) states into which the lead term can be transformed by one of the 3 interactions. All possible states are always present, either as real states or vacuum ones, e.g.: (ikE + ip + jm)  (ikE + ip + jm) s i (ikE + ip + jm)  (ikE – ip + jm) w k (ikE + ip + jm)  (– ikE + ip + jm) e j (ikE + ip + jm)  (– ikE – ip + jm) Slide147:  Coordinate systems Transformation of coordinate systems is an important technique in physics, precisely because coordinate systems that are transformable into each other are not necessarily physically equivalent. e.g. Polar coordinates privilege a particular point as the origin of the radial coordinate, and are ideal for defining a system with a singularity, e.g. gravitating planet plus satellite or point charge. If we use a system like this where there is no physical singularity, we may succeed in creating an artificial one. Slide148:  Matrices and rotation This is precisely what happens when we use matrix methods to model rotation via the O(3) group, rather than, say, quaternions. We create an artificial singularity. A singularity of this kind can be a mathematical inconvenience. It also often produces a great deal of redundancy, as the system repeatedly tries to cross the singuarity barrier. This is what happens with the conventional Dirac equation. Slide149:  Problems with the Dirac equation (gmm + im) y = 0 In addition, the equation, though apparently compact, contains a great deal of redundancy: The matrix representation produces a faultline, which is most clearly represented in the momentum operators. They are not rotation symmetric, unlike physical momentum. Essentially, a 3-D quantity is represented by a 2-D construct. Slide150:  Fermionic spin Fermionic spin is a routine derivation from the p component of the nilpotent structure. If we mathematically define a quantity s = –1, then [s, H] = [–1, i (ip1 + jp2 + kp3) + ijm] = [–1, i (ip1 + jp2 + kp3)] = –2i (ijp2 + ikp3 + jip1+ jkp3 + kip1 + kjp2) = –2ii (k(p2 ­– p1) + j(p1 ­– p3) + i(p3 ­– p2)) = –2ii1  p Slide151:  Fermionic spin If L is the orbital angular momentum r  p, then [L, H] = [r  p, i (ip1 + jp2 + kp3) + ikm] = [r  p, i (ip1 + jp2 + kp3)] = i [r, (ip1 + jp2 + kp3)]  p But [r, (ip1 + jp2 + kp3)]y = i1 y . Hence [L, H] = ii 1  p , and L + s / 2 is a constant of the motion, because [L + s / 2, H] = 0. Slide152:  Helicity Helicity (s.p) is another constant of the motion because [s.p, H] = [–p, i (ip1 + jp2 + kp3) + ijm] = 0 For fermion / antifermion with zero mass, (kE + ii s.p + ijm)  (kE – iip) (–kE + ii s.p + ijm)  (–kE – iip) Each of these is associated with a single sign of helicity, (kE + iip) and (– kE + iip) being excluded, if we choose the same sign conventions for p. Slide153:  Helicity Numerically,  E = p, so we can express the allowed states as  E(k – ii) Multiplication from the left by the projection operator (1 – ij) / 2  (1 – g5) / 2 leaves the allowed states unchanged while zeroing the excluded ones.

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