Jump to Undergraduate, Graduate student, Post-doc at Rutherford Lab., Fellow then Staff at CERN and Prof at Liverpool.
I went up to University (Jesus College, Oxford) at short notice (since I had won a Meyricke scholarship awarded on A-level results) in 1960; and I was offered the choice to start immediately in Physics or wait a year to do mathematics (to cover the material that was missing from the A-level syllabus but assumed by the Oxford course -- such as complex numbers). I decided to do physics - since that was quite mathematical. I subsequently discovered that the mathematics needed for the physics degree was quite close in spirit to that I had done at A-level: so I did not regret the choice.
I found that I fitted in well with other students at Jesus College: it had a Welsh connection then and 4 of the 9 students doing physics in my year were from South Wales grammar schools.
The first year of study (with an exam at the end called Moderations) had quite a big fraction of mathematics which suited me well. The practical exam involved a box with 3 terminals - which I assumed contained a transistor which I had plenty of experience of from my electronics hobby. Both my tutorial partner (Tony Hughes) and myself did really well in the exam (coming joint first in Honours Moderations).
Oxford had a system at that time where the final honours exam was set by a board of examiners who were not the same as those who gave lectures. So there was little point in attending lectures, better was to study the appropriate textbooks (choice guided by college tutors). Jesus College had two Physics tutors: Claude Hurst who had a mathematical bent and who was no longer active in research and John Houghton (atmospheric physicist, later head of the Met Office). Hurst was especially good at teaching. I found that by studying in the mornings, I could cover the material needed: so afternoons and evenings were mostly free.
In my final year, I took the Theoretical Physics Option. I also had
some tutorials in Nuclear and Particle Physics with Arthur Clegg
(experimentalist, soon to move to Lancaster as Prof).
My tutorial partner, Tony Hughes, got the top first, while I was a few
behind him in the list of 150 or so students doing physics in my year.
[Stephen Hawking had done the same course the previous year -
narrowly gaining a first class degree]
During my final year, I made some inquiries about possible supervisors for postgraduate study. I was interested in "what it's all made of" and Roger Blin-Stoyle was the lecturer who best fitted that. I discovered that he was moving to the University of Sussex and considered that option. Another potential supervisor was David Brink (nuclear/spin expert). I was also told that Oxford would be recruiting some new theorists.
So I decided to stay on at Oxford in 1963. In those days anyone with a first-class physics degree could get postgraduate funding (£450 p.a.) on appeal. This meant that there were a lot of students starting and not so many active supervisors. I remember that the first time I went into the Theoretical Physics Building (then at 12 Parks Road), I knocked on the secretary's door. A voice from across the corridor said come in: it was Prof Rudy Peierls who had just arrived from Birmingham as head. He welcomed me and then said he would introduce me to the secretary who was a bit strict.
My assigned supervisor was Dr Gordon Screaton (who was a tutor at University College and who also had a presence in the Mathematics Institute). I saw little of him (maybe three times) and relied on the rest of the group for inspiration.
One of the new people hired (as a Royal Society Professor) was Prof Richard (Dick) Dalitz from Chicago. He had brought some of his students and co-workers with him as well as attracting visiting physicists. The other significant supervisor in the "elementary particle" area was Dr J. C. (John) Taylor.
Quantum mechanics and relativity were firmly established.
Electromagnetic interactions were well understood through the work of Dirac, Feynman and others. Electrons, muons and photons were treated as point-like. A quantum field theory approach (Quantum Electro Dynamics) was successful in describing experimental results.
Weak interactions were understood empirically using the Fermi 4-point (current-current) interaction and it was known that parity (space reflection) was not conserved. [CP violation came later]. It was assumed that the 4-point interaction was mediated by some massive vector boson but no quantum field theory of a massive vector particle existed at that time.
Particle physics (then called elementary particle physics) only began to separate out from Nuclear Physics around that date. Proton accelerators (such as 7 GeV Nimrod at Rutherford Laboratory near Harwell south of Oxford) could produce beams of protons and charged pi-mesons and K-mesons. Scattering such particles off targets (typically of hydrogen: protons) then gave some access to what was called strong (or hadronic) interactions.
My interest was strong interactions. Historically, Yukawa had predicted the presence of the pi-meson to give a mechanism to bind protons (and neutrons) into nuclei. The idea of "pion exchange" was based on quantum field theory -- but no consistent such theory existed.
One of the main avenues of research in hadronic physics was phenomenology: using the data to extract what was going on. One built in everything known: conservation of probability (known as unitarity of the scattering (S) matrix); conservation of (relativistic) energy-momentum and of angular momentum; spatial inversion (parity) symmetry. An important step was the concept of isospin. This treated particles of different electric charge (e.g. neutron and proton; charged and neutral pions,..) as different representations of a basic entity (the nucleon; pion,..). This symmetry could not be exact: the neutron and proton have different masses. What was assumed was that, in a world with electromagnetism turned off, the symmetry would be exact. [we now know that this was wrong - as well as electromagnetism, the up-down quark mass difference also breaks isospin symmetry and this is crucial to explain why the neutron is heavier than the proton].
As an example, pion-nucleon scattering was described by phase-shifts in different partial waves (labelled by spin, isospin and angular momentum). The salient feature of low energy pion nucleon scattering was the N*(1236) [now Δ(1232)]. This was a resonance: the amplitude in that partial wave passed through a maximum which could be characterised by a mass and width. One economical way to express this was as a pole in the scattering amplitude at a complex value of the energy.
Many more of these resonances were found (listed in Rosenfeld's Tables -- forerunner of the Particle Data Group).
Although experimentalists studied scattering at real energy values, it was profitable to discuss the properties of scattering amplitudes at complex energy values. This was partly motivated by quantum field theory which indicated that amplitudes should be analytic functions of energy with only specific singularities (poles from single particle states and branch points, with associated cuts, from two particle states). Since an analytic function is basically determined by its singularities, this was seen as a way to make progress in studying strong interactions.
As well as resonances at low energy, the other salient feature of scattering (here two particles scattering to give two particles in the final state) was that the amplitude was peaked in the forward direction. Explanation was sought from quantum field theory where the exchange of a particle would give a possible mechanism.
What was fairly certain was that at large distance from the centre of a particle, the virtual cloud of particles would be dominated by the lightest particle allowed. Then as such particles collided, if they were at large impact parameter (a measure of their transverse separation as they collided from opposite directions) the exchange of the lightest particle allowed would be the dominant contribution. These collisions (known as peripheral) would then be dominated by pion exchange (if allowed) and next by the exchange of two pions.
Calculating pion exchange by quantum field theory rules (treating the proton, pion etc as point-like) then gave some encouraging results but also some problems. The problems could be swept away by modifying the pion exchange contribution at small impact parameter (central collisions) where it was not justified anyway.
A more severe problem arose when considering exchange of other light mesons (rho and omega vector mesons) since the quantum field theory result gave a totally wrong energy dependence. A clever way around this was found by Chew and Frautschi: Regge poles. Instead of treating a vector meson, one treated the Regge trajectory which included the vector meson but also heavier mesons of higher spin and, importantly, an effective exchanged particle of "lower spin" which gave a sensible energy dependence to the scattering amplitude.
A concept (associated with Geoffrey Chew at Berkeley) that was influential was the idea of "bootstrap". Pion-pion scattering has a major resonance: the rho-meson. Pion-pion scattering also has contributions from the exchange of a rho-meson. So maybe the rho-meson exchange is responsible for the rho-meson. This is "pulling oneself up by ones bootstraps". So there is no need of an underlying quantum field theory at all and there are no underlying point-like constituents: the rho-meson makes itself and is an extended object.
I remember that there was a debate broadcast in Britain between Geoffrey Chew and Dick Feynman. Chew advocated "bootstrap" while Feynman stuck to "quantum field theory works for electromagnetism so it will work for strong interactions". The feeling among my contemporary graduate students was that Chew was more likely to be correct.
We now know that they were both right to some extent: there are fundamental constituents but string theories are the lineal descendant of the "bootstrap".
My own research for my D. Phil at Oxford was focussed almost totally on resonances. The influence of nearby two-particle thresholds on resonances (Λ(1405) S01) and bound states (deuteron); overlapping resonances (N(1535) S11); modelling the line-shape of the Δ resonance. I received some advice from Prof Dalitz as well as from other graduate students (especially David Sutherland). My determination of the properties of the N(1535) was quoted by the Particle Data Group in annual reviews until 1982.
I remember being enthusiastic about what I was doing and determined that Prof Peierls would not fall asleep in a talk I gave (he usually stayed awake long enough to think of a question to ask at the end then nodded off). Failure: he was asleep in my talk too.
I had a car (an Austin A30) and came in by car from Woodstock. I would park it at the front of the building, next to Peierls's MG and Dalitz's Ford Thunderbird. Nobody ever complained. I inherited the job of coffee organiser for a year - the car helped with bringing in supplies of instant coffee. I passed the job on to Chris Llewellyn Smith: so I can claim to have taught him basic managerial skills.
Peierls ran the group in a very personal way: he went to all talks and he and his family invited students home and organised does in the department. His wife was quite an extrovert and would sometimes arrive by car to pick him up and would open the front door and scream "Rudy". She once told me how they met: they were both in a train compartment on some long trip and Rudy Peierls climbed into the luggage rack and went to sleep. She said to herself: that is the man I want.
The seminar on particle physics was run by Dick Dalitz and took place in the evening so that experimentalists from Rutherford Laboratory could attend. I remember one or two speakers who were quite garrulous after being wined and dined at All Souls beforehand.
In September 1965, an International Conference of High Energy Physics was held at Oxford: the plenary sessions were held in the Playhouse Theatre. As a local graduate student, I was allowed to attend. One of the surprises for me was the talk by Dick Dalitz: in which he proposed a quark model of hadrons. This was the first time we at Oxford had heard of his work on quark models.
Up to that date, the quark (proposed in 1964 by Gell-Mann and, as "aces", by George Zweig) had been considered to be a theoretical construct to help with calculations using the approximate hadronic symmetry SU(3) (an extension of isospin). Dalitz's approach needed "constituent" quarks which would be quite heavy (about 300 MeV), not point-like, and which would be treated as being bound in a potential, much like nucleons are bound in a nucleus.
Another anecdote from that Oxford Conference: the "banquet" was an open air ox-roast held in a field near the Thames at Wallingford. The roasting allowed meat to be served at a steady rate, but many participants pushed forward. The roasters told me that they were in danger of being pushed onto the fire themselves. My wife and I had eaten already -- so we could observe the mad-house without joining in. This was the first of many occasions I saw in which physicists' competitive instincts were aroused in the pursuit of food at meetings.
Directly after the Oxford Conference, I attended a summer school at Herceg Novi in, what was then, Yugoslavia. A session was proposed presenting highlights from the very recent Oxford conference and I agreed to contribute. As a result, the organiser, Milan Nicolic, invited me to give a series of talks at the summer school the next year. I am not sure whether he realised that I was still a graduate student. The form was to prepare a booklet before the talks and, I believe, he asked Tini Veltman to cast an eye over my effort named "S matrix theory" -- and it was judged to be OK.
I was making reasonable progress on my D. Phil thesis and decided to seek a post doc position. I had no close links to anyone at Oxford, so was not steered in any particular direction. My publications (3 single-author papers at that time) were not particularly significant. I applied to the USA, but received no interest. Within Britain, I was offered a teaching job at Trinity College (Dublin), a temporary lecturing job at Durham and a post-doc at Birmingham. I did not want to have to teach at that stage, preferring to try to establish my career through research. I then discovered from Dalitz that the Rutherford Lab (near Harwell) was setting up a particle theory group and had post-doc vacancies, one of which I was offered and accepted. I managed to complete my thesis by May (a secretary typed it up in one day as I recall and I added the equations by hand using carbon copy sheets) and I started at Rutherford in June 1966, aged 24. Although, at that (by today's standards) young age, I had a rather narrow range of knowledge, I was able to take advantage of a post-doc position which allowed me the freedom to do what excited me. This is an excellent way to extend ones expertise. I consider the tendency, in some countries (eg. USA and Germany), to award a Ph D at a much later age, to not be the optimum way to encourage innovation.
A model of such a dual picture (embodying the finite energy sum rules) was produced about that time (1968) by Gabriele Veneziano. This model was a simple algebraic formula for the scattering amplitude which had poles in the "resonance" channels and in the "Regge exchange" channels. This model was an example of a bootstrap. It came to be seen as coming from an underlying string description (so no point particles but a fundamental 2D-surface). String theory applied to hadronic physics is just an approximate theory that has some limited use for phenomenology. Applied at a deeper level, it is one of the few ways to attempt to incorporate gravity into particle physics.
There had been a series of national meetings held near Christmas at the Rutherford and funded by the lab. There was no meeting in Dec 1966 and Dick Dalitz decided to hold a meeting in Dec 1967 and asked me (as a post doc at Rutherford) to organise it. A few days later he changed his mind - but as I had already invited people, we decided to go ahead. So "I saved the Xmas meeting". This was the first meeting with a distinguished overseas invitee - John Bell from CERN. There were about 30 talks - by a mixture of established researchers (Squires, Leader, Byers, Ranft, Moorhouse, Oades, Barbour, Drummond, Wilkin, Donnachie, Branson, Morgan, JC Taylor) and juniors (Buttimore, Froggatt, Davies, Lyth, Ross, and others).
At Rutherford, I worked on understanding scattering using Regge
pole theory. Because of the limitations of Regge poles, I worked on
models that went beyond poles: absorption models, Regge cuts,.. One of
the main tools of such analyses was comparing the experimental data
with models which had a number of parameters that should be fitted to
best describe the data. This computer "fitting" of data by a model
with a set of parameters was to become a recurrent theme in my
research.
I also worked with Colin Wilkin on pion deuteron scattering.
Roger Phillips worked closely with Vernon Barger of Madison, Wisconsin. Vernon then invited me to visit Wisconsin for several months in 1969. We (wife and two small children) arrived in the winter when it was really cold (lakes frozen etc.). It was quite a challenge to get established: buying essential things for our apartment (furnished just meant furniture); getting a second-hand car; taking our elder son to pre-school, learning to drive on icy roads,.. I remember that I had an international driving licence which had a photo but no serial number. To pass a cheque (check) in the US, one needed a driving licence as ID and the serial number had to be noted. I solved this by typing a number on the international driving licence cover -- which was happily copied down by checkout staff.
Toward the end of our US stay, we were invited to Aspen for a month. Here I introduced myself to Murray Gell-Mann who told me that he had heard of me (great) and then fell on the floor writhing in agony (not great). I stood, embarrassed, until someone, who knew more, helped Gell-Mann to reset his shoulder (which mishap was apparently quite a frequent occurrence). After Aspen, we spent a month at Argonne National Lab. (where we saw the moon landing on prime-time US TV), before returning to Europe.
Our start at CERN was unpromising: no suitable accommodation was available and we lived in a hotel room and then in a colleague's flat while they were on holiday. After quite a few weeks, we were allocated a CERN flat in Meyrin.
Since I had met the head secretary at TP division (Tanya Fabergé) at Herceg Novi, I seemed to get a better than average office allocation for a Fellow. I had a room to myself.
Since I was at an experimental lab., I felt I should work on topics related to the experimental programme: especially scattering. The theory group had about 100 researchers and I found like-minded people to work with.
One of them was Francis Halzen, a Belgian from Leuven. I suggested two possible projects to him and he picked one: untangling the underlying scattering amplitudes in pion-nucleon scattering. This "amplitude analysis" allowed to compare the shape and phase of the amplitudes with Regge pole models -- and showed clear departures from such models.
This pioneering work was done using data with a pion beam energy of 6 GeV -- I was later to say that I had worked on high energy scattering, intermediate energy scattering and then low energy scattering -- all the while sticking to 6 GeV.
One definite advantage of working at CERN was that any research got good publicity and was widely disseminated. In those days there was no internet and information spread via preprints (i.e. text of papers before they were published) posted to different research groups.
One meeting to which I was invited was the "Rencontres de Moriond" organised annually around March by Tran Thanh Van (from Orsay) and held in a ski resort in the French Alps. This focussed on scattering and combined experimental and theory talks. I became a regular and helped with organisation. It started off as a quite small French-speaking meeting but gradually became more international as it moved from Mèribel to Les Arcs and then to La Plagne.
After two years as a fellow, I was invited to stay on at CERN with a (fixed term) staff appointment. To keep the family happy, we moved out of a Meyrin apartment to a house with a garden in Coppet (and later Commugny) in the canton of Vaud.
I worked with a lot of different people at CERN: Chris Schmit, Frances Halzen, Jacques Weyers, Alan D Martin, Penny Estabrooks, Alan Irving, Halstein Hogaasen, Roberto Oderico, Vesa Ruuskanen, Noel Cottingham, among others.
In 1972, I received invitations to talk at two major conferences: at Oxford (IVth International Conference on High Energy Collisions), and at FermiLab/Chicago (XVIth International Conference on High Energy Physics). My topic was models for two-body scattering but my message was, unfortunately, that there were no really good models -- just a few approximate approaches. These approximate models (Regge - absorption models) were better than nothing, but frustratingly limited.
One concern I had as a young staff member at CERN was that the permanent staff were getting older and more remote from the fixed-term staff (such as myself). The permanent staff also took rather little interest in the numerous visitors and fellows. I made several suggestions; some of which were successfully implemented: a seating area around the mail boxes where people could gather and meet; a brief talk of introduction (1 minute or so) by everyone at the start of each year; a summer picnic.
I was on the CERN computing resources panel: at that time theorists used rather limited computing power. In fact someone later quoted back to me that I had said that "a theorist who couldn't get a computer program to have a run time of less than 3 minutes was not trying hard enough". See more.
I also served on the experimental selection panel for several years. I don't think my input helped decisively, although I did succeed in pointing out that a proposal to measure Coulomb interference effects in pion-pion scattering using virtual pions as a target was likely to be thwarted by the consequences of gauge invariance.
In 1972 I was invited to spend a few months at CalTech. At that time Murray Gell-Mann was on leave at CERN, so I guess they had some funds available. The theoretical particle physics group was small (Feynman, Gell-Mann, Frautschi and Zachariasen, with Zweig moving to biological science). Geoff Fox from the experimental group was there also (a UK phenomenologist) and Tony Hey was a post-doc.
Lunchtime often included Feynman who was a great source of anecdotes; many of them have since appeared elsewhere. Fred Zachariasen was most helpful: he lent me his second car for a while, invited us around Christmas and advised me on sights to see (which would have left no time for research if fully explored). We had a second hand car (details) and made excursions to Death Valley and the Grand Canyon; to Mexico (Ensenada) and to Malibu. We took the children to Disneyland in Anaheim and to Marineland in San Diego.
While I was at CERN, I had various inquiries about whether I would be interested in jobs. Madison Wisconsin offered me an assistant professorship which I declined. I suggested they hire Frances Halzen which they did and he has done well (spokesperson for Ice Cube). Marseilles and Orsay both made encouraging noises. I preferred to return to the UK; primarily because I would feel more confident as a parent in a country I knew. I was told that there was a vacancy at Liverpool as Prof Fröhlich was retiring. He had been a condensed matter physicist so I had assumed that they would appoint someone in that area. I was told, however, that the large experimental particle group at Liverpool was keen on getting a phenomenologist, so I applied. The presence of a sizeable theory group at Daresbury, nearby, was also an inducement. I was duly short-listed, interviewed and offered the position. After negotiating a delay of a year in starting, I accepted the position.
Around this date, Quantum Chromodynamics (QCD) became established as the theory of hadronic interactions. Point-like quarks had been seen as constituents of hadrons experimentally using deep inelastic scattering. Non-Abelian Gauge theories were found to have the counter-intuitive property that forces got weaker at large energies: so weakly bound quarks would be seen at high energies (asymptotic freedom), whereas these quarks would interact too strongly at low energies to be produced as free particles (quark confinement).
QCD did not suggest any new way to deal with scattering, except for certain extreme cases (high transverse momentum, heavy di-leptons,..).
While I had a lot to show for my research (which had been my full-time activity for 8 years since my D. Phil.), I had experience of giving seminars, conference talks and talks at graduate summer schools, but none of teaching undergraduates. In a way this proved an advantage: since I could not remember my own undergraduate training in any detail, I was forced to think things through afresh. In fact I would come to enjoy teaching in new areas where I was able to learn more myself: general relativity; game theory; computer modelling; stochastic systems; financial mathematics.
I started full-time at Liverpool as Professor of Theoretical Physics in October 1974. This was actually an exciting time in particle physics: the J/ψ was discovered: a narrow resonance at around 3 GeV decaying primarily to lepton pairs. Possible explanations were as a weak vector boson, or as a bound state of charm quarks (heavier quarks postulated to keep flavour-changing neutral currents under control). The latter explanation won the day and potential models of bound states of charm-anticharm were a good way to explain the J/ψ and subsequent further charmonium states that were discovered.
What was needed was to evaluate the potential between heavy (essentially static) quarks from first principles using QCD. Ken Wilson's suggestion of discretising QCD to a space-time lattice provided a way forward. Initially strong-coupling methods were used (which gave a confining potential but had serious limitations). Mike Creutz was one of the first to show (in 1980) how Monte Carlo simulation was a very effective way to do a much better job.
My work on scattering (including now multi-particle production) was still very phenomenological. I was looking for new avenues and strong interaction calculation from first principles (using QCD) was a very attractive prospect.
A more elementary introduction to lattice QCD.
I had a lot of experience with computers including Monte Carlo methods, and although our resources were quite limited, I could see lots of interesting things to do. In those days, we ignored the light quarks and concentrated on the gluonic sector of QCD. It allowed purely gluonic states, the glueballs, to be evaluated and also allowed lots of other useful evaluations: the potential between static quarks; the spin dependence of the long-range potential; the excited potential with non-trivial symmetry (relevant for gluonic excitations -- hybrid mesons); adjoint (colour octet) sources (gluino with gluon field or 2 light quarks); 4-quark mesons with 2 b-quarks, and string-breaking.
One avenue I explored was to use a discrete group for colour -- after all space-time was being made discrete too. The main computational bottleneck was multiplying colour-group matrices which used many (then slow) floating point operations. For SU(2) (2-colour quarks) the icosahedral group providing a very quick way to do group multiplications: using a look-up table. For SU3) (3-colour quarks as in the real world) the largest subgroup was too "thin". We managed to find a way to interpolate between points in that rather sparse subgroup -- which was a feasible method. Improvements in computer power (and fast floating point multipliers) soon rendered this obsolete.
After a decade at Liverpool, I sought to get some new ideas by taking a year's leave of absence in 1985/6. I arranged to go, as a visiting professor, to the University of Illinois at Champaign-Urbana and then spend the summer at CERN. The lattice theorists at Illinois (John Kogut and John Stack) had already established close links to collaborators, so I did not end up collaborating directly with them myself. I used to take lunch with a group that included John Bardeen (Nobel Prizes for the transistor and for superconductivity). I taught two graduate courses (the physics department and associated research institutes had about 600 graduate students then) and got quite a lot of stimulation from the students and their questions. I did learn to use a supercomputer (a CRAY) as well as use electronic communication channels (early e-mail) to keep in touch with my graduate students still in Liverpool. I also bought an IBM personal computer.
On my personal computer, I wrote code in assembler to simulate 16 versions of the 3D Ising model in parallel (using the 16 bit word of that computer). There were some puzzling results from another simulation of the Ising model, so I was able to use a poor-mans supercomputer (25 IBM PC-AT computers in a teaching lab that I set running all night). This achieved a trillion (1012) spin updates -- enough to show the curious previous result to be wrong. See details here .
In the summer of 1986, I participated in a meeting at Imperial College (a sort of UK "Aspen" to facilitate collaboration). There I exchanged ideas with Mike Teper (based at Oxford) on improving the determination of glueball masses using lattice QCD. This was the start of a fruitful collaboration between us that led to big progress, helped by access to a CRAY supercomputer at Rutherford Lab.
Another long running collaboration was with Tony Green at Helsinki. He was interested in making models of the potential energy of several static quarks (typically 2 quarks and 2 antiquarks in different geometrical configurations). This was readily evaluated by lattice methods and provided the input he needed. I also worked with other Finns (Janne Peisa, Petrus Pennanen and Jonna Koponen) especially in later work on heavy-light mesons.
Lattice gauge theory (or QCD) proceeds in two stages: one creates samples of the vacuum (gauge configurations) and then "measures" in them the properties of states of interest. My expertise was mainly directed to the latter activity: fitting the parameters of states to the correlations between suitable operators measured in the gauge configurations.
As I told my non-physicist friends: I used a powerful computer
to study "nothing". But of course the vacuum fluctuations contain
everything in principle.
The difficulty to be overcome was that where the correlation was
strongest (at short space-time separation) many states contributed,
whereas at large separation (with a noisy signal) only the lightest
would contribute. Lots of techniques could be used to get the best from
the gauge configurations: appropriate choice of operators to use in
constructing correlators (including variational methods and
classification using discrete sub-groups); variance reduction
techniques (multi-hit; all-to-all evaluation; maximal variance
reduction); sophisticated fitting techniques that took into account
correlated data sets; etc.
As well as the difficulty of extracting accurate mass values (or
coupling strengths) from the correlators, it was necessary to
extrapolate to the physical régime (zero lattice spacing; huge
space-time volume; and physical values for quark masses). The latter
was the major constraint for many years. Early work (as described
above) treated the "pure gauge" version of QCD in which quark loops
were absent from the vacuum. This was generally known as the "quenched"
approximation, though I prefer to call it "gluodynamics".
For QCD in a small spatial volume (the "femto-universe") with
periodic boundary conditions, the non-zero momentum modes will have
high energy and can be treated perturbatively. This small volume does
not have rotational symmetry (because of the boundary) but can be
classified using the cubic group. The small number of zero-momentum
modes then can be analysed numerically (for example using variational
bases) as was championed by Pierre Van Baal. We were able check by
lattice methods those continuum evaluations of glueball spectra -
finding some small discrepancies and establishing the region of
validity.
With Joachim Kripfganz, we realised that massless fermions with
spatial antiperiodic boundary conditions also have large energy in a
small volume, so can also be included perturbatively. In a small spatial
volume, the lightest states are torelons (gluonic flux looping over
the periodic spatial boundary) which have a non-trivial representation
under the centre-group (Z3 for 3 colours). With no fermions
included, these low-lying torelons do not mix with glueballs. When
fermions are included, even taking the limit of their number to zero,
the fermionic term in the action breaks the Z3 symmetry and
dramatically changes the glueball results. So the spectrum without
fermions is not the same as that with fermions in the limit where the
number of fermion species goes to zero.
Another area of study was resonance decays. One can determine
energies on a lattice for stable particles and for two body stable
states. Lüscher showed how to use the small changes in two-body
energies, as the spatial volume varies, to deduce scattering
phase-shifts, which enables study of resonances.
If the quark masses and lattice size are such that a state (such
as the rho-meson) is coincident in energy with its lightest decay channel
(two pions); then one can study the transition matrix element directly
in large volumes and so deduce properties of the decay of rho to pi pi.
These evaluations were done with Craig McNeile.
On a lattice, introducing the quark degrees of freedom required a big increase in computer power, basically because the quarks effectively induced a long-range interaction which slowed down the simulation.
Another key issue was "fermion doubling". On a space-time lattice, quark degrees of freedom came with spurious extra "doublers". The early approaches were either to add terms to make these doublers heavy (and so irrelevant) following Wilson; or to combine quark degrees of freedom into suitable pairs staggered in space-time following Kogut-Susskind. The Wilson approach had a major drawback: the lattice "artifacts" were stronger, although the introduction of a "clover" term could rectify this. The other drawback of Wilson-like formulations was that chiral symmetry (present in the continuum theory formulation although spontaneously broken in nature) was not respected. The staggered approach was unsuitable, per se, for treating real-world quarks which do not come in neat pairs.
This situation prevailed for many years until several technical
advances were made. Overlap fermions (Neuberger) avoided the problem but
at the expense of a greatly increased computational effort. Domain wall
fermions (using an extra dimension) were recognised to be a specific
implementation of overlap fermions and gained support.
Another fertile approach was twisted mass fermions: a variant of
Wilson fermions that has automatically reduced lattice "artifacts".
This is computationally much quicker (similar to clover-Wilson) but
quark mass parameters still have to be tuned empirically until the
suitable value is found.
As an example of this fermion choice, of the three Lattice QCD researchers at Liverpool: I was a twisted mass exponent; Alan Irving was a staggered man and Paul Rakow was a clover-Wilson aficionado.
One particular interest of mine has been "disconnected diagrams". This refers to quark loops that are connected only by gluonic fields: so the fermion (quark) lines are "disconnected". These are important for the ninth pseudoscalar meson (the η') whose heavier mass comes from such contributions. I have also explored contributions of disconnected diagrams in decays, glueball-meson mixing, etc. It turns out that twisted-mass fermions are especially suited to such studies, since a very powerful variance reduction technique makes such evaluations more accurate. Disconnected diagrams are one of the areas where full QCD has features that go beyond the simple quark model. Twisted mass fermions have been collaborations involving Carsten Urbach and others.
For info on publications of Chris Michael see here.
For a list of most frequent co-authors .
List of places visited for Physics .
Some info about people at Liverpool in Theoretical Physics.
Some reminiscences about getting papers produced and published.