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1932 |
Multi-time formalism
Dirac, Fock, and Podolsky formulate the so-called "multi-time
formalism," the most popular form of the description of a field with
particles until the appearance of Schwinger's papers.
P.A.M. Dirac, V.A. Fock, B. Podolsky, "On quantum electrodynamics,"
Phys. Zeits. Sowjetunion 2 (1932); 468-479.
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1934 |
Fock functionals
Fock develops a method for describing a system with an indeterminate
number of particles by means of a generating functional, a method now
known as the method of Fock functionals. The first application (in
quantum electrodynamics) of the apparatus of variational
differentiation.
V. Fock, "Zur Quantenelektrodynamik," Phys. Zeits. Sowjetunion 6
(1934): 425-469.
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May 27, 1935 - Feb 04, 1937 |
The order parameter
Beginning with a paper on anomalous specific heats in
lambda transitions, Lev Landau systematizes the notion of
the "order parameter" (first employed by Felix Bloch). In
his general theory of phase transitions Landau employs the
order parameter as a measure of the degree of order in a
physical system. The reliance on symmetries as indicators
of common characteristics of otherwise distinct phenomena
subsequently becomes an indispensible tool of the theory of
critical phenomena.
L. Landau, "Zur Theorie der Anomalien der spezifischen
Waerme," Phys. Zeits. Sowjetunion 8 (1935): 113-118.
L. Landau, "Zur Theorie der Phasenumwandlungen. I, II,"
Phys. Zeits. Sowjetunion 11 (1937): 26-47, 545-555; "K
teorii fazovykh perekhodov I, II," ZhETF 7 (1937): 19-32,
627-632.
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1941 |
2-D Ising model exhibits phase transition
Kramers and Wannier are the first to locate a transition
point in the two-dimensional Ising model of the
ferromagnet. Montroll shows that the lattice must be
infinite in at least two dimensions in order for such a
transition to take place. Wannier (Montroll's teacher)
serves as the intermediary between these independent sets
of calculations. In a sense these are classic "precursor"
papers, since neither attempted an actual solution to the
partition function. But they continue to be read even
after the appearance of the Onsager solution. Kadanoff,
for example, employed their transfer matrix method in his
1966 paper providing a theoretical justification for
scaling. Are there other cases in which their techniques
were retained and further exploited?
Primary:
H. A. Kramers and G. H. Wannier, "Statistics of the two-
dimensional ferromagnet. I, II," Phys. Rev. 60 (1941): 252-
262, 263-276.
E. Montroll, "Statistical mechanics of nearest neighbor
systems," J. Chem. Phys. 9 (1941): 706-721.
Secondary:
Brush 1983, 244.
Domb 1996, 132.
Hoddeson et al. 1992, 528.
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1944 |
The Onsager solution to the Ising model
The canonical paper offering an exact solution to the two-
dimensional Ising model of a ferromagnet. This is usually
taken as a moment of crisis for classical mean field
theory, since the specific heat for the Onsager solution
was logarithmically infinite rather than merely
discontinuously finite, and the partition function was non-
analytic at the critical temperature. Though a virtuoso
calculation, the Onsager solution was based on a very
abstract model far from laboratory measurement. Extending
it to a physical three dimensions seemed out of the
question at the time, but for obvious reasons it became the
gold standard for comparison with non-classical examples of
critical behavior. Perturbation series expansions by Domb,
Sykes, Fisher, and others subsequently became an especially
fruitful domain of calculation in which reference to the
Onsager solution was crucial to developing adequate
models. One might also ask, what was the relation of the
Onsager solution to the eventual development of
non-perturbative approaches to critical phenomena? Where
and when did the Onsager solution take on heuristic
significance of the sort it could not claim when it was
first published?
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1949 - 1952 |
Spontaneous magnetization of planar square lattice
In the two-dimensional Ising model the spontaneous
magnetization is equivalent to the long-range correlation
function [theta] evaluated at infinity. Onsager somewhat
cryptically announced the exact formula for [theta] at a
conference in Florence in 1949, while Yang employed
perturbation theory in an explicit demonstration. The
formula implies that the critical exponent [beta] = 1/8, in
marked contrast to the [beta] = 1/2 value for all classical
theories. One might term this a "practical" result of
Onsager's theory, but it bore little relation to
contemporary experimental values for magnetic materials.
Primary:
L. Onsager, "Discussion remark," Nuovo Cimento, supplement,
6 (1949): 249.
C. N. Yang, "The spontaneous magnetization of a two-
dimensional Ising model," Phys. Rev. 85 (1952): 808-816.
C. H. Chang, "The spontaneous magnetization of a two-
dimensional rectangular Ising model," Phys. Rev. 88 (1952):
1422.
Secondary:
Brush 1983, 245.
Domb 1996, 146.
Fisher 1967, 670.
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1949 - 1959 |
Exact series expansions
For Ising ferromagnets in finite fields the challenge
became to find reliable methods of successive
approximation. Domb introduced and advocated the use of
exact series expansions of the free energy. Provided one
carefully demonstrated the regular and rapid convergence of
the solutions, it was possible to obtain fairly accurate
values for the critical exponents not only in the two-
dimensional case, but for physically meaningful three-
dimensional cases. With a center of gravity at King's
College, Domb, Sykes, Fisher, Rushbrooke, Wakefield, and
others exploited these techniques to their fullest extent,
and by the mid-1960s their work started to gain the
attention of other physicists who had originally come to
study critical phenomena from quite different perspectives.
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1950 - 1964 |
Critical behavior of mixtures and pure fluids
Not only pure fluids exhibit critical behavior between
phases. Mixtures of fluids can also exhibit critical
behavior between the two components, as Zimm, Gopal, Rice,
and Thompson demonstrated with increasing accuracy,
measuring coexistence curves quite close to the critical
point for CCl4-C7F14 systems.
Primary:
B. H. Zimm, "Opalescence of a two-component liquid system
near the critical mixing point," J. Phys. Colloid Chem. 54
(1950): 1306-1317.
R. Gopal and O. K. Rice, "Shape of the coexistence curve in
the perfluoromethylcyclohexane-carbon tetrachloride
system," J. Chem. Phys. 23 (1955): 2428-2431.
D. R. Thompson and O. K. Rice, "Shape of the coexistence
curve in the perfluoromethylcyclohexane-carbon
tetrachloride system. II. Measurements accurate to
0.0001°," J. Amer. Chem. Soc. 86 (1964): 3547-3553.
Secondary:
Brush 1983, 249.
Heller 1967, 759.
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1952 |
Spherical model of a ferromagnet
Let the spin values of the Ising ferromagnet take on
continuous values, and the partition function becomes a
tractable integral. Berlin and Kac constrained the sum of
the squares of the spin values to be equal to the number of
lattice sites, and obtained an exactly soluble three-
dimensional model that exhibited a phase transition in non-
zero field. The model did not approximate any known
physical systems, but it did later prove useful as a first
step in Wilson's application of the renormalization group.
Primary:
T. H. Berlin and M. Kac, "The spherical model of a
ferromagnet," Phys. Rev. 86 (1952): 821-835.
Secondary:
Domb 1996, 179.
Hoddeson et al. 1992, 574.
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1952 |
Lattice gas model of Yang and Lee
Yang and Lee lend further credibility to the lattice gas
model, which they show can reproduce the crucial features
of liquid-gas equilibrium.
Primary:
C. N. Yang and T. D. Lee, "Statistical theory of equations
of state and phase transitions. I. Theory of condensation,"
Phys. Rev. 87 (1952): 404-409.
C. N. Yang and T. D. Lee, "Statistical theory of equations
of state and phase transitions. II. Lattice gas and Ising
model," Phys. Rev. 87 (1952): 410-419.
Secondary:
Brush 1983, 252.
Domb 1996, 201.
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1953 |
Sharpness of critical point: Optical density measurements of Lorentzen
When mapping out the curves experimentally for equations of
state, it turns out that the shape of the isotherms can be
remarkably sensitive to variations in density. Lorentzen
arranged a vertical glass tube filled with CO2 at extremely
stable temperatures so that slits at each end would define
a cylindrical lens whose focal length depended on the fluid
density. Near the critical point tiny differences in
height could lead to large discrepancies in density. That
most previous tests of equations of state did not take this
density inhomogeneity into account meant that coexistence
curves had been unduly flattened, implying that the
critical transition was more gradual than it actually was.
Primary:
H. L. Lorentzen, "Studies of critical phenomena in carbon
dioxide contained in vertical tubes," Acta Chem. Scand. 7
(1953): 1335-1346.
Secondary:
Brush 1983, 249
Heller 1967, 735
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1953 - 1954 |
Magnetic analogue of critical opalescence
Shoot a beam of neutrons at a magnetic material, and part
of the observed scattering will be due to the interactions
of the atomic magnetic moments with the moment of the
neutron. Near the critical temperature region the
scattering cross section increases sharply, the magnetic
analogue of critical opalescence. Palevsky & Hughes and
Squires first observe the effect in iron.
Primary:
H. Palevsky and D. J. Hughes, "Magnetic inelastic
scattering of slow neutrons," Phys. Rev. 92 (1953): 202-203.
G. L. Squires, "The scattering of slow neutron by
ferromagnetic crystals," Proc. Phys. Soc. 67A (1954): 248-
253.
Secondary:
Heller 1967, 742.
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1954 - 1956 |
The Moscow Zero
Landau, Abrikosov, Khalatnikov, Pomeranchuk, and
(independently) Fradkin publish papers in which the
ultraviolet asymptotic forms of quantum electrodynamics are
derived in the "leading-lograrithm" approximation.
Primary:
L. D. Landau, A. A. Abrikosov, I. M
Khalatnikov, "Asimptoticheskoe vyrazhenie dlia funktsii
Grina elektrona v kvantovoi elektrodinamike," Dokl. Akad.
Nauk SSSR 95 (1954): 773.
L.D. Landau, A. A. Abrikosov, I.M.
Khalatnikov, "Asimptoticheskoe
vyrazhenie dlia funktsii Grina fotona v kvantovoi
elektrodinamike,"
Dokl. Akad. Nauk SSSR 95 (1954): 1177.
L.D. Landau, A. A. Abrikosov, I.M. Khalatnikov, "Massa
elektrona v kvantovoi elektrodinamike," Dokl. Akad. Nauk
SSSR 96 (1954): 261.
L.D. Landau and I.Ia. Pomeranchuk, "O tochechnom
vzaimodeistvii v kvantovoi elektrodinamike," Dokl. Akad.
Nauk SSSR 102 (1955): 489.
I.Ia. Pomeranchuk, "Ravenstvo nuliu perenormirovannogo
zariada v kvantovoi elektrodinamike," Dokl. Akad. Nauk SSSR
103 (1955): 1005.
L.D. Landau, "Quantum Field Theory," in Niels Bohr and the
Development of Physics (London, 1955), 52.
L.D. Landau, "Fundamental problems," in M. Fierz and V.F.
Weisskopf, Theoretical Physics in the Twentieth Century: A
Memorial Volume to Wolfgang Pauli (New York, 1960), 245-248.
E.S. Fradkin, "Ob asimptotike funktsii Grina v kvantovoi
elektrodinamike," ZhETF 28 (1955): 750.
Secondary:
V. B. Berestetskii, "Nul' zariad i asimptoticheskaia
svoboda," Uspekhi fizicheskikh nauk 120 (1976): 439-
454; "Zero-charge and asymptotic freedom," Soviet Physics
Uspekhi 19 (1976): 934-943.
D.A. Kirzhnits, "On the 'Moscow-Zero' Problem," in I. A.
Batalin, C. J. Isham, and G. A .Vilkovisky, eds., Quantum
Field Theory and Quantum Statistics: Essays in Honour of
the Sixtieth Birthday of E S Fradkin (Bristol: Adam Hilger,
1987), vol. 1, 349-369.
E.L. Feinberg, "The Importance of (Sometimes) being
Conservative," in ibid., vol. 2, 3-14.
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1954 - 1957 |
Lattice model applied to liquid helium
Kikuchi in Chicago and Matsubara and Matsuda in Kyoto
developed the first lattice models which could be applied
to liquid helium and gave plausibility arguments for the
second-order phase transition long known experimentally.
Primary:
R. Kikuchi, "[Lambda] transition of liquid helium," Phys.
Rev. 96 (1954): 563-568.
T. Matsubara and H. Matsuda, "A lattice model of liquid
helium, I," Prog. Theoret. Phys. 16 (1956): 569-582.
H. Matsuda and T. Matsubara, "A lattice model of liquid
helium, II," Prog. Theoret. Phys. 17 (1957): 19-29.
H. Matsuda, "A lattice model of liquid helium, III," Prog.
Theoret. Phys. 18 (1957): 357-366.
Secondary:
Brush 1983, 194.
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1955 |
Functional renormalization-group equations of QED
Bogoliubov and Shirkov derive functional renormalization-
group equations of quantum electrodynamics for the general
case. The relationship between the work of Stueckelberg
and Petermann (1953) and Gell-Mann and Low (1954) is
established. Differential group equations are constructed
for the first time, and a program for systematically
improving the results of ordinary perturbation theory is
formulated.
N. Bogoliubov and D. Shirkov, "O renormalizatsionnoi gruppe
v kvantovoi elektrodinamike," Dokl. Akad. Nauk SSSR 103
(1955): 203; "Primenenie renormalizatsionnoi gruppy k
uluchsheniiu formul teorii vozmushchenii," Dokl. Akad. Nauk
SSSR 103 (1955): 391; "Gruppa mul'tiplikativnykh
perenormirovok v kvantovoi teorii polia," ZhETF 30
(1956): 77.
N.N. Bogoliubov and D.V. Shirkov, "Voprosy kvantovoi teorii
polia," Uspekhi fizicheskikh nauk 55 (1955): 149-214; 57
(1955): 1-92.
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1955 |
Temperature Green's functions
Feynman diagrams are an extraordinarily powerful way to
calculate Green's functions for individual particles in
quantum field theory, but to carry over diagram techniques
usefully into statistical physics, one eventually has to be
able to introduce non-zero temperatures. Takeo Matsubara
suggested one highly effective way to do so by constructing
so-called "temperature Green's functions," which do not
depend on time t, but rather on a fictitious "imaginary
time" [tau] = it. This made it easier to calculate the
partition function in close parallel with the techniques
one would use to obtain the vacuum expectation values of
the S-matrix in quantum field theory.
Primary:
T. Matsubara, "A new approach to quantum statistical
mechanics," Prog. Theoret. Phys. 14 (1955): 351-378.
Secondary:
Hoddeson et al. 1992, 582.
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1955 |
Microscopic causality condition for scattering matrix
Bogoliubov introduces an explicit formulation of the microscopic
causality condition for the scattering matrix expressed in terms of
its variational derivatives.
N.N. Bogoliubov, Izv. Akad. Nauk SSSR Ser. fiz. 19 (1955): 237 [Bull.
Acad. Sci. USSR. Phys. Ser. 19 (1955): 215].
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1955 |
Functional renormalization-group equations of QED
Bogoliubov and Shirkov derive functional renormalization-group
equations of quantum electrodynamics for the general case. The
relationship between the work of Stueckelberg and Petermann (1953)
and
Gell-Mann and Low (1954) is established. Differential group
equations
are constructed for the first time, and a program for systematically
improving the results of ordinary perturbation theory is formulated.
N. Bogoliubov and D. Shirkov, "O renormalizatsionnoi gruppe v
kvantovoi elektrodinamike," Dokl. Akad. Nauk SSSR 103 (1955): 203;
"Primenenie renormalizatsionnoi gruppy k uluchsheniiu formul teorii
vozmushchenii," Dokl. Akad. Nauk SSSR 103 (1955): 391; "Gruppa
mul'tiplikativnykh perenormirovok v kvantovoi teorii polia," ZhETF 30
(1956): 77.
N.N. Bogoliubov and D.V. Shirkov, "Voprosy kvantovoi teorii polia,"
Uspekhi fizicheskikh nauk 55 (1955): 149-214; 57 (1955): 1-92.
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1956 - 1963 |
R-operations
Bogoliubov, Parasiuk, and Stepanov develop R-operations demonstrating
self-consistency of perturbation theory in QFT.
N.N. Bogoliubov, O.S. Parasiuk, "Ueber die Multiplikation der Kausal
Funktionen in der Quantentheorie der Felder," Acta Mathematica 97
(1957): 227; "Umnozhenie prichinnykh funktsii pri nesovpadaiushchikh
argumentakh," Izv. Akad. Nauk SSSR ser. mat. 20 (1956): 843.
B.M. Stepanov, "Abstraktnaia teoriia R-operatsii," Izv. AN SSSR, ser.
mat. 27 (1963): 819.
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Sep 1956 - 1958 |
Dispersion relations
Causality, Lorentz invariance, spectral conditions, unitarity, and
other conditions for the axiomatic foundations of QFT are shown by
Bogoliubov using dispersion relations for pion-nucleon scattering.
Generalizations of dispersions relations by Bogoliubov, Vladimirov,
Logunov, Stepanov, Tavkhelidze.
N.N. Bogoliubov, paper in Seattle (1956).
N.N. Bogoliubov, B.V. Medvedev, and M. K. Polivanov, Voprosy teorii
dispersionnykh sootnoshenii (Moscow, 1958).
N.N. Bogoliubov and V.S. Vladimirov, "Ob analiticheskom prodolzhenii
obobshchennykh funktsii," Izv. Akad. Nauk SSSR ser. mat. 22 (1958):
15.
V.S. Vladimirov and A.A. Logunov, "Dokazatel'stvo nekotorykh
dispersionnykh sootnoshenii v kvantovoi teorii polia," Izv. AN SSSR
ser. mat. 23 (1959): 661.
A.A. Logunov and B.M. Stepanov, "Dispersionnye sootnosheniia dlia
reaktsii fotorozhdeniia pi-mezonov," Dokl. Akad. Nauk SSSR 110 (1956):
368; 117 (1957): 492.
A.A. Logunov, I.T. Todorov, "On the proof of dispersion relations for
inelastic scattering," Nucl. Phys. 10 (1959): 552.
A.A. Logunov and A.N. Tavkhelidze, "Generalised dispersion relations,"
Nuovo Cim. 10 (1958): 913.
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