Abstract
It has been known for a long time, that certain non-linear two-dimensional** wave equations possess “soliton” solutions, describing solitary waves, that travel without changing shape or size.
I apologize if, for the sake of clarity, one finds in these lectures some repetition of matters already covered in S. Coleman’s “classical lumps and their quantum descendants,” lectures held at this school in 1975. I have kept overlap with these lectures (quoted as Coleman in the following) to a minimum, I hope, by: a) a different emphasis on subjects b) using recent material and c) referring to Coleman for more detail (especially in my second lecture) whenever this was possible without interrupting the flow of the main argument.
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References
See e.g. Footnote 3 in Coleman. See also W. Blaschke “Different-ialgeometrie” Vol. I. p. 207 (Chelsea Publ. Co. N. York, 1967)
This is a consequence of the Lorentz-contraction course, relativistic.. Eq. (S. G.) is, of course, relativistic.
See for example, G. B. Whitham,“Linear and Nonlinear Waves” J. Wiley and Sons 1974, Chapters 13 and 17.
Another transformation is (φ, ψ, λ, x) → (− ψ,φ,λ−1 −x).
Strictly speaking ψ is defined only up to an integration constant. The notation (1.3’) must be used with some caution.
See Bianchi’s textbook cited by Coleman. The theorem assumes, of course, a suitable choice of integration constants. (See earlier remarks!)
G. L. Lamb, Physics Letters 25A, 181.
See e.g. N. Christ, and T. D. Lee, Phys. Rev. D11, 1606 (1975)
R. F. Dashen, B. Hasslacher and A. Neveu Phys. Rev. D11, 3424 (1975)
J. L. Gervais and A. Jevicki, (1976) CCNY-HEP-76/2 preprint. L. D. Faddev, L. A. Takhtajan, T. M. P.21, 1046 (1975)
See e.g. B. Yoon, Phys. Rev. D13, 3440 (1976).
T. Skyrme, Proc. Roy. Soc. A247, 260(1958); A262, 237 (1961)
G. Mie Ann. Phys. 37, 511 (1912) 39, 1(1912) 40, 1 (1913); See also W. Paul i“Theory of Relativity” (Pergamon Press 1958) Part V S64.
R. Friedberg, T. D. Lee and A. Sirlin, Phys. Rev. D13, 2739 (1976), The fields A and ψ in the original paper difefom those used here by proportionality constants. We omit an overall factor g−2 in the Lagrangian. This factor is irrelevant at the classical level.
Coleman (loc. cit.) refers to this argument as “Derrick’s theorem” comp. G. H. Derrick Journ. Math. Phys. 5, 1252 (1964). See also J. Goldstone and R. Jackiw, Phys. lev. D11, 1486(1975)
There are, of course, two possible vacuum values for the field A (= ± 1). This “degeneracy,” however, will not play any significant role in the following.
As indicated earlier, the omissions are deliberate. The reader will find much of the needed complementary information in Coleman, Section 3. It should be possible to read the following, however, independently of this reference.
A discussion of solitons involving Fermions (anticommuting fields) is beyond the scope of these lectures
Our viewpoint is somewhat different. from Coleman’s. He claims to be discussing a set of initial conditions (this includes velocities).
In general G can act in the space of q through a representation D(G). The representation need not be faithful. We shall also occasionally assume that q is a complex vector, and G is SU(n).
In the S. G. theory we have encountered an example of infinite but denumerable set of minima of V(φ), i.e. φ = 0, ± 2π, ±4π,… etc.
Other, physically equivalent, solutions with Au + 0 are obtained by gauge-transformations. See Third Lecture.
We gloss over questions connected with A0 components of the gauge field, see Coleman.
In ordinary field theories we have, of course, no other choice. That interesting solutions arise if we allow a more general behavior, Eq. (2.2) was pointed out by H. Nielsen and P. Olesen Nucl. Phys. B61, 45 (1973).
That the two problems are nevertheless equivalent, when E is ordinary space and B its boundary, is intuitively obvious, but we shall not prove it.
This conclusion presumably extends to the quantised theory, since there is an infinite potential barrier to be crossed.
There are quite simple cases of manifolds with non-abelian fundamental group, e.g. the surface of a doughnut with two holes.
A sufficient condition for this has been stated by S. Eilenberg, Fund. Math. 32, 167 (1939). It is satisfied if M is a sphere. We cannot go into this, however. See also: N. Steenrod “The Topology of Fibre Bundles” S 16.5.
A. A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu. S. Tyupkin, Phys. L. 59B, 85 (1975).
G.’t Hooft Phys. Rev. L. 37, 8(1976) and Nucl. Phys. to be published. R. Jackies ând C. Rebbi, Phys. Rev. D14,517(1976), and to be published. C. G. Callan, R. F. Dashen, D. J. Gross, to be published.
To obtain the customary expression for the Lagrangian take twice the trace. Remember Tr (TaTb) = 2 Gab.
Unlike (3.4) the transformation law for the G-field: Gμv(x) → g−1 (x) Gμv (x) g(x) does not contain an inhomogeneous terms
Please notice, although g(x) is ambiguously defined at the origin and its derivatives are singular, the solution (3.9) (3. 10) is regular everywhere.
See also ‘t Hooft’s η symbol. One can write σμv = ηaμv σa. See especially Eq. (A.4) and (A. 13) in t Hooft’s paper.
In pedantic detail: is there a gauge function f(x), well defined and non singular over the whole Euclidean four -space such that B’μ = f−1 Bf + 1/γ f−1 ∂μ f has the required properties?
Interesting references may be found in J. Arafune, P. G. O. Freund and C. J. Goebel Journ. Math Phys. 16, 433 (1975)
See also W. Bardeen Nucl. Phys. B75, 246 (1974)
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Wick, G.C. (1978). Three Lectures on Solitons. In: Zichichi, A. (eds) Understanding the Fundamental Constituents of Matter. The Subnuclear Series, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-0931-4_3
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