Summary
The connection between integrals over conserved local currents and generators of symmetry transformations is discussed. It is shown that this connection is simpler in the case of the Hamiltonian than in other cases. It is also shown that conserved symmetric-tensor currents of second rank never lead to spontaneously broken symmetries.
Riassunto
Si discute il legame fra integrali su correnti locali conservate e generatori di trasformazioni della simmetria. Si dimostra che questo legame è più semplice nel caso dell’hamiltoniana che in altri casi. Si dimostra anche che correnti tensoriali simmetriche conservate dal secondo ordine non portano mai a simmetrie spontaneamente infrante.
Реэюме
Обсуждается свяэь между интегралами от сохраняюшихся локальных токов и генераторами преобраэований симметрий. Покаэывается, что зта свяэь является более простой в случае Гамильтониана, чем в других случаях. Также покаэывается, что сохраняюшиеся симметричные тенэорные токи второго ранга никогда не приводят к спонтанному нарущению симметрии.
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Abbreviations
- μ, ν, ϰ, σ :
-
indices running from 0 to 3, specify Lorentz tensors
- i :
-
index 1, 2 or 3
- g 00 = 1,g ii = −1:
-
all otherg μν = 0
- R n :
-
n-dimensional Euclidean space
- x = (x 0,x)∈R 4 :
-
x 2 =x 02−x 2
- p = (p 0,P):
-
xp =x 0 p 0−xp
- ℛ:
-
algebra of unbounded field operators = polynomials of the fields smeared with test functions of compact support
- ℛ(O):
-
polynomials of the fields smeared with test functions with support in a finite regionO⊂ℛ4
- A(O):
-
= ℛ(O)“ von Neumann algebra of bounded operators generated by ℛ(O)
- A :
-
algebraic union of allA O,O⊂⊂R 4
- Ω :
-
vacuum vector
- ℋ:
-
completion ofA Ω and ℛΩ
- (·|·):
-
scalar product on ℋ
- [·, ·]:
-
commutator of two operators
- w-lim, s-lim:
-
limits in the weak and strong topology on ℋ and for bounded operators on ℋ
- D(R n):
-
space of infinitely often differentiable test functions onR 4 with compact support
- t[f]:
-
value of a distribution fromD* for the test functionf∈D
- E(R n):
-
space of infinitely often differentiable functions onR n
- ⊄:
-
the equation is correct after ⊄ being replaced by some finite nonzero number
- ∼ f (p):
-
Fourier transform of the functionf(x)
References
R. F. Streater andA. S. Wightman:PCT, Spin and Statistics and All That (New York, Amsterdam, 1964).
Comparee.g. (3,4) for a review and other references.
D. W. Robinson: Istanbul lectures (1966);B. Schroer andP. Stichel:Comm. Math. Phys.,3, 258 (1966);J. A. Swieca:Cargèse Lectures (1969);C. A. Orzalesi:Charges and generators of symmetry transformations in quantum field theory, preprint Columbia University NYO-1932(2)-161.
H. Reeh:Fortschr. Phys.,16, 687 (1968).
H. Araki, K. Hepp andD. Ruelle:Helv. Phys. Acta,35, 164 (1962).
H. J. Borchers:Nuovo Cimento,33, 1600 (1964).
D. Maison: to be published.
Cf.,e.g.,D. W. Robinson:Comm. Math. Phys.,1, 89 (1965);H. J. Borchers andW. Zimmermann:Nuovo Cimento,31, 1047 (1964).
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Maison, D., Reeh, H. Properties of conserved local currents and symmetry transformations. Nuov Cim A 1, 78–88 (1971). https://doi.org/10.1007/BF02722612
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DOI: https://doi.org/10.1007/BF02722612