Previous Works
H.B. Nielsen, S. Chadha
“On how to count Goldstone bosons”
Nucl. Phys. B105, 445 (1976)
Ω µ k2 n+1 (n Î Z) as k ® 0 type-I NGB, Ω µ k2 n (n Î Z) as k ® 0 type-II NGB.
nI+ 2 nI I³ dimHGHL
nI (nI I): the number of type-I (II) NGBs.
Very complicated proof by the analiticity of correlation functions. Comments:
- Assumed the spatial rotation.
- Classifies NGBs based on commutation relations
- Just an inequality. Doesn’t say anything about the upper limit of nNGB= nI+ nI I. - Usually the equality holds, but there are artificial exceptions:
Lfree boson= ¶tΨ¾ ¶tΨ - C2Ñ Ψ¾ Ñ Ψ - C4Ñ2Ψ¾ Ñ2Ψ, Ω2= C2k2+ C4k4 can be seen NGB of shift symmetries
Q1: Ψ ® Ψ + Ε1, Q2: Ψ ® Ψ + ä Ε2,
which prohibit the mass term µ Ψ¾Ψ.
- C2¹ 0 Þ nI= 2, nI I= 0. nI+ 2 nI I= dimHGHL = 2 - C2= 0 Þ nI= 0, nI I= 2. nI+ 2 nI I= 4 > dimHGHL = 2 Fine-tuning of the parameter may generally give the same (?)
T. Schäfer, D.T. Son, M.A. Stephanov, D. Toublan, J.J.M. Verbaarschot,
“Kaon condensation and Goldstone’s theorem”
Phys. Lett. B, 522, 67 (2001)
Theorem
If Qa, a = 1, …, n is the full set of broken generators, and if X@Qa, QbD\ = 0 for any pair Ha, bL, then the number of Goldstone bosons is equal to n, i.e., the number of broken generators.
Proof (?) of the contraposition:
Suppose that a\ = Qa 0\ are linearly dependent, i.e., ÚaCa a\ = ÚaCaQa 0\ = 0.
If all Ca' s are real or pure imaginary, ÚaCaQa is unbroken generator. Þ Q1=ÚaRe@CaD Qa¹ 0, Q2=ÚaIm@CaD Qa¹ 0 are broken generators.
We define
Q1 0\ = B\, ä Q2 0\ = - B\, so that HQ1+ ä Q2L 0\ = ÚaCa a\ = 0. Then,
ÈÈ
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\ \
\ Then,
X0¤@Q1, Q2D 0\ = ä ÈÈ B\ ÈÈ2 ¹ 0.
Comments:
- They themselves said in the paper that the inverse is not necessarily true by their argument. - Not quite rigorous, since Qa is not well-behaving.
Y. Nambu,
“Spontaneous Breaking of Lie and Current Algebras”
J. Stat. Phys. 115, 7, (2004) ¬ 83 years old!!
Photo from Wikipedia
Yoichiro Nambu at a Franklin Institute ceremony in Philadelphia (2005) where he was awarded the Franklin Medal
Suppose a symmetry generator (charge) Q develops a vacuum expectation value XQ\ = C. If two other charges Qi , Qj
are such that their commutator AQi, QjE = ä Q, then their corresponding zero modes Zi , Zj behave like canonical conjugates of each other: AZi, ZjE = ä C. Hence they belong to the same dynamical degree of freedom, and the number of NG bosons is thereby reduced to one per each such pair.
Comments:
- He didn’t prove this statement generally. Just observed by an example. - What if
X@Q1, Q2D\ ¹ 0, X@Q2, Q3D\ ¹ 0, X@Q3, Q1D\ ¹ 0? Three reduce to one?
HW and T. Brauner,
“Number of Nambu-Goldstone bosons and its relation to charge
densities”
PRD 84, 125013 (2011)
We conjectured
nNGB= dimHGHL -1
2rank Ρ,
where ä Ρa b= limh®+0limV®¥X@Qa ,QbD\
V =YAQa, jb
0H0LE] = ä fa bcYjc0H0L] in the absence of central extentions. - Lorentz invariance ® XjiΜHxL\ = 0 Þ Ρa b= 0.
- Consistent with Schäfer et al: Ρ = 0 Þ nNGB= dimHGHL.
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L\
L
We tried to prove this formula from the operator approach (inserting complete set, spectral decomposition, …) but failed, since it is difficult to give an upper bound for the number of NGBs. (Easy to say we need at least some gapless modes, i.e., the lower bound.)
We will see this counting rule from an alternative approach: the effective field theory ® HW&HM, PRL (2012).
Other recent works
(I will not talk about them today)
- Series of works by Y. Hidaka (Riken). Especially, PRL (2013) is an alternative “proof” of the above counting rule from a completely different approach. (Mori’s projection operator?) See also Y. Hidaka and T. Hayata (Riken).
- M. Nitta (Keio), M. Kobayashi (Kyoto), D. Takahashi (Riken), …
- T. Brauner (Vienna), S. Moroz (Colorado), …
- A. Nicolis (Columbia), F. Piazza (Paris, Columbia), R. Penco (Columbia), …
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