2568:
2126:
5243:
296:
dissipative quantum model of the brain). Applications of spontaneous symmetry breaking and of
Goldstone's theorem to biological systems, in general, have been published by E. Del Giudice, S. Doglia, M. Milani, and G. Vitiello, and by E. Del Giudice, G. Preparata and G. Vitiello. Mari Jibu and Kunio Yasue and Giuseppe Vitiello, based on these findings, discussed the implications for consciousness.
2563:{\displaystyle {\begin{aligned}\langle 0|QQ|0\rangle &=\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Q\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}\right|0\right\rangle \\&=\int d^{3}x\left\langle 0\left|j_{0}(0)Q\right|0\right\rangle \end{aligned}}}
840:
1812:
3250:, and the only superpartners, at that. That is, to say, two non-goldstino particles are connected to only goldstinos through supersymmetry transformations, and not to each other, even if they were so connected before the breaking of supersymmetry. As a result, the masses and spin multiplicities of such particles are then arbitrary.
208:, the situation is more complicated; the Goldstone bosons are the longitudinal and transverse phonons and they happen to be the Goldstone bosons of spontaneously broken Galilean, translational, and rotational symmetry with no simple one-to-one correspondence between the Goldstone modes and the broken symmetries.
1529:
Thus, if the vacuum is not invariant under the symmetry, action of the charge operator produces a state which is different from the vacuum chosen, but which has zero frequency. This is a long-wavelength oscillation of a field which is nearly stationary: there are physical states with zero frequency,
1189:
which takes into account quantum effects, however it is equal to the classical potential to first approximation. Symmetry dictates that all variations of the potential with respect to the fields in all symmetry directions vanish. The vacuum value of the first order variation in any direction vanishes
295:
Ricciardi and
Umezawa proposed in 1967 a general theory (quantum brain) about the possible brain mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons. This theory was subsequently extended in 1995 by Giuseppe Vitiello taking into account that the brain is an "open" system (the
246:
correspond to the
Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons are absorbed by the three gauge bosons corresponding to the three broken
3154:
is the number of
Goldstone modes that arise. For internal symmetries, the above equation has no non-trivial solutions, so the usual Goldstone theorem holds. When solutions do exist, this is because the Goldstone modes are linearly dependent among themselves, in that the resulting mode can be
622:
2972:, conformal, rotational, or translational symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the number of independent massless modes may be fewer than the number of spontaneously broken generators. For a theory with an order parameter
2077: = 0. However this argument fails when the symmetry is gauged, because then the symmetry generator is only performing a gauge transformation. A gauge transformed state is the same exact state, so that acting with a symmetry generator does not get one out of the vacuum (see
1316:
1553:
108:
of these. They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and are
234:
that result from the spontaneous breakdown of the chiral-flavor symmetries of QCD effected by quark condensation due to the strong interaction. These symmetries are further explicitly broken by the masses of the quarks so that the pions are not massless, but their mass is
2061:
1928:
1431:, as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above. Actually, here, the charge itself is ill-defined, cf. the Fabri–Picasso argument below.
952:
177:
By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum ("soft") Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish ("Adler zeros").
3149:
1525:
in its
Fourier transform. (This ensures that, inserting a complete set of intermediate states in a nonvanishing current commutator can lead to vanishing time-evolution only when one or more of these states is massless.)
835:{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi -m^{2}\phi ^{*}\phi ={\frac {1}{2}}(-ive^{-i\theta }\partial ^{\mu }\theta )(ive^{i\theta }\partial _{\mu }\theta )-m^{2}v^{2},}
3618:
L.M. Ricciardi, H. Umezawa (1967). Brain and physics of many-body problems. Kybernetik, 4, 44–8. Reprinted in: Globus GG, Pribram K.H., Vitiello G., publishers. Brain and being. Amsterdam: John
Benjamins. P. 255–66
3193:
is in the direction of the unbroken generators, when all translation generators are broken, no non-trivial solutions exist and the number of
Goldstone modes is once again exactly the number of broken generators.
1807:{\displaystyle {d \over dt}Q_{A}={d \over dt}\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}J^{0}(x)=-\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}\nabla \cdot J=\int _{x}\nabla \left(e^{-{\frac {x^{2}}{2A^{2}}}}\right)\cdot J,}
1512:
3779:
3750:
2876:
As a first example an antiferromagnet has 2 goldstone bosons, a ferromagnet has 1 goldstone bosons, where in both cases we are breaking symmetry from SO(3) to SO(2), for the antiferromagnet the dispersion is
1218:
215:, the original rotational symmetry (present in the absence of an external magnetic field) is spontaneously broken such that the magnetization points in a specific direction. The Goldstone bosons then are the
1418:
165:
particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the
3007:
1063:
434:
2131:
1190:
as just seen; while the vacuum value of the second order variation must also vanish, as follows. Vanishing vacuum values of field symmetry transformation increments add no new information.
2960:
Still in regards to symmetry breaking there is also a close analogy between gapless modes in condensed matter and the Higgs boson, e.g. in the paramagnet to ferromagnet phase transition
2624:
2676:
which is (partially) spontaneously broken. Some of the "Goldstone bosons" associated with this spontaneous symmetry breaking are charged under the unbroken gauge group and hence, these
1957:
593:
3191:
3072:
1329:
The principle behind
Goldstone's argument is that the ground state is not unique. Normally, by current conservation, the charge operator for any symmetry current is time-independent,
3637:
E. Del
Giudice, S. Doglia, M. Milani, G. Vitiello (1985). A quantum field theoretical approach to the collective behavior of biological systems. Nucl. Phys., B251 (FS 13), 375 - 400.
1823:
358:
259:, which served as the original source of inspiration for Nambu, namely, the photon develops a dynamical mass (expressed as magnetic flux exclusion from a superconductor), cf. the
513:
3646:
E. Del
Giudice, S. Doglia, M. Milani, G. Vitiello (1986). Electromagnetic field and spontaneous symmetry breaking in biological matter. Nucl. Phys., B275 (FS 17), 185 - 199.
2934:
2748:
2901:
994:
853:
3315:. The latter become massive and their new, longitudinal polarization is provided by the would-be Goldstone boson, in an elaborate rearrangement of degrees of freedom .
545:
3783:
3747:
3034:
2803:
5166:
2829:
2774:
3080:
4916:
5287:
5154:
247:
generators; this gives these three gauge bosons a mass and the associated necessary third polarization degree of freedom. This is described in the
3682:
3766:
Haruki Watanabe, Hitoshi Murayama, Unified Description of Nambu Goldstone Bosons without Lorentz invariance Phys. Rev. Lett. 108,251602,2012,
4085:
1311:{\displaystyle \left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,}
161:
is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact)
4196:
1543:
This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region
1455:
3603:
3465:
Anderson, P. W. (1958-05-15). "Coherent Excited States in the Theory of Superconductivity: Gauge Invariance and the Meissner Effect".
2717:. In condensed matter these goldstone bosons are also called gapless modes (i.e. states where the energy dispersion relation is like
5052:
1335:
996:
in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.
3748:
https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html
120:
If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone
5133:
2936:
and the expectation value of the ground state is not zero, i.e. there is a spontaneously broken symmetry for the ground state
1141:, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is
2975:
1009:
5277:
5272:
374:
154:
62:
3673:
Giuseppe Vitiello, My Double Unveiled - The dissipative quantum model of brain. John Benjamins Publ. Co., Amsterdam 2001.
5149:
2645:
There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-
260:
3655:
E. Del Giudice, G. Preparata, G. Vitiello (1988). Water as a free electron laser. Phys. Rev. Lett., 61, 1085 – 1088.
2056:{\displaystyle \left\|{\frac {d}{dt}}|\theta \rangle \right\|=\|H|\theta \rangle \|\geq m_{0}\||\theta \rangle \|.}
114:
5225:
3274:
5282:
4637:
2579:
274:
3628:
G. Vitiello, (1995). Memory dissipation and capacity in the quantum brain model. Int. J. Mod. Phys. B9, 973-989.
563:
3279:
3259:
3235:, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.
3197:
In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken translation symmetry.
2661:
1923:{\displaystyle \left\|{d \over dt}Q_{A}|0\rangle \right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle \right\|.}
231:
126:
90:
42:
4455:
3551:
Bogoljubov, N. N.; Tolmachov, V. V.; Širkov, D. V. (1958). "A New Method in the Theory of Superconductivity".
3238:
Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristic
3158:
3039:
2776:), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is also
5193:
4828:
4189:
266:
3815:
Hoinka, Sascha; Dyke, Paul; Lingham, Marcus G.; Kinnunen, Jami J.; Bruun, Georg M.; Vale, Chris J. (2017).
5128:
4465:
2665:
320:
31:
479:
104:
bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the
5215:
3325:
3239:
270:
3767:
609:
Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.
4153:
4118:
4022:
3963:
3914:
3838:
3703:
3560:
3517:
3474:
3439:
3396:
3357:
3307:, the Goldstone bosons are absent. Their degrees of freedom are absorbed ("eaten", gauged out) by
3284:
2903:
and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is
2709:
theories. It essentially states that, for each spontaneously broken symmetry, there corresponds some
86:
2906:
2720:
5230:
4213:
309:
289:
162:
150:
66:
5242:
3328:
where the tip and the sides shoot to infinity, preserving the location of the minimum at its base.
2968:
In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such as
947:{\displaystyle ={\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}
5173:
4182:
4091:
4065:
4046:
4012:
3854:
3828:
3412:
2677:
613:
365:
278:
2880:
1434:
But its better behaved commutators with fields, that is, the nonvanishing transformation shifts
962:
3508:
Anderson, P. W. (1958-12-15). "Random-Phase Approximation in the Theory of Superconductivity".
5210:
5123:
4602:
4347:
4274:
4081:
4038:
3981:
3932:
3719:
3664:
M. Jibu, K. Yasue (1995). Quantum brain dynamics and consciousness. Amsterdam: John Benjamins.
3599:
3576:
3533:
3490:
2950:
2685:
1942:, the frequency of any state like the above, which is orthogonal to the vacuum, is at least
1138:
527:
256:
201:
110:
82:
5220:
4991:
4766:
4627:
4612:
4294:
4205:
4161:
4126:
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3922:
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3711:
3568:
3525:
3482:
3447:
3404:
3365:
2969:
1186:
74:
38:
3694:
Fabri, E.; Picasso, L. E. (1966-03-07). "Quantum Field Theory and Approximate Symmetries".
3348:
Nambu, Y (1960). "Quasiparticles and Gauge Invariance in the Theory of Superconductivity".
3012:
5267:
5188:
5113:
5097:
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3155:
expressed as a gradients of another mode. Since the spacetime dependence of the solutions
2779:
2706:
2673:
2078:
1156:
252:
171:
27:
Massless boson that must be present in a quantum system with spontaneously broken symmetry
4000:
2808:
2753:
4157:
4122:
4026:
3967:
3918:
3842:
3707:
3564:
3521:
3478:
3443:
3400:
3361:
17:
5247:
5161:
5118:
4854:
4642:
4415:
4337:
4332:
4254:
3304:
3144:{\displaystyle c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}
306:
248:
243:
105:
101:
70:
3873:
273:
can be viewed as Goldstone bosons arising due to the spontaneous symmetry breaking of
124:
are not massless, though they typically remain relatively light; they are then called
5261:
5205:
5057:
5024:
4816:
4786:
4718:
4577:
4357:
4284:
4269:
4165:
4130:
4095:
3416:
3206:
2710:
2689:
2681:
94:
4050:
3858:
5183:
4733:
4723:
4713:
4474:
4425:
4367:
4289:
4244:
3803:
3736:
2646:
464:
285:
167:
158:
4034:
1101:. The current connects the original vacuum with the Nambu–Goldstone boson state,
4077:
3715:
170:. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding
5198:
4966:
4869:
4864:
4781:
4776:
4706:
4660:
4617:
4582:
4541:
4433:
4397:
4259:
3308:
3009:
that spontaneously breaks a spacetime symmetry, the number of broken generators
2684:
with arbitrarily small masses but yet there is no Goldstone boson with exactly
4940:
4834:
4824:
4806:
4696:
4597:
4532:
4249:
3215:
2940:
2714:
1078:
and the ground state to a new, degenerate, ground state. Thus, a vacuum with
78:
3723:
3580:
3572:
3537:
3529:
3494:
606:—the hallmark of invariance), as evident in the charge of the current below.
30:"Goldstone theorem" redirects here. For the theorem on Feynman diagrams, see
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5072:
5042:
4935:
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4894:
4771:
4761:
4756:
4728:
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4279:
3816:
3486:
3451:
3369:
2654:
361:
360:, a constant. One way to impose a constraint of this sort is by including a
4042:
4109:
Volkov, D.V.; Akulov, V. (1973). "Is the neutrino a Goldstone particle?".
3817:"Goldstone mode and pair-breaking excitations in atomic Fermi superfluids"
3430:
Goldstone, J; Salam, Abdus; Weinberg, Steven (1962). "Broken Symmetries".
2831:). Note that the energy in the non relativistic condensed matter case is
2573:
so the integrand in the right hand side does not depend on the position.
200:
is longitudinal and it is the Goldstone boson of the spontaneously broken
5178:
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4945:
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4879:
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4537:
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4362:
4352:
4342:
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4312:
4239:
3210:
2669:
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1537:
282:
221:, i.e., spin waves in which the local magnetization direction oscillates.
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5001:
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Spontaneously broken global fermionic symmetries, which occur in some
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217:
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197:
58:
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https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.251602
3384:
1170:, at some ground state |0〉 chosen at the minimum of the potential,
4001:"Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem"
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5092:
5032:
4884:
4743:
4622:
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1817:
a state with approximately vanishing time derivative is produced,
1507:{\displaystyle {\frac {d\langle \delta \phi _{g}\rangle }{dt}}=0,}
205:
193:
121:
3242:
of broken supersymmetry, so that goldstinos are superpartners of
4859:
4791:
4751:
4327:
4322:
4066:"Spontaneous Symmetry Breaking and Phonon as the Goldstone Mode"
2945:
226:
4178:
1413:{\displaystyle {d \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0.}
2120:
Consider the correlation function of the charge with itself,
1195:
nonvanishing vacuum expectations of transformation increments
449:
The constraint, and the action, below, are invariant under a
2576:
Thus, its value is proportional to the total space volume,
628:
2649:
with arbitrarily small energies. Take for example a chiral
2869:
spontaneously broken generators may now give rise to the
3002:{\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }
4144:
Salam, A.; et al. (1974). "On Goldstone fermion".
1185:. In principle the vacuum should be the minimum of the
1058:{\displaystyle J_{\mu }=v^{2}\partial _{\mu }\theta ~.}
3161:
3083:
3042:
3015:
2978:
2909:
2883:
2811:
2782:
2756:
2723:
2582:
2129:
2102:
The argument requires both the vacuum and the charge
2091:
does not properly exist in the Hilbert space, unless
1960:
1826:
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1458:
1423:
Acting with the charge operator on the vacuum either
1338:
1221:
1012:
965:
856:
625:
566:
530:
482:
446:. This is called the "Abelian nonlinear σ-model".
377:
323:
2865:
as it would be in a relativistic case. However, two
2070:
become large leads to a contradiction. Consequently
1119:
In general, in a theory with several scalar fields,
429:{\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}
85:, and systematically generalized in the context of
5142:
5106:
5023:
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3036:minus the number non-trivial independent solutions
1321:and hence the corresponding zero-mass eigenvalues.
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3143:
3066:
3028:
3001:
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988:
946:
834:
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507:
428:
352:
3516:(6). American Physical Society (APS): 1900–1916.
2705:A version of Goldstone's theorem also applies to
113:if the spontaneously broken symmetry is not also
3702:(10). American Physical Society (APS): 408–410.
3683:Scholarpedia proof of goldstone theorem - kibble
551:(1) symmetry transformation effects a shift on
242:The longitudinal polarization components of the
3473:(4). American Physical Society (APS): 827–835.
3385:"Field Theories with Superconductor Solutions"
2637:does not properly exist in the Hilbert space.
4190:
602:(i.e. the above infinitesimal transformation
463:. The field can be redefined to give a real
61:that appear necessarily in models exhibiting
8:
4070:Gauge Invariance Approach to Acoustic Fields
3598:. Cambridge University Press. pp. 5–8.
3132:
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157:; i.e., its currents are conserved, but the
97:and are known as Anderson–Bogoliubov modes.
2688:. In other words, the Goldstone bosons are
2619:{\displaystyle \|Q|0\rangle \|^{2}=\infty }
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4921:
4674:
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4479:
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4393:
4227:
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4197:
4183:
4175:
588:{\displaystyle \delta \theta =\epsilon ~,}
81:mechanism, and subsequently elucidated by
4016:
3975:
3950:Lykken, Joseph; Spiropulu, Maria (2013).
3926:
3901:Lykken, Joseph; Spiropulu, Maria (2013).
3832:
3594:Baumann, D.; McAllister, L. (2015). "1".
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2626:— unless the symmetry is unbroken,
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3999:Low, I.; Manohar, A.V. (February 2002).
3186:{\displaystyle c_{a}({\boldsymbol {r}})}
3067:{\displaystyle c_{a}({\boldsymbol {r}})}
598:but does not preserve the ground state
3340:
3296:
3176:
3125:
3098:
3057:
2989:
2953:are spontaneously broken. However, the
524:is the Nambu–Goldstone boson (actually
2713:which is typically a boson and has no
1072:, resulting from this current shifts
7:
1536:, so that the theory cannot have a
1210:null eigenvectors of the mass matrix
353:{\displaystyle \phi ^{*}\phi =v^{2}}
255:. An analogous phenomenon occurs in
1208:, specify the relevant (Goldstone)
508:{\displaystyle \phi =ve^{i\theta }}
2613:
1933:Assuming a nonvanishing mass gap
1745:
1723:
1257:
1244:
1230:
1037:
900:
881:
791:
753:
673:
650:
25:
2957:is the Goldstone boson for both.
2106:to be translationally invariant,
1427:, if that is symmetric; else, if
5241:
5134:Timeline of particle discoveries
3209:models, lead to Nambu–Goldstone
2964:Breaking of spacetime symmetries
5288:Subatomic particles with spin 0
3952:"The future of the Higgs boson"
3903:"The future of the Higgs boson"
1143:nonvanishing vacuum expectation
999:The symmetry-induced conserved
3180:
3172:
3129:
3121:
3102:
3094:
3061:
3053:
2993:
2985:
2929:{\displaystyle E\propto p^{2}}
2743:{\displaystyle E\propto p^{n}}
2590:
2537:
2531:
2409:
2403:
2300:
2294:
2219:
2212:
2206:
2192:
2152:
2141:
2037:
2007:
1993:
1982:
1962:
1913:
1902:
1887:
1869:
1858:
1828:
1665:
1659:
1401:
1395:
912:
896:
893:
877:
803:
768:
765:
724:
669:
646:
411:
381:
1:
4035:10.1103/PhysRevLett.88.101602
3874:"Phonons as Goldstone bosons"
2949:particle number symmetry and
467:(i.e., a spin-zero particle)
132:pseudo–Nambu–Goldstone bosons
5150:History of subatomic physics
4166:10.1016/0370-2693(74)90637-6
4131:10.1016/0370-2693(73)90490-5
4078:10.1007/978-981-13-8751-7_11
3716:10.1103/physrevlett.16.408.2
3326:Goldstone sombrero potential
3246:particles in the theory, of
959:Note that the constant term
281:. These fluctuations in the
3596:Inflation and String Theory
2672:. The chiral symmetry is a
1128:, the Nambu–Goldstone mode
453:(1) phase transformation,
317:, with the constraint that
239:than typical hadron masses.
5304:
2939:As a second example, in a
2896:{\displaystyle E\propto p}
1547:is applied to the vacuum,
989:{\displaystyle m^{2}v^{2}}
473:without any constraint by
290:cosmic structure formation
77:within the context of the
69:. They were discovered by
29:
5239:
4924:
3559:(11–12). Wiley: 605–682.
2680:bosons have a continuous
439:and taking the limit as
275:time translation symmetry
5167:mathematical formulation
4762:Eta and eta prime mesons
4064:Gan, Woon Siong (2019).
3573:10.1002/prop.19580061102
3530:10.1103/physrev.112.1900
3280:Vacuum expectation value
3201:Nambu–Goldstone fermions
2701:Nonrelativistic theories
1324:
540:{\displaystyle v\theta }
364:interaction term in its
91:condensed matter physics
43:condensed matter physics
18:Goldstone's theorem
4829:Double-charm tetraquark
3696:Physical Review Letters
3553:Fortschritte der Physik
3487:10.1103/physrev.110.827
3452:10.1103/PhysRev.127.965
3370:10.1103/PhysRev.117.648
2873:Nambu–Goldstone boson.
267:Primordial fluctuations
232:pseudo-Goldstone bosons
141:
127:pseudo-Goldstone bosons
3324:It corresponds to the
3260:Pseudo-Goldstone boson
3240:nonlinear realizations
3187:
3145:
3068:
3030:
3003:
2930:
2897:
2825:
2799:
2770:
2744:
2620:
2564:
2086:Fabri–Picasso Theorem.
2057:
1924:
1808:
1508:
1425:annihilates the vacuum
1414:
1312:
1193:By contrast, however,
1059:
990:
948:
836:
604:does not annihilate it
589:
541:
509:
430:
354:
261:Ginzburg–Landau theory
51:Nambu–Goldstone bosons
32:linked-cluster theorem
5226:Wave–particle duality
5216:Relativistic particle
4353:Electron antineutrino
4011:(10): 101602–101605.
3383:Goldstone, J (1961).
3275:Mermin–Wagner theorem
3188:
3146:
3069:
3031:
3029:{\displaystyle T^{a}}
3004:
2931:
2898:
2826:
2800:
2771:
2745:
2657:model with a nonzero
2621:
2565:
2058:
1925:
1809:
1509:
1445:, are, nevertheless,
1415:
1313:
1060:
991:
949:
837:
590:
542:
510:
431:
355:
237:significantly smaller
79:BCS superconductivity
67:continuous symmetries
63:spontaneous breakdown
5278:Mathematical physics
5273:Quantum field theory
4456:Faddeev–Popov ghosts
4206:Particles in physics
3804:Fabri dispense 1965
3737:Fabri dispense 1965
3159:
3081:
3040:
3013:
2976:
2907:
2881:
2809:
2798:{\displaystyle E=pc}
2780:
2754:
2721:
2580:
2127:
1958:
1824:
1554:
1456:
1336:
1325:Goldstone's argument
1219:
1010:
963:
854:
623:
564:
528:
480:
375:
321:
155:spontaneously broken
87:quantum field theory
5231:Particle chauvinism
5174:Subatomic particles
4158:1974PhLB...49..465S
4123:1973PhLB...46..109V
4027:2002PhRvL..88j1602L
3968:2013PhT....66l..28L
3919:2013PhT....66l..28L
3843:2017NatPh..13..943H
3708:1966PhRvL..16..408F
3565:1958ForPh...6..605B
3522:1958PhRv..112.1900A
3479:1958PhRv..110..827A
3444:1962PhRv..127..965G
3401:1961NCim...19..154G
3362:1960PhRv..117..648N
2824:{\displaystyle p=0}
2769:{\displaystyle p=0}
1187:effective potential
151:continuous symmetry
149:examines a generic
147:Goldstone's theorem
142:Goldstone's theorem
4072:. pp. 59–62.
3753:2022-01-14 at the
3409:10.1007/BF02812722
3219:. These have spin
3183:
3141:
3064:
3026:
2999:
2926:
2893:
2821:
2795:
2766:
2740:
2616:
2560:
2558:
2053:
1920:
1804:
1517:thus generating a
1504:
1410:
1308:
1055:
986:
944:
832:
614:Lagrangian density
612:The corresponding
585:
537:
505:
426:
366:Lagrangian density
350:
288:subsequently seed
279:de Sitter universe
5255:
5254:
5211:Massless particle
5019:
5018:
5015:
5014:
4980:
4979:
4843:
4842:
4655:
4654:
4651:
4650:
4603:Magnetic monopole
4551:
4550:
4442:
4441:
4383:
4382:
4363:Muon antineutrino
4348:Electron neutrino
4087:978-981-13-8750-0
3977:10.1063/PT.3.2212
3928:10.1063/PT.3.2212
3851:10.1038/nphys4187
3303:In theories with
3285:Noether's theorem
2951:Galilean symmetry
2856:
2847:
2633:. Consequently,
1979:
1884:
1845:
1787:
1719:
1645:
1598:
1570:
1493:
1373:
1352:
1304:
1271:
1051:
940:
875:
722:
644:
581:
422:
257:superconductivity
202:Galilean symmetry
115:broken explicitly
83:Jeffrey Goldstone
16:(Redirected from
5295:
5283:Physics theorems
5245:
5221:Virtual particle
4992:Mesonic molecule
4926:
4922:
4767:Bottom eta meson
4675:
4666:
4638:W′ and Z′ bosons
4628:Sterile neutrino
4613:Majorana fermion
4480:
4471:
4394:
4373:Tau antineutrino
4228:
4219:
4199:
4192:
4185:
4176:
4170:
4169:
4141:
4135:
4134:
4106:
4100:
4099:
4061:
4055:
4054:
4020:
3996:
3990:
3989:
3979:
3947:
3941:
3940:
3930:
3898:
3892:
3891:
3889:
3887:
3878:
3869:
3863:
3862:
3836:
3812:
3806:
3801:
3795:
3794:
3792:
3791:
3782:. Archived from
3776:
3770:
3764:
3758:
3745:
3739:
3734:
3728:
3727:
3691:
3685:
3680:
3674:
3671:
3665:
3662:
3656:
3653:
3647:
3644:
3638:
3635:
3629:
3626:
3620:
3616:
3610:
3609:
3591:
3585:
3584:
3548:
3542:
3541:
3505:
3499:
3498:
3462:
3456:
3455:
3427:
3421:
3420:
3380:
3374:
3373:
3345:
3329:
3322:
3316:
3301:
3234:
3232:
3231:
3228:
3225:
3192:
3190:
3189:
3184:
3179:
3171:
3170:
3150:
3148:
3147:
3142:
3128:
3114:
3113:
3101:
3093:
3092:
3073:
3071:
3070:
3065:
3060:
3052:
3051:
3035:
3033:
3032:
3027:
3025:
3024:
3008:
3006:
3005:
3000:
2992:
2935:
2933:
2932:
2927:
2925:
2924:
2902:
2900:
2899:
2894:
2864:
2858:
2857:
2854:
2848:
2845:
2830:
2828:
2827:
2822:
2804:
2802:
2801:
2796:
2775:
2773:
2772:
2767:
2750:and is zero for
2749:
2747:
2746:
2741:
2739:
2738:
2636:
2632:
2625:
2623:
2622:
2617:
2609:
2608:
2593:
2569:
2567:
2566:
2561:
2559:
2555:
2551:
2547:
2543:
2530:
2529:
2504:
2503:
2485:
2481:
2477:
2473:
2469:
2468:
2467:
2449:
2448:
2430:
2429:
2402:
2401:
2392:
2391:
2360:
2359:
2341:
2337:
2333:
2329:
2325:
2321:
2320:
2293:
2292:
2283:
2282:
2251:
2250:
2232:
2222:
2205:
2204:
2195:
2181:
2180:
2155:
2144:
2116:
2112:
2105:
2097:
2090:
2073:
2069:
2062:
2060:
2059:
2054:
2040:
2032:
2031:
2010:
1996:
1992:
1985:
1980:
1978:
1967:
1950:
1941:
1929:
1927:
1926:
1921:
1916:
1912:
1905:
1900:
1899:
1885:
1877:
1872:
1868:
1861:
1856:
1855:
1846:
1844:
1833:
1813:
1811:
1810:
1805:
1794:
1790:
1789:
1788:
1786:
1785:
1784:
1771:
1770:
1761:
1744:
1743:
1722:
1721:
1720:
1718:
1717:
1716:
1703:
1702:
1693:
1683:
1682:
1658:
1657:
1648:
1647:
1646:
1644:
1643:
1642:
1629:
1628:
1619:
1609:
1608:
1599:
1597:
1586:
1581:
1580:
1571:
1569:
1558:
1546:
1535:
1524:
1513:
1511:
1510:
1505:
1494:
1492:
1484:
1480:
1479:
1460:
1444:
1419:
1417:
1416:
1411:
1394:
1393:
1384:
1383:
1374:
1372:
1361:
1353:
1351:
1340:
1317:
1315:
1314:
1309:
1302:
1292:
1291:
1276:
1272:
1270:
1269:
1268:
1256:
1255:
1242:
1238:
1237:
1227:
1207:
1184:
1169:
1159:, for vanishing
1154:
1136:
1127:
1115:
1100:
1088:different vacuum
1086:will shift to a
1085:
1077:
1064:
1062:
1061:
1056:
1049:
1045:
1044:
1035:
1034:
1022:
1021:
1003:(1) current is
995:
993:
992:
987:
985:
984:
975:
974:
953:
951:
950:
945:
938:
937:
936:
927:
926:
908:
907:
889:
888:
876:
871:
870:
861:
841:
839:
838:
833:
828:
827:
818:
817:
799:
798:
789:
788:
761:
760:
751:
750:
723:
715:
707:
706:
697:
696:
681:
680:
668:
667:
658:
657:
645:
637:
632:
631:
601:
594:
592:
591:
586:
579:
556:
546:
544:
543:
538:
523:
514:
512:
511:
506:
504:
503:
472:
462:
445:
435:
433:
432:
427:
420:
419:
418:
409:
408:
393:
392:
359:
357:
356:
351:
349:
348:
333:
332:
316:
93:such bosons are
75:particle physics
47:Goldstone bosons
21:
5303:
5302:
5298:
5297:
5296:
5294:
5293:
5292:
5258:
5257:
5256:
5251:
5235:
5189:Nuclear physics
5138:
5102:
5038:Davydov soliton
5011:
4976:
4950:
4911:
4839:
4811:
4738:
4647:
4547:
4506:
4460:
4438:
4420:
4379:
4299:
4208:
4203:
4173:
4146:Physics Letters
4143:
4142:
4138:
4111:Physics Letters
4108:
4107:
4103:
4088:
4063:
4062:
4058:
4005:Phys. Rev. Lett
3998:
3997:
3993:
3949:
3948:
3944:
3900:
3899:
3895:
3885:
3883:
3876:
3871:
3870:
3866:
3827:(10): 943–946.
3814:
3813:
3809:
3802:
3798:
3789:
3787:
3778:
3777:
3773:
3765:
3761:
3755:Wayback Machine
3746:
3742:
3735:
3731:
3693:
3692:
3688:
3681:
3677:
3672:
3668:
3663:
3659:
3654:
3650:
3645:
3641:
3636:
3632:
3627:
3623:
3617:
3613:
3606:
3593:
3592:
3588:
3550:
3549:
3545:
3510:Physical Review
3507:
3506:
3502:
3467:Physical Review
3464:
3463:
3459:
3432:Physical Review
3429:
3428:
3424:
3382:
3381:
3377:
3350:Physical Review
3347:
3346:
3342:
3338:
3333:
3332:
3323:
3319:
3313:Higgs mechanism
3302:
3298:
3293:
3270:Higgs mechanism
3256:
3229:
3226:
3223:
3222:
3220:
3203:
3162:
3157:
3156:
3105:
3084:
3079:
3078:
3043:
3038:
3037:
3016:
3011:
3010:
2974:
2973:
2966:
2916:
2905:
2904:
2879:
2878:
2860:
2853:
2844:
2832:
2807:
2806:
2778:
2777:
2752:
2751:
2730:
2719:
2718:
2707:nonrelativistic
2703:
2698:
2674:global symmetry
2652:
2643:
2634:
2627:
2600:
2578:
2577:
2557:
2556:
2521:
2520:
2516:
2512:
2508:
2495:
2483:
2482:
2450:
2434:
2412:
2393:
2377:
2376:
2372:
2368:
2364:
2351:
2339:
2338:
2303:
2284:
2268:
2267:
2263:
2259:
2255:
2242:
2230:
2229:
2196:
2172:
2162:
2125:
2124:
2114:
2107:
2103:
2092:
2088:
2079:Higgs mechanism
2076:
2071:
2067:
2023:
1971:
1965:
1961:
1956:
1955:
1949:
1943:
1940:
1934:
1891:
1890:
1886:
1847:
1837:
1831:
1827:
1822:
1821:
1776:
1772:
1762:
1752:
1748:
1735:
1708:
1704:
1694:
1684:
1674:
1649:
1634:
1630:
1620:
1610:
1600:
1590:
1572:
1562:
1552:
1551:
1544:
1531:
1518:
1485:
1471:
1461:
1454:
1453:
1442:
1435:
1385:
1375:
1365:
1344:
1334:
1333:
1327:
1319:
1283:
1260:
1247:
1243:
1229:
1228:
1222:
1217:
1216:
1205:
1198:
1182:
1171:
1166:
1160:
1157:order parameter
1151:
1145:
1135:
1129:
1126:
1120:
1109:
1102:
1091:
1079:
1073:
1036:
1026:
1013:
1008:
1007:
976:
966:
961:
960:
928:
918:
899:
880:
862:
852:
851:
819:
809:
790:
777:
752:
736:
698:
688:
672:
659:
649:
621:
620:
599:
562:
561:
552:
526:
525:
519:
492:
478:
477:
468:
454:
440:
410:
400:
384:
373:
372:
340:
324:
319:
318:
312:
303:
253:Higgs mechanism
189:
184:
172:order parameter
144:
106:quantum numbers
35:
28:
23:
22:
15:
12:
11:
5:
5301:
5299:
5291:
5290:
5285:
5280:
5275:
5270:
5260:
5259:
5253:
5252:
5248:Physics portal
5240:
5237:
5236:
5234:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5202:
5201:
5191:
5186:
5181:
5176:
5171:
5170:
5169:
5162:Standard Model
5159:
5158:
5157:
5146:
5144:
5140:
5139:
5137:
5136:
5131:
5129:Quasiparticles
5126:
5121:
5116:
5110:
5108:
5104:
5103:
5101:
5100:
5095:
5090:
5085:
5080:
5075:
5070:
5065:
5060:
5055:
5050:
5045:
5040:
5035:
5029:
5027:
5025:Quasiparticles
5021:
5020:
5017:
5016:
5013:
5012:
5010:
5009:
5004:
4999:
4994:
4988:
4986:
4982:
4981:
4978:
4977:
4975:
4974:
4969:
4964:
4958:
4956:
4952:
4951:
4949:
4948:
4943:
4938:
4932:
4930:
4919:
4913:
4912:
4910:
4909:
4904:
4899:
4898:
4897:
4892:
4887:
4882:
4877:
4872:
4862:
4857:
4851:
4849:
4845:
4844:
4841:
4840:
4838:
4837:
4832:
4821:
4819:
4817:Exotic hadrons
4813:
4812:
4810:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4759:
4754:
4748:
4746:
4740:
4739:
4737:
4736:
4731:
4726:
4721:
4716:
4711:
4710:
4709:
4704:
4699:
4694:
4683:
4681:
4672:
4663:
4657:
4656:
4653:
4652:
4649:
4648:
4646:
4645:
4643:X and Y bosons
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4565:
4559:
4557:
4553:
4552:
4549:
4548:
4546:
4545:
4535:
4530:
4525:
4520:
4514:
4512:
4508:
4507:
4505:
4504:
4499:
4494:
4488:
4486:
4477:
4468:
4462:
4461:
4459:
4458:
4452:
4450:
4444:
4443:
4440:
4439:
4437:
4436:
4430:
4428:
4422:
4421:
4419:
4418:
4416:W and Z bosons
4413:
4408:
4402:
4400:
4391:
4385:
4384:
4381:
4380:
4378:
4377:
4376:
4375:
4370:
4365:
4360:
4355:
4350:
4340:
4335:
4330:
4325:
4320:
4315:
4309:
4307:
4301:
4300:
4298:
4297:
4292:
4287:
4282:
4277:
4272:
4270:Strange (quark
4267:
4262:
4257:
4252:
4247:
4242:
4236:
4234:
4225:
4216:
4210:
4209:
4204:
4202:
4201:
4194:
4187:
4179:
4172:
4171:
4152:(5): 465–467.
4136:
4117:(1): 109–110.
4101:
4086:
4056:
4018:hep-th/0110285
3991:
3942:
3893:
3872:Leutwyler, H.
3864:
3821:Nature Physics
3807:
3796:
3771:
3759:
3740:
3729:
3686:
3675:
3666:
3657:
3648:
3639:
3630:
3621:
3611:
3605:978-1107089693
3604:
3586:
3543:
3500:
3457:
3438:(3): 965–970.
3422:
3395:(1): 154–164.
3375:
3356:(3): 648–663.
3339:
3337:
3334:
3331:
3330:
3317:
3311:, through the
3305:gauge symmetry
3295:
3294:
3292:
3289:
3288:
3287:
3282:
3277:
3272:
3267:
3262:
3255:
3252:
3207:supersymmetric
3202:
3199:
3182:
3178:
3174:
3169:
3165:
3152:
3151:
3140:
3137:
3134:
3131:
3127:
3123:
3120:
3117:
3112:
3108:
3104:
3100:
3096:
3091:
3087:
3063:
3059:
3055:
3050:
3046:
3023:
3019:
2998:
2995:
2991:
2987:
2984:
2981:
2965:
2962:
2923:
2919:
2915:
2912:
2892:
2889:
2886:
2820:
2817:
2814:
2794:
2791:
2788:
2785:
2765:
2762:
2759:
2737:
2733:
2729:
2726:
2702:
2699:
2697:
2694:
2690:infraparticles
2650:
2642:
2641:Infraparticles
2639:
2615:
2612:
2607:
2603:
2599:
2596:
2592:
2588:
2585:
2571:
2570:
2554:
2550:
2546:
2542:
2539:
2536:
2533:
2528:
2524:
2519:
2515:
2511:
2507:
2502:
2498:
2494:
2491:
2488:
2486:
2484:
2480:
2476:
2472:
2466:
2463:
2460:
2457:
2453:
2447:
2444:
2441:
2437:
2433:
2428:
2425:
2422:
2419:
2415:
2411:
2408:
2405:
2400:
2396:
2390:
2387:
2384:
2380:
2375:
2371:
2367:
2363:
2358:
2354:
2350:
2347:
2344:
2342:
2340:
2336:
2332:
2328:
2324:
2319:
2316:
2313:
2310:
2306:
2302:
2299:
2296:
2291:
2287:
2281:
2278:
2275:
2271:
2266:
2262:
2258:
2254:
2249:
2245:
2241:
2238:
2235:
2233:
2231:
2228:
2225:
2221:
2217:
2214:
2211:
2208:
2203:
2199:
2194:
2190:
2187:
2184:
2179:
2175:
2171:
2168:
2165:
2163:
2161:
2158:
2154:
2150:
2147:
2143:
2139:
2136:
2133:
2132:
2100:
2099:
2074:
2064:
2063:
2052:
2049:
2046:
2043:
2039:
2035:
2030:
2026:
2022:
2019:
2016:
2013:
2009:
2005:
2002:
1999:
1995:
1991:
1988:
1984:
1977:
1974:
1970:
1964:
1947:
1938:
1931:
1930:
1919:
1915:
1911:
1908:
1904:
1898:
1894:
1889:
1883:
1880:
1875:
1871:
1867:
1864:
1860:
1854:
1850:
1843:
1840:
1836:
1830:
1815:
1814:
1803:
1800:
1797:
1793:
1783:
1779:
1775:
1769:
1765:
1759:
1755:
1751:
1747:
1742:
1738:
1734:
1731:
1728:
1725:
1715:
1711:
1707:
1701:
1697:
1691:
1687:
1681:
1677:
1673:
1670:
1667:
1664:
1661:
1656:
1652:
1641:
1637:
1633:
1627:
1623:
1617:
1613:
1607:
1603:
1596:
1593:
1589:
1584:
1579:
1575:
1568:
1565:
1561:
1515:
1514:
1503:
1500:
1497:
1491:
1488:
1483:
1478:
1474:
1470:
1467:
1464:
1447:time-invariant
1440:
1421:
1420:
1409:
1406:
1403:
1400:
1397:
1392:
1388:
1382:
1378:
1371:
1368:
1364:
1359:
1356:
1350:
1347:
1343:
1326:
1323:
1307:
1301:
1298:
1295:
1290:
1286:
1282:
1279:
1275:
1267:
1263:
1259:
1254:
1250:
1246:
1241:
1236:
1232:
1225:
1214:
1203:
1180:
1164:
1149:
1133:
1124:
1107:
1066:
1065:
1054:
1048:
1043:
1039:
1033:
1029:
1025:
1020:
1016:
983:
979:
973:
969:
957:
956:
955:
954:
943:
935:
931:
925:
921:
917:
914:
911:
906:
902:
898:
895:
892:
887:
883:
879:
874:
869:
865:
859:
843:
842:
831:
826:
822:
816:
812:
808:
805:
802:
797:
793:
787:
784:
780:
776:
773:
770:
767:
764:
759:
755:
749:
746:
743:
739:
735:
732:
729:
726:
721:
718:
713:
710:
705:
701:
695:
691:
687:
684:
679:
675:
671:
666:
662:
656:
652:
648:
643:
640:
635:
630:
596:
595:
584:
578:
575:
572:
569:
536:
533:
516:
515:
502:
499:
495:
491:
488:
485:
437:
436:
425:
417:
413:
407:
403:
399:
396:
391:
387:
383:
380:
347:
343:
339:
336:
331:
327:
302:
299:
298:
297:
293:
264:
249:Standard Model
244:W and Z bosons
240:
222:
209:
188:
185:
183:
180:
143:
140:
95:quasiparticles
71:Yoichiro Nambu
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5300:
5289:
5286:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5265:
5263:
5250:
5249:
5244:
5238:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5206:Exotic matter
5204:
5200:
5197:
5196:
5195:
5194:Eightfold way
5192:
5190:
5187:
5185:
5184:Antiparticles
5182:
5180:
5177:
5175:
5172:
5168:
5165:
5164:
5163:
5160:
5156:
5153:
5152:
5151:
5148:
5147:
5145:
5141:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5111:
5109:
5105:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5054:
5051:
5049:
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5030:
5028:
5026:
5022:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4989:
4987:
4983:
4973:
4970:
4968:
4965:
4963:
4960:
4959:
4957:
4953:
4947:
4944:
4942:
4939:
4937:
4934:
4933:
4931:
4927:
4923:
4920:
4918:
4914:
4908:
4905:
4903:
4900:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4867:
4866:
4863:
4861:
4858:
4856:
4855:Atomic nuclei
4853:
4852:
4850:
4846:
4836:
4833:
4830:
4826:
4823:
4822:
4820:
4818:
4814:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4787:Upsilon meson
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4749:
4747:
4745:
4741:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4719:Lambda baryon
4717:
4715:
4712:
4708:
4705:
4703:
4700:
4698:
4695:
4693:
4690:
4689:
4688:
4685:
4684:
4682:
4680:
4676:
4673:
4671:
4667:
4664:
4662:
4658:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4578:Dual graviton
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4560:
4558:
4554:
4543:
4539:
4536:
4534:
4531:
4529:
4526:
4524:
4521:
4519:
4516:
4515:
4513:
4509:
4503:
4500:
4498:
4495:
4493:
4490:
4489:
4487:
4485:
4481:
4478:
4476:
4475:Superpartners
4472:
4469:
4467:
4463:
4457:
4454:
4453:
4451:
4449:
4445:
4435:
4432:
4431:
4429:
4427:
4423:
4417:
4414:
4412:
4409:
4407:
4404:
4403:
4401:
4399:
4395:
4392:
4390:
4386:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4358:Muon neutrino
4356:
4354:
4351:
4349:
4346:
4345:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4310:
4308:
4306:
4302:
4296:
4293:
4291:
4290:Bottom (quark
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4251:
4248:
4246:
4243:
4241:
4238:
4237:
4235:
4233:
4229:
4226:
4224:
4220:
4217:
4215:
4211:
4207:
4200:
4195:
4193:
4188:
4186:
4181:
4180:
4177:
4167:
4163:
4159:
4155:
4151:
4147:
4140:
4137:
4132:
4128:
4124:
4120:
4116:
4112:
4105:
4102:
4097:
4093:
4089:
4083:
4079:
4075:
4071:
4067:
4060:
4057:
4052:
4048:
4044:
4040:
4036:
4032:
4028:
4024:
4019:
4014:
4010:
4006:
4002:
3995:
3992:
3987:
3983:
3978:
3973:
3969:
3965:
3962:(12): 28–33.
3961:
3957:
3956:Physics Today
3953:
3946:
3943:
3938:
3934:
3929:
3924:
3920:
3916:
3913:(12): 28–33.
3912:
3908:
3907:Physics Today
3904:
3897:
3894:
3882:
3875:
3868:
3865:
3860:
3856:
3852:
3848:
3844:
3840:
3835:
3830:
3826:
3822:
3818:
3811:
3808:
3805:
3800:
3797:
3786:on 2022-01-14
3785:
3781:
3775:
3772:
3769:
3763:
3760:
3756:
3752:
3749:
3744:
3741:
3738:
3733:
3730:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3690:
3687:
3684:
3679:
3676:
3670:
3667:
3661:
3658:
3652:
3649:
3643:
3640:
3634:
3631:
3625:
3622:
3615:
3612:
3607:
3601:
3597:
3590:
3587:
3582:
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3547:
3544:
3539:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3504:
3501:
3496:
3492:
3488:
3484:
3480:
3476:
3472:
3468:
3461:
3458:
3453:
3449:
3445:
3441:
3437:
3433:
3426:
3423:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3389:Nuovo Cimento
3386:
3379:
3376:
3371:
3367:
3363:
3359:
3355:
3351:
3344:
3341:
3335:
3327:
3321:
3318:
3314:
3310:
3306:
3300:
3297:
3290:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3257:
3253:
3251:
3249:
3245:
3241:
3236:
3218:
3217:
3212:
3208:
3200:
3198:
3195:
3167:
3163:
3138:
3135:
3118:
3110:
3106:
3089:
3085:
3077:
3076:
3075:
3048:
3044:
3021:
3017:
2982:
2971:
2963:
2961:
2958:
2956:
2952:
2948:
2947:
2942:
2937:
2921:
2917:
2913:
2910:
2890:
2887:
2884:
2874:
2872:
2868:
2863:
2852:
2843:
2839:
2835:
2818:
2815:
2812:
2805:and zero for
2792:
2789:
2786:
2783:
2763:
2760:
2757:
2735:
2731:
2727:
2724:
2716:
2712:
2711:quasiparticle
2708:
2700:
2695:
2693:
2691:
2687:
2683:
2682:mass spectrum
2679:
2675:
2671:
2667:
2663:
2660:
2656:
2648:
2647:vacuum states
2640:
2638:
2630:
2610:
2605:
2594:
2586:
2574:
2552:
2548:
2544:
2540:
2534:
2526:
2522:
2517:
2513:
2509:
2505:
2500:
2496:
2492:
2489:
2487:
2478:
2474:
2470:
2464:
2461:
2458:
2455:
2451:
2445:
2442:
2439:
2435:
2431:
2426:
2423:
2420:
2417:
2413:
2406:
2398:
2394:
2388:
2385:
2382:
2378:
2373:
2369:
2365:
2361:
2356:
2352:
2348:
2345:
2343:
2334:
2330:
2326:
2322:
2317:
2314:
2311:
2308:
2304:
2297:
2289:
2285:
2279:
2276:
2273:
2269:
2264:
2260:
2256:
2252:
2247:
2243:
2239:
2236:
2234:
2223:
2215:
2209:
2201:
2197:
2188:
2182:
2177:
2173:
2169:
2166:
2164:
2156:
2148:
2145:
2137:
2123:
2122:
2121:
2118:
2110:
2095:
2087:
2084:
2083:
2082:
2080:
2050:
2041:
2028:
2024:
2020:
2011:
2003:
1997:
1986:
1975:
1972:
1968:
1954:
1953:
1952:
1946:
1937:
1917:
1906:
1896:
1892:
1881:
1878:
1873:
1862:
1852:
1848:
1841:
1838:
1834:
1820:
1819:
1818:
1801:
1798:
1795:
1791:
1781:
1777:
1773:
1767:
1763:
1757:
1753:
1749:
1740:
1736:
1732:
1729:
1726:
1713:
1709:
1705:
1699:
1695:
1689:
1685:
1679:
1675:
1671:
1668:
1662:
1654:
1650:
1639:
1635:
1631:
1625:
1621:
1615:
1611:
1605:
1601:
1594:
1591:
1587:
1582:
1577:
1573:
1566:
1563:
1559:
1550:
1549:
1548:
1541:
1539:
1534:
1527:
1522:
1501:
1498:
1495:
1489:
1486:
1476:
1472:
1468:
1462:
1452:
1451:
1450:
1448:
1439:
1432:
1430:
1426:
1407:
1404:
1398:
1390:
1386:
1380:
1376:
1369:
1366:
1362:
1357:
1354:
1348:
1345:
1341:
1332:
1331:
1330:
1322:
1318:
1305:
1299:
1296:
1288:
1284:
1280:
1273:
1265:
1261:
1252:
1248:
1239:
1234:
1223:
1213:
1211:
1202:
1196:
1191:
1188:
1179:
1175:
1167:
1158:
1152:
1144:
1140:
1132:
1123:
1117:
1113:
1106:
1099:
1095:
1089:
1083:
1076:
1071:
1052:
1046:
1041:
1031:
1027:
1023:
1018:
1014:
1006:
1005:
1004:
1002:
997:
981:
977:
971:
967:
941:
933:
929:
923:
919:
915:
909:
904:
890:
885:
872:
867:
863:
857:
850:
849:
848:
847:
846:
829:
824:
820:
814:
810:
806:
800:
795:
785:
782:
778:
774:
771:
762:
757:
747:
744:
741:
737:
733:
730:
727:
719:
716:
711:
708:
703:
699:
693:
689:
685:
682:
677:
664:
660:
654:
641:
638:
633:
619:
618:
617:
615:
610:
607:
605:
582:
576:
573:
570:
567:
560:
559:
558:
555:
550:
534:
531:
522:
500:
497:
493:
489:
486:
483:
476:
475:
474:
471:
466:
461:
457:
452:
447:
443:
423:
415:
405:
401:
397:
394:
389:
385:
378:
371:
370:
369:
367:
363:
345:
341:
337:
334:
329:
325:
315:
311:
308:
300:
294:
291:
287:
284:
280:
276:
272:
268:
265:
262:
258:
254:
250:
245:
241:
238:
233:
229:
228:
223:
220:
219:
214:
210:
207:
203:
199:
195:
191:
190:
186:
181:
179:
175:
173:
169:
164:
160:
156:
152:
148:
139:
137:
134:(abbreviated
133:
129:
128:
123:
118:
116:
112:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
5246:
4917:Hypothetical
4865:Exotic atoms
4734:Omega baryon
4724:Sigma baryon
4714:Delta baryon
4466:Hypothetical
4448:Ghost fields
4434:Higgs boson
4368:Tau neutrino
4260:Charm (quark
4149:
4145:
4139:
4114:
4110:
4104:
4069:
4059:
4008:
4004:
3994:
3959:
3955:
3945:
3910:
3906:
3896:
3884:. Retrieved
3880:
3867:
3824:
3820:
3810:
3799:
3788:. Retrieved
3784:the original
3774:
3762:
3757:- min. 30-60
3743:
3732:
3699:
3695:
3689:
3678:
3669:
3660:
3651:
3642:
3633:
3624:
3614:
3595:
3589:
3556:
3552:
3546:
3513:
3509:
3503:
3470:
3466:
3460:
3435:
3431:
3425:
3392:
3388:
3378:
3353:
3349:
3343:
3320:
3309:gauge bosons
3299:
3247:
3243:
3237:
3214:
3204:
3196:
3153:
2967:
2959:
2944:
2938:
2875:
2870:
2866:
2861:
2850:
2841:
2837:
2833:
2704:
2644:
2628:
2575:
2572:
2119:
2108:
2101:
2093:
2085:
2065:
1944:
1935:
1932:
1816:
1542:
1532:
1528:
1520:
1516:
1446:
1437:
1433:
1428:
1424:
1422:
1328:
1320:
1215:
1209:
1200:
1194:
1192:
1177:
1173:
1162:
1147:
1142:
1130:
1121:
1118:
1111:
1104:
1097:
1093:
1087:
1081:
1074:
1069:
1068:The charge,
1067:
1000:
998:
958:
844:
616:is given by
611:
608:
603:
597:
553:
548:
547:is) and the
520:
517:
469:
465:scalar field
459:
455:
450:
448:
441:
438:
313:
310:scalar field
304:
286:scalar field
251:through the
236:
225:
216:
176:
168:ground state
159:ground state
146:
145:
135:
131:
125:
119:
99:
54:
50:
46:
36:
5199:Quark model
4967:Theta meson
4870:Positronium
4782:Omega meson
4777:J/psi meson
4707:Antineutron
4618:Dark photon
4583:Graviphoton
4542:Stop squark
4250:Down (quark
3881:cds.cern.ch
2943:, both the
305:Consider a
5262:Categories
4941:Heptaquark
4902:Superatoms
4835:Pentaquark
4825:Tetraquark
4807:Quarkonium
4697:Antiproton
4598:Leptoquark
4533:Neutralino
4295:antiquark)
4285:antiquark)
4280:Top (quark
4275:antiquark)
4265:antiquark)
4255:antiquark)
4245:antiquark)
4214:Elementary
3886:4 November
3834:1707.00406
3790:2022-01-14
3336:References
3216:goldstinos
2941:superfluid
2715:energy gap
2696:Extensions
1114:〉≠ 0
845:and thus
557:, namely
5179:Particles
5124:Particles
5083:Polariton
5073:Plasmaron
5043:Dropleton
4936:Hexaquark
4907:Molecules
4895:Protonium
4772:Phi meson
4757:Rho meson
4729:Xi baryon
4661:Composite
4497:Gravitino
4240:Up (quark
4096:201256113
3724:0031-9007
3581:0015-8208
3538:0031-899X
3495:0031-899X
3417:120409034
3133:⟩
3119:ϕ
3116:⟨
2997:⟩
2983:ϕ
2980:⟨
2914:∝
2888:∝
2867:different
2728:∝
2686:zero mass
2678:composite
2666:conformal
2664:which is
2655:super QCD
2614:∞
2602:‖
2598:⟩
2584:‖
2493:∫
2456:−
2418:−
2349:∫
2309:−
2240:∫
2227:⟩
2186:⟨
2170:∫
2160:⟩
2135:⟨
2048:‖
2045:⟩
2042:θ
2034:‖
2021:≥
2018:‖
2015:⟩
2012:θ
2001:‖
1990:⟩
1987:θ
1910:⟩
1874:≈
1866:⟩
1796:⋅
1758:−
1746:∇
1737:∫
1727:⋅
1724:∇
1690:−
1676:∫
1672:−
1616:−
1602:∫
1482:⟩
1473:ϕ
1469:δ
1466:⟨
1377:∫
1294:⟩
1285:ϕ
1281:δ
1278:⟨
1262:ϕ
1258:∂
1249:ϕ
1245:∂
1231:∂
1047:θ
1042:μ
1038:∂
1019:μ
916:−
910:θ
905:μ
901:∂
891:θ
886:μ
882:∂
807:−
801:θ
796:μ
792:∂
786:θ
763:θ
758:μ
754:∂
748:θ
742:−
728:−
709:ϕ
704:∗
700:ϕ
686:−
683:ϕ
678:μ
674:∂
665:∗
661:ϕ
655:μ
651:∂
577:ϵ
571:θ
568:δ
535:θ
501:θ
484:ϕ
398:−
395:ϕ
390:∗
386:ϕ
379:λ
362:potential
335:ϕ
330:∗
326:ϕ
271:inflation
153:which is
5155:timeline
5007:R-hadron
4962:Glueball
4946:Skyrmion
4880:Tauonium
4593:Inflaton
4588:Graviton
4568:Curvaton
4538:Sfermion
4528:Higgsino
4523:Chargino
4484:Gauginos
4343:Neutrino
4328:Antimuon
4318:Positron
4313:Electron
4223:Fermions
4051:15997403
4043:11909340
3859:59392755
3780:"min 42"
3751:Archived
3254:See also
3248:any spin
3211:fermions
2859:and not
2553:⟩
2510:⟨
2479:⟩
2366:⟨
2335:⟩
2257:⟨
2111:|0〉 = 0
2066:Letting
1994:‖
1963:‖
1914:‖
1888:‖
1870:‖
1829:‖
1538:mass gap
1274:⟩
1224:⟨
1139:massless
283:inflaton
230:are the
182:Examples
111:massless
102:spinless
39:particle
5143:Related
5114:Baryons
5088:Polaron
5078:Plasmon
5053:Fracton
5048:Exciton
5002:Diquark
4997:Pomeron
4972:T meson
4929:Baryons
4890:Pionium
4875:Muonium
4802:D meson
4797:B meson
4702:Neutron
4687:Nucleon
4679:Baryons
4670:Hadrons
4633:Tachyon
4608:Majoron
4573:Dilaton
4502:Photino
4338:Antitau
4305:Leptons
4154:Bibcode
4119:Bibcode
4023:Bibcode
3986:1131296
3964:Bibcode
3937:1131296
3915:Bibcode
3839:Bibcode
3704:Bibcode
3619:(2004).
3561:Bibcode
3518:Bibcode
3475:Bibcode
3440:Bibcode
3397:Bibcode
3358:Bibcode
3265:Majoron
3233:
3221:
2970:Lorentz
2668:in the
2631:|0〉 = 0
2096:|0〉 = 0
307:complex
269:during
218:magnons
213:magnets
187:Natural
57:) are
5268:Bosons
5119:Mesons
5068:Phonon
5063:Magnon
4985:Others
4955:Mesons
4848:Others
4744:Mesons
4692:Proton
4556:Others
4511:Others
4492:Gluino
4426:Scalar
4406:Photon
4389:Bosons
4232:Quarks
4094:
4084:
4049:
4041:
3984:
3935:
3857:
3722:
3602:
3579:
3536:
3493:
3415:
2955:phonon
2659:squark
1303:
1050:
939:
580:
518:where
421:
301:Theory
206:solids
198:phonon
196:, the
194:fluids
163:scalar
122:bosons
100:These
59:bosons
5107:Lists
5098:Trion
5093:Roton
5033:Anyon
4860:Atoms
4623:Preon
4563:Axion
4518:Axino
4411:Gluon
4398:Gauge
4092:S2CID
4047:S2CID
4013:arXiv
3877:(PDF)
3855:S2CID
3829:arXiv
3413:S2CID
3291:Notes
3213:, or
1183:〉 = 0
1168:〉 = 0
1155:, an
1090:with
1084:〉 = 0
277:of a
227:pions
204:. In
136:PNGBs
89:. In
5058:Hole
4885:Onia
4792:Kaon
4752:Pion
4323:Muon
4082:ISBN
4039:PMID
3982:OSTI
3933:OSTI
3888:2023
3720:ISSN
3600:ISBN
3577:ISSN
3534:ISSN
3491:ISSN
2946:U(1)
2871:same
2653:= 1
1449:,
1110:(0)|
1096:〉 =
224:The
55:NGBs
41:and
4333:Tau
4162:doi
4150:B49
4127:doi
4115:B46
4074:doi
4031:doi
3972:doi
3923:doi
3847:doi
3712:doi
3569:doi
3526:doi
3514:112
3483:doi
3471:110
3448:doi
3436:127
3405:doi
3366:doi
3354:117
3244:all
3074:to
2662:VEV
2115:= 0
2113:,
2081:).
1429:not
1137:is
1103:〈0|
600:|0〉
444:→ ∞
368:,
211:In
192:In
138:).
130:or
73:in
65:of
49:or
37:In
5264::
4160:.
4148:.
4125:.
4113:.
4090:.
4080:.
4068:.
4045:.
4037:.
4029:.
4021:.
4009:88
4007:.
4003:.
3980:.
3970:.
3960:66
3958:.
3954:.
3931:.
3921:.
3911:66
3909:.
3905:.
3879:.
3853:.
3845:.
3837:.
3825:13
3823:.
3819:.
3718:.
3710:.
3700:16
3698:.
3575:.
3567:.
3555:.
3532:.
3524:.
3512:.
3489:.
3481:.
3469:.
3446:.
3434:.
3411:.
3403:.
3393:19
3391:.
3387:.
3364:.
3352:.
3230:2
3224:1
2838:μN
2692:.
2670:IR
2117:.
1951:,
1540:.
1519:δ(
1438:δϕ
1408:0.
1212:,
1201:δϕ
1197:,
1176:/∂
1172:〈∂
1148:δϕ
1116:.
460:εϕ
458:=i
456:δϕ
174:.
117:.
45:,
4831:)
4827:(
4544:)
4540:(
4198:e
4191:t
4184:v
4168:.
4164::
4156::
4133:.
4129::
4121::
4098:.
4076::
4053:.
4033::
4025::
4015::
3988:.
3974::
3966::
3939:.
3925::
3917::
3890:.
3861:.
3849::
3841::
3831::
3793:.
3726:.
3714::
3706::
3608:.
3583:.
3571::
3563::
3557:6
3540:.
3528::
3520::
3497:.
3485::
3477::
3454:.
3450::
3442::
3419:.
3407::
3399::
3372:.
3368::
3360::
3227:/
3181:)
3177:r
3173:(
3168:a
3164:c
3139:0
3136:=
3130:)
3126:r
3122:(
3111:a
3107:T
3103:)
3099:r
3095:(
3090:a
3086:c
3062:)
3058:r
3054:(
3049:a
3045:c
3022:a
3018:T
2994:)
2990:r
2986:(
2922:2
2918:p
2911:E
2891:p
2885:E
2862:H
2855:→
2851:P
2849:⋅
2846:→
2842:α
2840:−
2836:−
2834:H
2819:0
2816:=
2813:p
2793:c
2790:p
2787:=
2784:E
2764:0
2761:=
2758:p
2736:n
2732:p
2725:E
2651:N
2635:Q
2629:Q
2611:=
2606:2
2595:0
2591:|
2587:Q
2549:0
2545:|
2541:Q
2538:)
2535:0
2532:(
2527:0
2523:j
2518:|
2514:0
2506:x
2501:3
2497:d
2490:=
2475:0
2471:|
2465:x
2462:P
2459:i
2452:e
2446:x
2443:P
2440:i
2436:e
2432:Q
2427:x
2424:P
2421:i
2414:e
2410:)
2407:0
2404:(
2399:0
2395:j
2389:x
2386:P
2383:i
2379:e
2374:|
2370:0
2362:x
2357:3
2353:d
2346:=
2331:0
2327:|
2323:Q
2318:x
2315:P
2312:i
2305:e
2301:)
2298:0
2295:(
2290:0
2286:j
2280:x
2277:P
2274:i
2270:e
2265:|
2261:0
2253:x
2248:3
2244:d
2237:=
2224:0
2220:|
2216:Q
2213:)
2210:x
2207:(
2202:0
2198:j
2193:|
2189:0
2183:x
2178:3
2174:d
2167:=
2157:0
2153:|
2149:Q
2146:Q
2142:|
2138:0
2109:P
2104:Q
2098:.
2094:Q
2089:Q
2075:0
2072:m
2068:A
2051:.
2038:|
2029:0
2025:m
2008:|
2004:H
1998:=
1983:|
1976:t
1973:d
1969:d
1948:0
1945:m
1939:0
1936:m
1918:.
1907:0
1903:|
1897:A
1893:Q
1882:A
1879:1
1863:0
1859:|
1853:A
1849:Q
1842:t
1839:d
1835:d
1802:,
1799:J
1792:)
1782:2
1778:A
1774:2
1768:2
1764:x
1754:e
1750:(
1741:x
1733:=
1730:J
1714:2
1710:A
1706:2
1700:2
1696:x
1686:e
1680:x
1669:=
1666:)
1663:x
1660:(
1655:0
1651:J
1640:2
1636:A
1632:2
1626:2
1622:x
1612:e
1606:x
1595:t
1592:d
1588:d
1583:=
1578:A
1574:Q
1567:t
1564:d
1560:d
1545:A
1533:k
1523:)
1521:k
1502:,
1499:0
1496:=
1490:t
1487:d
1477:g
1463:d
1443:〉
1441:g
1436:〈
1405:=
1402:)
1399:x
1396:(
1391:0
1387:J
1381:x
1370:t
1367:d
1363:d
1358:=
1355:Q
1349:t
1346:d
1342:d
1306:,
1300:0
1297:=
1289:j
1266:j
1253:i
1240:V
1235:2
1206:〉
1204:g
1199:〈
1181:i
1178:ϕ
1174:V
1165:g
1163:ϕ
1161:〈
1153:〉
1150:g
1146:〈
1134:g
1131:ϕ
1125:j
1122:ϕ
1112:θ
1108:0
1105:J
1098:ε
1094:θ
1092:〈
1082:θ
1080:〈
1075:θ
1070:Q
1053:.
1032:2
1028:v
1024:=
1015:J
1001:U
982:2
978:v
972:2
968:m
942:.
934:2
930:v
924:2
920:m
913:)
897:(
894:)
878:(
873:2
868:2
864:v
858:=
830:,
825:2
821:v
815:2
811:m
804:)
783:i
779:e
775:v
772:i
769:(
766:)
745:i
738:e
734:v
731:i
725:(
720:2
717:1
712:=
694:2
690:m
670:)
647:(
642:2
639:1
634:=
629:L
583:,
574:=
554:θ
549:U
532:v
521:θ
498:i
494:e
490:v
487:=
470:θ
451:U
442:λ
424:,
416:2
412:)
406:2
402:v
382:(
346:2
342:v
338:=
314:ϕ
292:.
263:.
53:(
34:.
20:)
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