2557:
5232:
2115:
285:
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.
2552:{\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}}}
829:
1801:
3239:, 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.
197:, 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.
1518:
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,
1178:
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
284:
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
235:
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
3143:
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
611:
2961:, 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
2066: = 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
1305:
1542:
97:
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
223:
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
2050:
1917:
1420:, 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.
941:
166:
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").
3138:
1514:
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.)
824:{\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},}
3607:
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
3182:
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.
1796:{\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,}
1501:
3768:
3739:
2865:
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
1207:
204:, 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
1407:
154:
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
2996:
1052:
423:
2120:
1179:
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.
2949:
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
2613:
2665:
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
1946:
582:
3180:
3061:
1318:
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,
3626:
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.
1812:
347:
248:, 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
502:
3635:
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.
2923:
2737:
2890:
983:
842:
3304:. 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 .
534:
3772:
3736:
3023:
2792:
2818:
2763:
3069:
236:
generators; this gives these three gauge bosons a mass and the associated necessary third polarization degree of freedom. This is described in the
5155:
3671:
3755:
Haruki Watanabe, Hitoshi Murayama, Unified Description of Nambu Goldstone Bosons without Lorentz invariance Phys. Rev. Lett. 108,251602,2012,
3314:
1300:{\displaystyle \left\langle {\partial ^{2}V \over \partial \phi _{i}\partial \phi _{j}}\right\rangle \langle \delta \phi _{j}\rangle =0~,}
150:
is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact)
1532:
This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region
1444:
3454:
Anderson, P. W. (1958-05-15). "Coherent Excited States in the Theory of Superconductivity: Gauge Invariance and the Meissner Effect".
4905:
2706:. In condensed matter these goldstone bosons are also called gapless modes (i.e. states where the energy dispersion relation is like
1324:
5276:
5143:
985:
in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.
3737:
https://www.theorie.physik.uni-muenchen.de/activities/lectures/twentyfourth_series/murayama_2/video_murayama_colloquium/index.html
109:
If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the Nambu–Goldstone
2925:
and the expectation value of the ground state is not zero, i.e. there is a spontaneously broken symmetry for the ground state
1130:, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is
2964:
4074:
998:
363:
4185:
3662:
Giuseppe Vitiello, My Double Unveiled - The dissipative quantum model of brain. John Benjamins Publ. Co., Amsterdam 2001.
3592:
2634:
There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-
5041:
3644:
E. Del Giudice, G. Preparata, G. Vitiello (1988). Water as a free electron laser. Phys. Rev. Lett., 61, 1085 – 1088.
5122:
2045:{\displaystyle \left\|{\frac {d}{dt}}|\theta \rangle \right\|=\|H|\theta \rangle \|\geq m_{0}\||\theta \rangle \|.}
2568:
3617:
G. Vitiello, (1995). Memory dissipation and capacity in the quantum brain model. Int. J. Mod. Phys. B9, 973-989.
552:
3224:, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.
3186:
In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken translation symmetry.
1912:{\displaystyle \left\|{d \over dt}Q_{A}|0\rangle \right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle \right\|.}
3540:
Bogoljubov, N. N.; Tolmachov, V. V.; Širkov, D. V. (1958). "A New Method in the Theory of Superconductivity".
3227:
Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristic
3147:
3028:
5266:
5261:
2765:), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is also
143:
51:
5138:
3804:
Hoinka, Sascha; Dyke, Paul; Lingham, Marcus G.; Kinnunen, Jami J.; Bruun, Georg M.; Vale, Chris J. (2017).
309:
468:
93:
bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the
3756:
598:
Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.
249:
103:
5271:
5214:
3296:, the Goldstone bosons are absent. Their degrees of freedom are absorbed ("eaten", gauged out) by
3263:
2892:
and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is
2698:
theories. It essentially states that, for each spontaneously broken symmetry, there corresponds some
263:
2895:
2709:
4626:
3268:
3248:
2650:
220:
115:
79:
31:
3317:
where the tip and the sides shoot to infinity, preserving the location of the minimum at its base.
2957:
In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such as
936:{\displaystyle ={\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}
5182:
4817:
4444:
4178:
4054:
255:
2869:
1423:
But its better behaved commutators with fields, that is, the nonvanishing transformation shifts
951:
5117:
4454:
3497:
Anderson, P. W. (1958-12-15). "Random-Phase Approximation in the Theory of Superconductivity".
2654:
20:
3653:
M. Jibu, K. Yasue (1995). Quantum brain dynamics and consciousness. Amsterdam: John Benjamins.
1931:, the frequency of any state like the above, which is orthogonal to the vacuum, is at least
516:
5204:
3228:
259:
4142:
4107:
4011:
3952:
3903:
3827:
3692:
3683:
Fabri, E.; Picasso, L. E. (1966-03-07). "Quantum Field Theory and Approximate Symmetries".
3549:
3506:
3463:
3428:
3385:
3346:
3337:
Nambu, Y (1960). "Quasiparticles and Gauge Invariance in the Theory of Superconductivity".
3273:
3001:
75:
3144:
expressed as a gradients of another mode. Since the spacetime dependence of the solutions
2768:
16:
Massless boson that must be present in a quantum system with spontaneously broken symmetry
8:
5219:
4202:
3989:
2797:
2742:
298:
278:
151:
139:
55:
4146:
4111:
4015:
3956:
3907:
3831:
3696:
3553:
3510:
3467:
3432:
3389:
3350:
5162:
4171:
4080:
4035:
4001:
3843:
3817:
3401:
3133:{\displaystyle c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}
2666:
602:
354:
267:
3862:
262:
can be viewed as Goldstone bosons arising due to the spontaneous symmetry breaking of
113:
are not massless, though they typically remain relatively light; they are then called
5231:
5199:
5112:
4591:
4336:
4263:
4154:
4119:
4084:
4070:
4027:
3970:
3921:
3708:
3588:
3565:
3522:
3479:
3405:
2939:
2674:
1127:
245:
190:
99:
71:
4039:
3847:
5209:
4980:
4755:
4616:
4601:
4283:
4194:
4150:
4115:
4062:
4019:
3960:
3911:
3835:
3792:
3725:
3700:
3557:
3514:
3471:
3436:
3393:
3354:
2958:
1175:
63:
27:
4023:
1090:. The current connects the original vacuum with the Nambu–Goldstone boson state,
5256:
5177:
5102:
5086:
5026:
4436:
4361:
4351:
4341:
4253:
4066:
3743:
3704:
3301:
3258:
2695:
2662:
2067:
1145:
241:
160:
159:. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding
2998:
that spontaneously breaks a spacetime symmetry, the number of broken generators
5236:
5150:
5107:
4843:
4631:
4404:
4326:
4321:
4243:
3293:
2673:
with arbitrarily small masses but yet there is no Goldstone boson with exactly
295:
237:
232:
94:
90:
59:
1067:
and the ground state to a new, degenerate, ground state. Thus, a vacuum with
5250:
5194:
5046:
5013:
4805:
4775:
4707:
4566:
4346:
4273:
4258:
3712:
3569:
3561:
3526:
3518:
3483:
3195:
2699:
2678:
2670:
595:—the hallmark of invariance), as evident in the charge of the current below.
83:
19:"Goldstone theorem" redirects here. For the theorem on Feynman diagrams, see
3805:
3475:
3440:
3358:
349:, a constant. One way to impose a constraint of this sort is by including a
5172:
4722:
4712:
4702:
4463:
4414:
4356:
4278:
4233:
4031:
2635:
453:
274:
156:
147:
4098:
Volkov, D.V.; Akulov, V. (1973). "Is the neutrino a Goldstone particle?".
3806:"Goldstone mode and pair-breaking excitations in atomic Fermi superfluids"
3419:
Goldstone, J; Salam, Abdus; Weinberg, Steven (1962). "Broken Symmetries".
2820:). Note that the energy in the non relativistic condensed matter case is
2562:
so the integrand in the right hand side does not depend on the position.
189:
is longitudinal and it is the Goldstone boson of the spontaneously broken
5187:
4955:
4858:
4853:
4770:
4765:
4695:
4649:
4606:
4571:
4530:
4422:
4386:
4248:
3297:
210:, i.e., spin waves in which the local magnetization direction oscillates.
4929:
4823:
4813:
4795:
4685:
4586:
4521:
4238:
4163:
4006:
3397:
3204:
2929:
2703:
67:
3974:
3965:
3940:
3925:
3916:
3891:
3839:
3194:
Spontaneously broken global fermionic symmetries, which occur in some
5071:
5061:
5031:
4924:
4890:
4883:
4760:
4750:
4745:
4717:
4485:
4268:
2643:
350:
3757:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.108.251602
3373:
1159:, at some ground state |0〉 chosen at the minimum of the potential,
5167:
4995:
4950:
4934:
4895:
4868:
4581:
4576:
4556:
4526:
4516:
4511:
4331:
4306:
4301:
4228:
3990:"Spontaneously Broken Spacetime Symmetries and Goldstone's Theorem"
3822:
3199:
2658:
2647:
1526:
271:
1806:
a state with approximately vanishing time derivative is produced,
1496:{\displaystyle {\frac {d\langle \delta \phi _{g}\rangle }{dt}}=0,}
5076:
5066:
5036:
4990:
4985:
4960:
4878:
4863:
4790:
4785:
4690:
4675:
4621:
4596:
4561:
4490:
4472:
4211:
3253:
3231:
of broken supersymmetry, so that goldstinos are superpartners of
4055:"Spontaneous Symmetry Breaking and Phonon as the Goldstone Mode"
5056:
5051:
4680:
4667:
4658:
4480:
4394:
4293:
2943:
206:
201:
186:
47:
1402:{\displaystyle {d \over dt}Q={d \over dt}\int _{x}J^{0}(x)=0.}
5081:
5021:
4873:
4732:
4611:
4551:
4506:
4399:
4377:
4220:
2109:
Consider the correlation function of the charge with itself,
1184:
nonvanishing vacuum expectations of transformation increments
438:
The constraint, and the action, below, are invariant under a
194:
182:
110:
2565:
Thus, its value is proportional to the total space volume,
4848:
4780:
4740:
4316:
4311:
2934:
2638:
with arbitrarily small energies. Take for example a chiral
215:
3803:
3539:
2858:
spontaneously broken generators may now give rise to the
2991:{\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }
4133:
Salam, A.; et al. (1974). "On Goldstone fermion".
1174:. In principle the vacuum should be the minimum of the
1047:{\displaystyle J_{\mu }=v^{2}\partial _{\mu }\theta ~.}
3418:
3150:
3072:
3031:
3004:
2967:
2898:
2872:
2800:
2771:
2745:
2712:
2571:
2118:
2091:
The argument requires both the vacuum and the charge
2080:
does not properly exist in the Hilbert space, unless
1949:
1815:
1545:
1447:
1412:
Acting with the charge operator on the vacuum either
1327:
1210:
1001:
954:
845:
614:
555:
519:
471:
435:. This is called the "Abelian nonlinear σ-model".
366:
312:
2854:
as it would be in a relativistic case. However, two
2059:
become large leads to a contradiction. Consequently
1108:
In general, in a theory with several scalar fields,
418:{\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}
74:, and systematically generalized in the context of
3025:minus the number non-trivial independent solutions
2952:
1310:and hence the corresponding zero-mass eigenvalues.
3174:
3132:
3055:
3017:
2990:
2917:
2884:
2812:
2786:
2757:
2731:
2607:
2551:
2044:
1911:
1795:
1495:
1401:
1299:
1046:
977:
935:
823:
576:
528:
496:
417:
341:
3582:
3505:(6). American Physical Society (APS): 1900–1916.
2694:A version of Goldstone's theorem also applies to
102:if the spontaneously broken symmetry is not also
5248:
3691:(10). American Physical Society (APS): 408–410.
3672:Scholarpedia proof of goldstone theorem - kibble
540:(1) symmetry transformation effects a shift on
231:The longitudinal polarization components of the
3938:
3889:
3462:(4). American Physical Society (APS): 827–835.
3374:"Field Theories with Superconductor Solutions"
2626:does not properly exist in the Hilbert space.
4179:
591:(i.e. the above infinitesimal transformation
452:. The field can be redefined to give a real
50:that appear necessarily in models exhibiting
4097:
4059:Gauge Invariance Approach to Acoustic Fields
3682:
3587:. Cambridge University Press. pp. 5–8.
3121:
3104:
2985:
2968:
2590:
2586:
2572:
2215:
2174:
2148:
2123:
2036:
2033:
2022:
2006:
2003:
1989:
1978:
1898:
1854:
1470:
1454:
1282:
1266:
146:; i.e., its currents are conserved, but the
86:and are known as Anderson–Bogoliubov modes.
3189:
2689:
2677:. In other words, the Goldstone bosons are
2608:{\displaystyle \|Q|0\rangle \|^{2}=\infty }
4186:
4172:
3987:
577:{\displaystyle \delta \theta =\epsilon ~,}
70:mechanism, and subsequently elucidated by
4005:
3964:
3939:Lykken, Joseph; Spiropulu, Maria (2013).
3915:
3890:Lykken, Joseph; Spiropulu, Maria (2013).
3821:
3583:Baumann, D.; McAllister, L. (2015). "1".
3371:
2615:— unless the symmetry is unbroken,
4193:
3988:Low, I.; Manohar, A.V. (February 2002).
3496:
3453:
3175:{\displaystyle c_{a}({\boldsymbol {r}})}
3056:{\displaystyle c_{a}({\boldsymbol {r}})}
587:but does not preserve the ground state
3165:
3114:
3087:
3046:
2978:
2942:are spontaneously broken. However, the
1313:
513:is the Nambu–Goldstone boson (actually
5249:
4657:
2702:which is typically a boson and has no
130:
4167:
4132:
3860:
3336:
1061:, resulting from this current shifts
1525:, so that the theory cannot have a
1199:null eigenvectors of the mass matrix
342:{\displaystyle \phi ^{*}\phi =v^{2}}
244:. An analogous phenomenon occurs in
4052:
1197:, specify the relevant (Goldstone)
497:{\displaystyle \phi =ve^{i\theta }}
13:
2602:
1922:Assuming a nonvanishing mass gap
1734:
1712:
1246:
1233:
1219:
1026:
889:
870:
780:
742:
662:
639:
617:
14:
5288:
2946:is the Goldstone boson for both.
2629:
2095:to be translationally invariant,
1416:, if that is symmetric; else, if
5230:
5123:Timeline of particle discoveries
3198:models, lead to Nambu–Goldstone
2953:Breaking of spacetime symmetries
5277:Subatomic particles with spin 0
4126:
4091:
4046:
3981:
3941:"The future of the Higgs boson"
3932:
3892:"The future of the Higgs boson"
3883:
3854:
3797:
3786:
3761:
3749:
3730:
3719:
3676:
3665:
3656:
3647:
3638:
3629:
3620:
3611:
3307:
1132:nonvanishing vacuum expectation
988:The symmetry-induced conserved
3601:
3576:
3533:
3490:
3447:
3412:
3365:
3330:
3286:
3169:
3161:
3118:
3110:
3091:
3083:
3050:
3042:
2982:
2974:
2918:{\displaystyle E\propto p^{2}}
2732:{\displaystyle E\propto p^{n}}
2579:
2526:
2520:
2398:
2392:
2289:
2283:
2208:
2201:
2195:
2181:
2141:
2130:
2026:
1996:
1982:
1971:
1951:
1902:
1891:
1876:
1858:
1847:
1817:
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1648:
1390:
1384:
901:
885:
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866:
792:
757:
754:
713:
658:
635:
400:
370:
1:
4024:10.1103/PhysRevLett.88.101602
3863:"Phonons as Goldstone bosons"
3324:
2938:particle number symmetry and
2684:
456:(i.e., a spin-zero particle)
121:pseudo–Nambu–Goldstone bosons
5139:History of subatomic physics
4155:10.1016/0370-2693(74)90637-6
4120:10.1016/0370-2693(73)90490-5
4067:10.1007/978-981-13-8751-7_11
3705:10.1103/physrevlett.16.408.2
3315:Goldstone sombrero potential
3235:particles in the theory, of
948:Note that the constant term
270:. These fluctuations in the
7:
3585:Inflation and String Theory
3242:
2661:. The chiral symmetry is a
1117:, the Nambu–Goldstone mode
442:(1) phase transformation,
306:, with the constraint that
228:than typical hadron masses.
170:
10:
5293:
2928:As a second example, in a
2885:{\displaystyle E\propto p}
1536:is applied to the vacuum,
978:{\displaystyle m^{2}v^{2}}
462:without any constraint by
279:cosmic structure formation
175:
66:within the context of the
58:. They were discovered by
18:
5228:
5131:
5095:
5012:
4973:
4943:
4917:
4913:
4904:
4836:
4804:
4731:
4666:
4648:
4544:
4499:
4471:
4462:
4453:
4435:
4413:
4385:
4376:
4292:
4219:
4210:
4201:
3548:(11–12). Wiley: 605–682.
2669:bosons have a continuous
428:and taking the limit as
289:
264:time translation symmetry
5156:mathematical formulation
4751:Eta and eta prime mesons
4053:Gan, Woon Siong (2019).
3562:10.1002/prop.19580061102
3519:10.1103/physrev.112.1900
3279:
3269:Vacuum expectation value
3190:Nambu–Goldstone fermions
2690:Nonrelativistic theories
529:{\displaystyle v\theta }
353:interaction term in its
80:condensed matter physics
32:condensed matter physics
4818:Double-charm tetraquark
3685:Physical Review Letters
3542:Fortschritte der Physik
3476:10.1103/physrev.110.827
3441:10.1103/PhysRev.127.965
3359:10.1103/PhysRev.117.648
2862:Nambu–Goldstone boson.
256:Primordial fluctuations
221:pseudo-Goldstone bosons
116:pseudo-Goldstone bosons
3313:It corresponds to the
3249:Pseudo-Goldstone boson
3229:nonlinear realizations
3176:
3134:
3057:
3019:
2992:
2919:
2886:
2814:
2788:
2759:
2733:
2609:
2553:
2075:Fabri–Picasso Theorem.
2046:
1913:
1797:
1497:
1414:annihilates the vacuum
1403:
1301:
1182:By contrast, however,
1048:
979:
937:
825:
593:does not annihilate it
578:
530:
498:
419:
343:
250:Ginzburg–Landau theory
40:Nambu–Goldstone bosons
21:linked-cluster theorem
5215:Wave–particle duality
5205:Relativistic particle
4342:Electron antineutrino
4000:(10): 101602–101605.
3372:Goldstone, J (1961).
3264:Mermin–Wagner theorem
3177:
3135:
3058:
3020:
3018:{\displaystyle T^{a}}
2993:
2920:
2887:
2815:
2789:
2760:
2734:
2646:model with a nonzero
2610:
2554:
2047:
1914:
1798:
1498:
1434:, are, nevertheless,
1404:
1302:
1049:
980:
938:
826:
579:
531:
499:
420:
344:
226:significantly smaller
68:BCS superconductivity
56:continuous symmetries
52:spontaneous breakdown
5267:Mathematical physics
5262:Quantum field theory
4445:Faddeev–Popov ghosts
4195:Particles in physics
3793:Fabri dispense 1965
3726:Fabri dispense 1965
3148:
3070:
3029:
3002:
2965:
2896:
2870:
2798:
2787:{\displaystyle E=pc}
2769:
2743:
2710:
2569:
2116:
1947:
1813:
1543:
1445:
1325:
1314:Goldstone's argument
1208:
999:
952:
843:
612:
553:
517:
469:
364:
310:
144:spontaneously broken
76:quantum field theory
5220:Particle chauvinism
5163:Subatomic particles
4147:1974PhLB...49..465S
4112:1973PhLB...46..109V
4016:2002PhRvL..88j1602L
3957:2013PhT....66l..28L
3908:2013PhT....66l..28L
3832:2017NatPh..13..943H
3697:1966PhRvL..16..408F
3554:1958ForPh...6..605B
3511:1958PhRv..112.1900A
3468:1958PhRv..110..827A
3433:1962PhRv..127..965G
3390:1961NCim...19..154G
3351:1960PhRv..117..648N
2813:{\displaystyle p=0}
2758:{\displaystyle p=0}
1176:effective potential
140:continuous symmetry
138:examines a generic
136:Goldstone's theorem
131:Goldstone's theorem
4061:. pp. 59–62.
3742:2022-01-14 at the
3398:10.1007/BF02812722
3208:. These have spin
3172:
3130:
3053:
3015:
2988:
2915:
2882:
2810:
2784:
2755:
2729:
2605:
2549:
2547:
2042:
1909:
1793:
1506:thus generating a
1493:
1399:
1297:
1044:
975:
933:
821:
603:Lagrangian density
601:The corresponding
574:
526:
494:
415:
355:Lagrangian density
339:
277:subsequently seed
268:de Sitter universe
5244:
5243:
5200:Massless particle
5008:
5007:
5004:
5003:
4969:
4968:
4832:
4831:
4644:
4643:
4640:
4639:
4592:Magnetic monopole
4540:
4539:
4431:
4430:
4372:
4371:
4352:Muon antineutrino
4337:Electron neutrino
4076:978-981-13-8750-0
3966:10.1063/PT.3.2212
3917:10.1063/PT.3.2212
3840:10.1038/nphys4187
3292:In theories with
3274:Noether's theorem
2940:Galilean symmetry
2845:
2836:
2622:. Consequently,
1968:
1873:
1834:
1776:
1708:
1634:
1587:
1559:
1482:
1362:
1341:
1293:
1260:
1040:
929:
864:
711:
633:
570:
411:
246:superconductivity
191:Galilean symmetry
104:broken explicitly
72:Jeffrey Goldstone
5284:
5272:Physics theorems
5234:
5210:Virtual particle
4981:Mesonic molecule
4915:
4914:
4911:
4910:
4756:Bottom eta meson
4664:
4663:
4655:
4654:
4627:W′ and Z′ bosons
4617:Sterile neutrino
4602:Majorana fermion
4469:
4468:
4460:
4459:
4383:
4382:
4362:Tau antineutrino
4217:
4216:
4208:
4207:
4188:
4181:
4174:
4165:
4164:
4159:
4158:
4130:
4124:
4123:
4095:
4089:
4088:
4050:
4044:
4043:
4009:
3985:
3979:
3978:
3968:
3936:
3930:
3929:
3919:
3887:
3881:
3880:
3878:
3876:
3867:
3858:
3852:
3851:
3825:
3801:
3795:
3790:
3784:
3783:
3781:
3780:
3771:. Archived from
3765:
3759:
3753:
3747:
3734:
3728:
3723:
3717:
3716:
3680:
3674:
3669:
3663:
3660:
3654:
3651:
3645:
3642:
3636:
3633:
3627:
3624:
3618:
3615:
3609:
3605:
3599:
3598:
3580:
3574:
3573:
3537:
3531:
3530:
3494:
3488:
3487:
3451:
3445:
3444:
3416:
3410:
3409:
3369:
3363:
3362:
3334:
3318:
3311:
3305:
3290:
3223:
3221:
3220:
3217:
3214:
3181:
3179:
3178:
3173:
3168:
3160:
3159:
3139:
3137:
3136:
3131:
3117:
3103:
3102:
3090:
3082:
3081:
3062:
3060:
3059:
3054:
3049:
3041:
3040:
3024:
3022:
3021:
3016:
3014:
3013:
2997:
2995:
2994:
2989:
2981:
2924:
2922:
2921:
2916:
2914:
2913:
2891:
2889:
2888:
2883:
2853:
2847:
2846:
2843:
2837:
2834:
2819:
2817:
2816:
2811:
2793:
2791:
2790:
2785:
2764:
2762:
2761:
2756:
2739:and is zero for
2738:
2736:
2735:
2730:
2728:
2727:
2625:
2621:
2614:
2612:
2611:
2606:
2598:
2597:
2582:
2558:
2556:
2555:
2550:
2548:
2544:
2540:
2536:
2532:
2519:
2518:
2493:
2492:
2474:
2470:
2466:
2462:
2458:
2457:
2456:
2438:
2437:
2419:
2418:
2391:
2390:
2381:
2380:
2349:
2348:
2330:
2326:
2322:
2318:
2314:
2310:
2309:
2282:
2281:
2272:
2271:
2240:
2239:
2221:
2211:
2194:
2193:
2184:
2170:
2169:
2144:
2133:
2105:
2101:
2094:
2086:
2079:
2062:
2058:
2051:
2049:
2048:
2043:
2029:
2021:
2020:
1999:
1985:
1981:
1974:
1969:
1967:
1956:
1939:
1930:
1918:
1916:
1915:
1910:
1905:
1901:
1894:
1889:
1888:
1874:
1866:
1861:
1857:
1850:
1845:
1844:
1835:
1833:
1822:
1802:
1800:
1799:
1794:
1783:
1779:
1778:
1777:
1775:
1774:
1773:
1760:
1759:
1750:
1733:
1732:
1711:
1710:
1709:
1707:
1706:
1705:
1692:
1691:
1682:
1672:
1671:
1647:
1646:
1637:
1636:
1635:
1633:
1632:
1631:
1618:
1617:
1608:
1598:
1597:
1588:
1586:
1575:
1570:
1569:
1560:
1558:
1547:
1535:
1524:
1513:
1502:
1500:
1499:
1494:
1483:
1481:
1473:
1469:
1468:
1449:
1433:
1408:
1406:
1405:
1400:
1383:
1382:
1373:
1372:
1363:
1361:
1350:
1342:
1340:
1329:
1306:
1304:
1303:
1298:
1291:
1281:
1280:
1265:
1261:
1259:
1258:
1257:
1245:
1244:
1231:
1227:
1226:
1216:
1196:
1173:
1158:
1148:, for vanishing
1143:
1125:
1116:
1104:
1089:
1077:different vacuum
1075:will shift to a
1074:
1066:
1053:
1051:
1050:
1045:
1038:
1034:
1033:
1024:
1023:
1011:
1010:
992:(1) current is
984:
982:
981:
976:
974:
973:
964:
963:
942:
940:
939:
934:
927:
926:
925:
916:
915:
897:
896:
878:
877:
865:
860:
859:
850:
830:
828:
827:
822:
817:
816:
807:
806:
788:
787:
778:
777:
750:
749:
740:
739:
712:
704:
696:
695:
686:
685:
670:
669:
657:
656:
647:
646:
634:
626:
621:
620:
590:
583:
581:
580:
575:
568:
545:
535:
533:
532:
527:
512:
503:
501:
500:
495:
493:
492:
461:
451:
434:
424:
422:
421:
416:
409:
408:
407:
398:
397:
382:
381:
348:
346:
345:
340:
338:
337:
322:
321:
305:
82:such bosons are
64:particle physics
36:Goldstone bosons
5292:
5291:
5287:
5286:
5285:
5283:
5282:
5281:
5247:
5246:
5245:
5240:
5224:
5178:Nuclear physics
5127:
5091:
5027:Davydov soliton
5000:
4965:
4939:
4900:
4828:
4800:
4727:
4636:
4536:
4495:
4449:
4427:
4409:
4368:
4288:
4197:
4192:
4162:
4135:Physics Letters
4131:
4127:
4100:Physics Letters
4096:
4092:
4077:
4051:
4047:
3994:Phys. Rev. Lett
3986:
3982:
3937:
3933:
3888:
3884:
3874:
3872:
3865:
3859:
3855:
3816:(10): 943–946.
3802:
3798:
3791:
3787:
3778:
3776:
3767:
3766:
3762:
3754:
3750:
3744:Wayback Machine
3735:
3731:
3724:
3720:
3681:
3677:
3670:
3666:
3661:
3657:
3652:
3648:
3643:
3639:
3634:
3630:
3625:
3621:
3616:
3612:
3606:
3602:
3595:
3581:
3577:
3538:
3534:
3499:Physical Review
3495:
3491:
3456:Physical Review
3452:
3448:
3421:Physical Review
3417:
3413:
3370:
3366:
3339:Physical Review
3335:
3331:
3327:
3322:
3321:
3312:
3308:
3302:Higgs mechanism
3291:
3287:
3282:
3259:Higgs mechanism
3245:
3218:
3215:
3212:
3211:
3209:
3192:
3164:
3155:
3151:
3149:
3146:
3145:
3113:
3098:
3094:
3086:
3077:
3073:
3071:
3068:
3067:
3045:
3036:
3032:
3030:
3027:
3026:
3009:
3005:
3003:
3000:
2999:
2977:
2966:
2963:
2962:
2955:
2909:
2905:
2897:
2894:
2893:
2871:
2868:
2867:
2849:
2842:
2833:
2821:
2799:
2796:
2795:
2770:
2767:
2766:
2744:
2741:
2740:
2723:
2719:
2711:
2708:
2707:
2696:nonrelativistic
2692:
2687:
2663:global symmetry
2641:
2632:
2623:
2616:
2593:
2589:
2578:
2570:
2567:
2566:
2546:
2545:
2514:
2510:
2509:
2505:
2501:
2497:
2488:
2484:
2472:
2471:
2443:
2439:
2427:
2423:
2405:
2401:
2386:
2382:
2370:
2366:
2365:
2361:
2357:
2353:
2344:
2340:
2328:
2327:
2296:
2292:
2277:
2273:
2261:
2257:
2256:
2252:
2248:
2244:
2235:
2231:
2219:
2218:
2207:
2189:
2185:
2180:
2165:
2161:
2151:
2140:
2129:
2119:
2117:
2114:
2113:
2103:
2096:
2092:
2081:
2077:
2068:Higgs mechanism
2065:
2060:
2056:
2025:
2016:
2012:
1995:
1970:
1960:
1955:
1954:
1950:
1948:
1945:
1944:
1938:
1932:
1929:
1923:
1890:
1884:
1880:
1879:
1875:
1865:
1846:
1840:
1836:
1826:
1821:
1820:
1816:
1814:
1811:
1810:
1769:
1765:
1761:
1755:
1751:
1749:
1745:
1741:
1737:
1728:
1724:
1701:
1697:
1693:
1687:
1683:
1681:
1677:
1673:
1667:
1663:
1642:
1638:
1627:
1623:
1619:
1613:
1609:
1607:
1603:
1599:
1593:
1589:
1579:
1574:
1565:
1561:
1551:
1546:
1544:
1541:
1540:
1533:
1520:
1507:
1474:
1464:
1460:
1450:
1448:
1446:
1443:
1442:
1431:
1424:
1378:
1374:
1368:
1364:
1354:
1349:
1333:
1328:
1326:
1323:
1322:
1316:
1308:
1276:
1272:
1253:
1249:
1240:
1236:
1232:
1222:
1218:
1217:
1215:
1211:
1209:
1206:
1205:
1194:
1187:
1171:
1160:
1155:
1149:
1146:order parameter
1140:
1134:
1124:
1118:
1115:
1109:
1098:
1091:
1080:
1068:
1062:
1029:
1025:
1019:
1015:
1006:
1002:
1000:
997:
996:
969:
965:
959:
955:
953:
950:
949:
921:
917:
911:
907:
892:
888:
873:
869:
855:
851:
849:
844:
841:
840:
812:
808:
802:
798:
783:
779:
770:
766:
745:
741:
729:
725:
703:
691:
687:
681:
677:
665:
661:
652:
648:
642:
638:
625:
616:
615:
613:
610:
609:
588:
554:
551:
550:
541:
518:
515:
514:
508:
485:
481:
470:
467:
466:
457:
443:
429:
403:
399:
393:
389:
377:
373:
365:
362:
361:
333:
329:
317:
313:
311:
308:
307:
301:
292:
242:Higgs mechanism
178:
173:
161:order parameter
133:
95:quantum numbers
24:
17:
12:
11:
5:
5290:
5280:
5279:
5274:
5269:
5264:
5259:
5242:
5241:
5237:Physics portal
5229:
5226:
5225:
5223:
5222:
5217:
5212:
5207:
5202:
5197:
5192:
5191:
5190:
5180:
5175:
5170:
5165:
5160:
5159:
5158:
5151:Standard Model
5148:
5147:
5146:
5135:
5133:
5129:
5128:
5126:
5125:
5120:
5118:Quasiparticles
5115:
5110:
5105:
5099:
5097:
5093:
5092:
5090:
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5054:
5049:
5044:
5039:
5034:
5029:
5024:
5018:
5016:
5014:Quasiparticles
5010:
5009:
5006:
5005:
5002:
5001:
4999:
4998:
4993:
4988:
4983:
4977:
4975:
4971:
4970:
4967:
4966:
4964:
4963:
4958:
4953:
4947:
4945:
4941:
4940:
4938:
4937:
4932:
4927:
4921:
4919:
4908:
4902:
4901:
4899:
4898:
4893:
4888:
4887:
4886:
4881:
4876:
4871:
4866:
4861:
4851:
4846:
4840:
4838:
4834:
4833:
4830:
4829:
4827:
4826:
4821:
4810:
4808:
4806:Exotic hadrons
4802:
4801:
4799:
4798:
4793:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4743:
4737:
4735:
4729:
4728:
4726:
4725:
4720:
4715:
4710:
4705:
4700:
4699:
4698:
4693:
4688:
4683:
4672:
4670:
4661:
4652:
4646:
4645:
4642:
4641:
4638:
4637:
4635:
4634:
4632:X and Y bosons
4629:
4624:
4619:
4614:
4609:
4604:
4599:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4548:
4546:
4542:
4541:
4538:
4537:
4535:
4534:
4524:
4519:
4514:
4509:
4503:
4501:
4497:
4496:
4494:
4493:
4488:
4483:
4477:
4475:
4466:
4457:
4451:
4450:
4448:
4447:
4441:
4439:
4433:
4432:
4429:
4428:
4426:
4425:
4419:
4417:
4411:
4410:
4408:
4407:
4405:W and Z bosons
4402:
4397:
4391:
4389:
4380:
4374:
4373:
4370:
4369:
4367:
4366:
4365:
4364:
4359:
4354:
4349:
4344:
4339:
4329:
4324:
4319:
4314:
4309:
4304:
4298:
4296:
4290:
4289:
4287:
4286:
4281:
4276:
4271:
4266:
4261:
4259:Strange (quark
4256:
4251:
4246:
4241:
4236:
4231:
4225:
4223:
4214:
4205:
4199:
4198:
4191:
4190:
4183:
4176:
4168:
4161:
4160:
4141:(5): 465–467.
4125:
4106:(1): 109–110.
4090:
4075:
4045:
4007:hep-th/0110285
3980:
3931:
3882:
3861:Leutwyler, H.
3853:
3810:Nature Physics
3796:
3785:
3760:
3748:
3729:
3718:
3675:
3664:
3655:
3646:
3637:
3628:
3619:
3610:
3600:
3594:978-1107089693
3593:
3575:
3532:
3489:
3446:
3427:(3): 965–970.
3411:
3384:(1): 154–164.
3364:
3345:(3): 648–663.
3328:
3326:
3323:
3320:
3319:
3306:
3300:, through the
3294:gauge symmetry
3284:
3283:
3281:
3278:
3277:
3276:
3271:
3266:
3261:
3256:
3251:
3244:
3241:
3196:supersymmetric
3191:
3188:
3171:
3167:
3163:
3158:
3154:
3141:
3140:
3129:
3126:
3123:
3120:
3116:
3112:
3109:
3106:
3101:
3097:
3093:
3089:
3085:
3080:
3076:
3052:
3048:
3044:
3039:
3035:
3012:
3008:
2987:
2984:
2980:
2976:
2973:
2970:
2954:
2951:
2912:
2908:
2904:
2901:
2881:
2878:
2875:
2809:
2806:
2803:
2783:
2780:
2777:
2774:
2754:
2751:
2748:
2726:
2722:
2718:
2715:
2691:
2688:
2686:
2683:
2679:infraparticles
2639:
2631:
2630:Infraparticles
2628:
2604:
2601:
2596:
2592:
2588:
2585:
2581:
2577:
2574:
2560:
2559:
2543:
2539:
2535:
2531:
2528:
2525:
2522:
2517:
2513:
2508:
2504:
2500:
2496:
2491:
2487:
2483:
2480:
2477:
2475:
2473:
2469:
2465:
2461:
2455:
2452:
2449:
2446:
2442:
2436:
2433:
2430:
2426:
2422:
2417:
2414:
2411:
2408:
2404:
2400:
2397:
2394:
2389:
2385:
2379:
2376:
2373:
2369:
2364:
2360:
2356:
2352:
2347:
2343:
2339:
2336:
2333:
2331:
2329:
2325:
2321:
2317:
2313:
2308:
2305:
2302:
2299:
2295:
2291:
2288:
2285:
2280:
2276:
2270:
2267:
2264:
2260:
2255:
2251:
2247:
2243:
2238:
2234:
2230:
2227:
2224:
2222:
2220:
2217:
2214:
2210:
2206:
2203:
2200:
2197:
2192:
2188:
2183:
2179:
2176:
2173:
2168:
2164:
2160:
2157:
2154:
2152:
2150:
2147:
2143:
2139:
2136:
2132:
2128:
2125:
2122:
2121:
2089:
2088:
2063:
2053:
2052:
2041:
2038:
2035:
2032:
2028:
2024:
2019:
2015:
2011:
2008:
2005:
2002:
1998:
1994:
1991:
1988:
1984:
1980:
1977:
1973:
1966:
1963:
1959:
1953:
1936:
1927:
1920:
1919:
1908:
1904:
1900:
1897:
1893:
1887:
1883:
1878:
1872:
1869:
1864:
1860:
1856:
1853:
1849:
1843:
1839:
1832:
1829:
1825:
1819:
1804:
1803:
1792:
1789:
1786:
1782:
1772:
1768:
1764:
1758:
1754:
1748:
1744:
1740:
1736:
1731:
1727:
1723:
1720:
1717:
1714:
1704:
1700:
1696:
1690:
1686:
1680:
1676:
1670:
1666:
1662:
1659:
1656:
1653:
1650:
1645:
1641:
1630:
1626:
1622:
1616:
1612:
1606:
1602:
1596:
1592:
1585:
1582:
1578:
1573:
1568:
1564:
1557:
1554:
1550:
1504:
1503:
1492:
1489:
1486:
1480:
1477:
1472:
1467:
1463:
1459:
1456:
1453:
1436:time-invariant
1429:
1410:
1409:
1398:
1395:
1392:
1389:
1386:
1381:
1377:
1371:
1367:
1360:
1357:
1353:
1348:
1345:
1339:
1336:
1332:
1315:
1312:
1296:
1290:
1287:
1284:
1279:
1275:
1271:
1268:
1264:
1256:
1252:
1248:
1243:
1239:
1235:
1230:
1225:
1221:
1214:
1203:
1192:
1169:
1153:
1138:
1122:
1113:
1096:
1055:
1054:
1043:
1037:
1032:
1028:
1022:
1018:
1014:
1009:
1005:
972:
968:
962:
958:
946:
945:
944:
943:
932:
924:
920:
914:
910:
906:
903:
900:
895:
891:
887:
884:
881:
876:
872:
868:
863:
858:
854:
848:
832:
831:
820:
815:
811:
805:
801:
797:
794:
791:
786:
782:
776:
773:
769:
765:
762:
759:
756:
753:
748:
744:
738:
735:
732:
728:
724:
721:
718:
715:
710:
707:
702:
699:
694:
690:
684:
680:
676:
673:
668:
664:
660:
655:
651:
645:
641:
637:
632:
629:
624:
619:
585:
584:
573:
567:
564:
561:
558:
525:
522:
505:
504:
491:
488:
484:
480:
477:
474:
426:
425:
414:
406:
402:
396:
392:
388:
385:
380:
376:
372:
369:
336:
332:
328:
325:
320:
316:
291:
288:
287:
286:
282:
253:
238:Standard Model
233:W and Z bosons
229:
211:
198:
177:
174:
172:
169:
132:
129:
84:quasiparticles
60:Yoichiro Nambu
15:
9:
6:
4:
3:
2:
5289:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5254:
5252:
5239:
5238:
5233:
5227:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5195:Exotic matter
5193:
5189:
5186:
5185:
5184:
5183:Eightfold way
5181:
5179:
5176:
5174:
5173:Antiparticles
5171:
5169:
5166:
5164:
5161:
5157:
5154:
5153:
5152:
5149:
5145:
5142:
5141:
5140:
5137:
5136:
5134:
5130:
5124:
5121:
5119:
5116:
5114:
5111:
5109:
5106:
5104:
5101:
5100:
5098:
5094:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5019:
5017:
5015:
5011:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4978:
4976:
4972:
4962:
4959:
4957:
4954:
4952:
4949:
4948:
4946:
4942:
4936:
4933:
4931:
4928:
4926:
4923:
4922:
4920:
4916:
4912:
4909:
4907:
4903:
4897:
4894:
4892:
4889:
4885:
4882:
4880:
4877:
4875:
4872:
4870:
4867:
4865:
4862:
4860:
4857:
4856:
4855:
4852:
4850:
4847:
4845:
4844:Atomic nuclei
4842:
4841:
4839:
4835:
4825:
4822:
4819:
4815:
4812:
4811:
4809:
4807:
4803:
4797:
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4776:Upsilon meson
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4738:
4736:
4734:
4730:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4708:Lambda baryon
4706:
4704:
4701:
4697:
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4678:
4677:
4674:
4673:
4671:
4669:
4665:
4662:
4660:
4656:
4653:
4651:
4647:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4567:Dual graviton
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4549:
4547:
4543:
4532:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4504:
4502:
4498:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4476:
4474:
4470:
4467:
4465:
4464:Superpartners
4461:
4458:
4456:
4452:
4446:
4443:
4442:
4440:
4438:
4434:
4424:
4421:
4420:
4418:
4416:
4412:
4406:
4403:
4401:
4398:
4396:
4393:
4392:
4390:
4388:
4384:
4381:
4379:
4375:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4347:Muon neutrino
4345:
4343:
4340:
4338:
4335:
4334:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4299:
4297:
4295:
4291:
4285:
4282:
4280:
4279:Bottom (quark
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4226:
4224:
4222:
4218:
4215:
4213:
4209:
4206:
4204:
4200:
4196:
4189:
4184:
4182:
4177:
4175:
4170:
4169:
4166:
4156:
4152:
4148:
4144:
4140:
4136:
4129:
4121:
4117:
4113:
4109:
4105:
4101:
4094:
4086:
4082:
4078:
4072:
4068:
4064:
4060:
4056:
4049:
4041:
4037:
4033:
4029:
4025:
4021:
4017:
4013:
4008:
4003:
3999:
3995:
3991:
3984:
3976:
3972:
3967:
3962:
3958:
3954:
3951:(12): 28–33.
3950:
3946:
3945:Physics Today
3942:
3935:
3927:
3923:
3918:
3913:
3909:
3905:
3902:(12): 28–33.
3901:
3897:
3896:Physics Today
3893:
3886:
3871:
3864:
3857:
3849:
3845:
3841:
3837:
3833:
3829:
3824:
3819:
3815:
3811:
3807:
3800:
3794:
3789:
3775:on 2022-01-14
3774:
3770:
3764:
3758:
3752:
3745:
3741:
3738:
3733:
3727:
3722:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3679:
3673:
3668:
3659:
3650:
3641:
3632:
3623:
3614:
3604:
3596:
3590:
3586:
3579:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3536:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3493:
3485:
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3450:
3442:
3438:
3434:
3430:
3426:
3422:
3415:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3379:
3378:Nuovo Cimento
3375:
3368:
3360:
3356:
3352:
3348:
3344:
3340:
3333:
3329:
3316:
3310:
3303:
3299:
3295:
3289:
3285:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3246:
3240:
3238:
3234:
3230:
3225:
3207:
3206:
3201:
3197:
3187:
3184:
3156:
3152:
3127:
3124:
3107:
3099:
3095:
3078:
3074:
3066:
3065:
3064:
3037:
3033:
3010:
3006:
2971:
2960:
2950:
2947:
2945:
2941:
2937:
2936:
2931:
2926:
2910:
2906:
2902:
2899:
2879:
2876:
2873:
2863:
2861:
2857:
2852:
2841:
2832:
2828:
2824:
2807:
2804:
2801:
2794:and zero for
2781:
2778:
2775:
2772:
2752:
2749:
2746:
2724:
2720:
2716:
2713:
2705:
2701:
2700:quasiparticle
2697:
2682:
2680:
2676:
2672:
2671:mass spectrum
2668:
2664:
2660:
2656:
2652:
2649:
2645:
2637:
2636:vacuum states
2627:
2619:
2599:
2594:
2583:
2575:
2563:
2541:
2537:
2533:
2529:
2523:
2515:
2511:
2506:
2502:
2498:
2494:
2489:
2485:
2481:
2478:
2476:
2467:
2463:
2459:
2453:
2450:
2447:
2444:
2440:
2434:
2431:
2428:
2424:
2420:
2415:
2412:
2409:
2406:
2402:
2395:
2387:
2383:
2377:
2374:
2371:
2367:
2362:
2358:
2354:
2350:
2345:
2341:
2337:
2334:
2332:
2323:
2319:
2315:
2311:
2306:
2303:
2300:
2297:
2293:
2286:
2278:
2274:
2268:
2265:
2262:
2258:
2253:
2249:
2245:
2241:
2236:
2232:
2228:
2225:
2223:
2212:
2204:
2198:
2190:
2186:
2177:
2171:
2166:
2162:
2158:
2155:
2153:
2145:
2137:
2134:
2126:
2112:
2111:
2110:
2107:
2099:
2084:
2076:
2073:
2072:
2071:
2069:
2039:
2030:
2017:
2013:
2009:
2000:
1992:
1986:
1975:
1964:
1961:
1957:
1943:
1942:
1941:
1935:
1926:
1906:
1895:
1885:
1881:
1870:
1867:
1862:
1851:
1841:
1837:
1830:
1827:
1823:
1809:
1808:
1807:
1790:
1787:
1784:
1780:
1770:
1766:
1762:
1756:
1752:
1746:
1742:
1738:
1729:
1725:
1721:
1718:
1715:
1702:
1698:
1694:
1688:
1684:
1678:
1674:
1668:
1664:
1660:
1657:
1651:
1643:
1639:
1628:
1624:
1620:
1614:
1610:
1604:
1600:
1594:
1590:
1583:
1580:
1576:
1571:
1566:
1562:
1555:
1552:
1548:
1539:
1538:
1537:
1530:
1528:
1523:
1516:
1511:
1490:
1487:
1484:
1478:
1475:
1465:
1461:
1457:
1451:
1441:
1440:
1439:
1437:
1428:
1421:
1419:
1415:
1396:
1393:
1387:
1379:
1375:
1369:
1365:
1358:
1355:
1351:
1346:
1343:
1337:
1334:
1330:
1321:
1320:
1319:
1311:
1307:
1294:
1288:
1285:
1277:
1273:
1269:
1262:
1254:
1250:
1241:
1237:
1228:
1223:
1212:
1202:
1200:
1191:
1185:
1180:
1177:
1168:
1164:
1156:
1147:
1141:
1133:
1129:
1121:
1112:
1106:
1102:
1095:
1088:
1084:
1078:
1072:
1065:
1060:
1041:
1035:
1030:
1020:
1016:
1012:
1007:
1003:
995:
994:
993:
991:
986:
970:
966:
960:
956:
930:
922:
918:
912:
908:
904:
898:
893:
879:
874:
861:
856:
852:
846:
839:
838:
837:
836:
835:
818:
813:
809:
803:
799:
795:
789:
784:
774:
771:
767:
763:
760:
751:
746:
736:
733:
730:
726:
722:
719:
716:
708:
705:
700:
697:
692:
688:
682:
678:
674:
671:
666:
653:
649:
643:
630:
627:
622:
608:
607:
606:
604:
599:
596:
594:
571:
565:
562:
559:
556:
549:
548:
547:
544:
539:
523:
520:
511:
489:
486:
482:
478:
475:
472:
465:
464:
463:
460:
455:
450:
446:
441:
436:
432:
412:
404:
394:
390:
386:
383:
378:
374:
367:
360:
359:
358:
356:
352:
334:
330:
326:
323:
318:
314:
304:
300:
297:
283:
280:
276:
273:
269:
265:
261:
257:
254:
251:
247:
243:
239:
234:
230:
227:
222:
218:
217:
212:
209:
208:
203:
199:
196:
192:
188:
184:
180:
179:
168:
164:
162:
158:
153:
149:
145:
141:
137:
128:
126:
123:(abbreviated
122:
118:
117:
112:
107:
105:
101:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
5235:
4906:Hypothetical
4854:Exotic atoms
4723:Omega baryon
4713:Sigma baryon
4703:Delta baryon
4455:Hypothetical
4437:Ghost fields
4423:Higgs boson
4357:Tau neutrino
4249:Charm (quark
4138:
4134:
4128:
4103:
4099:
4093:
4058:
4048:
3997:
3993:
3983:
3948:
3944:
3934:
3899:
3895:
3885:
3873:. Retrieved
3869:
3856:
3813:
3809:
3799:
3788:
3777:. Retrieved
3773:the original
3763:
3751:
3746:- min. 30-60
3732:
3721:
3688:
3684:
3678:
3667:
3658:
3649:
3640:
3631:
3622:
3613:
3603:
3584:
3578:
3545:
3541:
3535:
3502:
3498:
3492:
3459:
3455:
3449:
3424:
3420:
3414:
3381:
3377:
3367:
3342:
3338:
3332:
3309:
3298:gauge bosons
3288:
3236:
3232:
3226:
3203:
3193:
3185:
3142:
2956:
2948:
2933:
2927:
2864:
2859:
2855:
2850:
2839:
2830:
2826:
2822:
2693:
2633:
2617:
2564:
2561:
2108:
2097:
2090:
2082:
2074:
2054:
1933:
1924:
1921:
1805:
1531:
1521:
1517:
1509:
1505:
1435:
1426:
1422:
1417:
1413:
1411:
1317:
1309:
1204:
1198:
1189:
1183:
1181:
1166:
1162:
1151:
1136:
1131:
1119:
1110:
1107:
1100:
1093:
1086:
1082:
1076:
1070:
1063:
1058:
1057:The charge,
1056:
989:
987:
947:
833:
605:is given by
600:
597:
592:
586:
542:
537:
536:is) and the
509:
506:
458:
454:scalar field
448:
444:
439:
437:
430:
427:
302:
299:scalar field
293:
275:scalar field
240:through the
225:
214:
205:
165:
157:ground state
148:ground state
135:
134:
124:
120:
114:
108:
88:
43:
39:
35:
25:
5188:Quark model
4956:Theta meson
4859:Positronium
4771:Omega meson
4766:J/psi meson
4696:Antineutron
4607:Dark photon
4572:Graviphoton
4531:Stop squark
4239:Down (quark
3870:cds.cern.ch
2932:, both the
294:Consider a
5251:Categories
4930:Heptaquark
4891:Superatoms
4824:Pentaquark
4814:Tetraquark
4796:Quarkonium
4686:Antiproton
4587:Leptoquark
4522:Neutralino
4284:antiquark)
4274:antiquark)
4269:Top (quark
4264:antiquark)
4254:antiquark)
4244:antiquark)
4234:antiquark)
4203:Elementary
3875:4 November
3823:1707.00406
3779:2022-01-14
3325:References
3205:goldstinos
2930:superfluid
2704:energy gap
2685:Extensions
1103:〉≠ 0
834:and thus
546:, namely
5168:Particles
5113:Particles
5072:Polariton
5062:Plasmaron
5032:Dropleton
4925:Hexaquark
4896:Molecules
4884:Protonium
4761:Phi meson
4746:Rho meson
4718:Xi baryon
4650:Composite
4486:Gravitino
4229:Up (quark
4085:201256113
3713:0031-9007
3570:0015-8208
3527:0031-899X
3484:0031-899X
3406:120409034
3122:⟩
3108:ϕ
3105:⟨
2986:⟩
2972:ϕ
2969:⟨
2903:∝
2877:∝
2856:different
2717:∝
2675:zero mass
2667:composite
2655:conformal
2653:which is
2644:super QCD
2603:∞
2591:‖
2587:⟩
2573:‖
2482:∫
2445:−
2407:−
2338:∫
2298:−
2229:∫
2216:⟩
2175:⟨
2159:∫
2149:⟩
2124:⟨
2037:‖
2034:⟩
2031:θ
2023:‖
2010:≥
2007:‖
2004:⟩
2001:θ
1990:‖
1979:⟩
1976:θ
1899:⟩
1863:≈
1855:⟩
1785:⋅
1747:−
1735:∇
1726:∫
1716:⋅
1713:∇
1679:−
1665:∫
1661:−
1605:−
1591:∫
1471:⟩
1462:ϕ
1458:δ
1455:⟨
1366:∫
1283:⟩
1274:ϕ
1270:δ
1267:⟨
1251:ϕ
1247:∂
1238:ϕ
1234:∂
1220:∂
1036:θ
1031:μ
1027:∂
1008:μ
905:−
899:θ
894:μ
890:∂
880:θ
875:μ
871:∂
796:−
790:θ
785:μ
781:∂
775:θ
752:θ
747:μ
743:∂
737:θ
731:−
717:−
698:ϕ
693:∗
689:ϕ
675:−
672:ϕ
667:μ
663:∂
654:∗
650:ϕ
644:μ
640:∂
566:ϵ
560:θ
557:δ
524:θ
490:θ
473:ϕ
387:−
384:ϕ
379:∗
375:ϕ
368:λ
351:potential
324:ϕ
319:∗
315:ϕ
260:inflation
142:which is
5144:timeline
4996:R-hadron
4951:Glueball
4935:Skyrmion
4869:Tauonium
4582:Inflaton
4577:Graviton
4557:Curvaton
4527:Sfermion
4517:Higgsino
4512:Chargino
4473:Gauginos
4332:Neutrino
4317:Antimuon
4307:Positron
4302:Electron
4212:Fermions
4040:15997403
4032:11909340
3848:59392755
3769:"min 42"
3740:Archived
3243:See also
3237:any spin
3200:fermions
2848:and not
2542:⟩
2499:⟨
2468:⟩
2355:⟨
2324:⟩
2246:⟨
2100:|0〉 = 0
2055:Letting
1983:‖
1952:‖
1903:‖
1877:‖
1859:‖
1818:‖
1527:mass gap
1263:⟩
1213:⟨
1128:massless
272:inflaton
219:are the
171:Examples
100:massless
91:spinless
28:particle
5132:Related
5103:Baryons
5077:Polaron
5067:Plasmon
5042:Fracton
5037:Exciton
4991:Diquark
4986:Pomeron
4961:T meson
4918:Baryons
4879:Pionium
4864:Muonium
4791:D meson
4786:B meson
4691:Neutron
4676:Nucleon
4668:Baryons
4659:Hadrons
4622:Tachyon
4597:Majoron
4562:Dilaton
4491:Photino
4327:Antitau
4294:Leptons
4143:Bibcode
4108:Bibcode
4012:Bibcode
3975:1131296
3953:Bibcode
3926:1131296
3904:Bibcode
3828:Bibcode
3693:Bibcode
3608:(2004).
3550:Bibcode
3507:Bibcode
3464:Bibcode
3429:Bibcode
3386:Bibcode
3347:Bibcode
3254:Majoron
3222:
3210:
2959:Lorentz
2657:in the
2620:|0〉 = 0
2085:|0〉 = 0
296:complex
258:during
207:magnons
202:magnets
176:Natural
46:) are
5257:Bosons
5108:Mesons
5057:Phonon
5052:Magnon
4974:Others
4944:Mesons
4837:Others
4733:Mesons
4681:Proton
4545:Others
4500:Others
4481:Gluino
4415:Scalar
4395:Photon
4378:Bosons
4221:Quarks
4083:
4073:
4038:
4030:
3973:
3924:
3846:
3711:
3591:
3568:
3525:
3482:
3404:
2944:phonon
2648:squark
1292:
1039:
928:
569:
507:where
410:
290:Theory
195:solids
187:phonon
185:, the
183:fluids
152:scalar
111:bosons
89:These
48:bosons
5096:Lists
5087:Trion
5082:Roton
5022:Anyon
4849:Atoms
4612:Preon
4552:Axion
4507:Axino
4400:Gluon
4387:Gauge
4081:S2CID
4036:S2CID
4002:arXiv
3866:(PDF)
3844:S2CID
3818:arXiv
3402:S2CID
3280:Notes
3202:, or
1172:〉 = 0
1157:〉 = 0
1144:, an
1079:with
1073:〉 = 0
266:of a
216:pions
193:. In
125:PNGBs
78:. In
5047:Hole
4874:Onia
4781:Kaon
4741:Pion
4312:Muon
4071:ISBN
4028:PMID
3971:OSTI
3922:OSTI
3877:2023
3709:ISSN
3589:ISBN
3566:ISSN
3523:ISSN
3480:ISSN
2935:U(1)
2860:same
2642:= 1
1438:,
1099:(0)|
1085:〉 =
213:The
44:NGBs
30:and
4322:Tau
4151:doi
4139:B49
4116:doi
4104:B46
4063:doi
4020:doi
3961:doi
3912:doi
3836:doi
3701:doi
3558:doi
3515:doi
3503:112
3472:doi
3460:110
3437:doi
3425:127
3394:doi
3355:doi
3343:117
3233:all
3063:to
2651:VEV
2104:= 0
2102:,
2070:).
1418:not
1126:is
1092:〈0|
589:|0〉
433:→ ∞
357:,
200:In
181:In
127:).
119:or
62:in
54:of
38:or
26:In
5253::
4149:.
4137:.
4114:.
4102:.
4079:.
4069:.
4057:.
4034:.
4026:.
4018:.
4010:.
3998:88
3996:.
3992:.
3969:.
3959:.
3949:66
3947:.
3943:.
3920:.
3910:.
3900:66
3898:.
3894:.
3868:.
3842:.
3834:.
3826:.
3814:13
3812:.
3808:.
3707:.
3699:.
3689:16
3687:.
3564:.
3556:.
3544:.
3521:.
3513:.
3501:.
3478:.
3470:.
3458:.
3435:.
3423:.
3400:.
3392:.
3382:19
3380:.
3376:.
3353:.
3341:.
3219:2
3213:1
2827:μN
2681:.
2659:IR
2106:.
1940:,
1529:.
1508:δ(
1427:δϕ
1397:0.
1201:,
1190:δϕ
1186:,
1165:/∂
1161:〈∂
1137:δϕ
1105:.
449:εϕ
447:=i
445:δϕ
163:.
106:.
34:,
4820:)
4816:(
4533:)
4529:(
4187:e
4180:t
4173:v
4157:.
4153::
4145::
4122:.
4118::
4110::
4087:.
4065::
4042:.
4022::
4014::
4004::
3977:.
3963::
3955::
3928:.
3914::
3906::
3879:.
3850:.
3838::
3830::
3820::
3782:.
3715:.
3703::
3695::
3597:.
3572:.
3560::
3552::
3546:6
3529:.
3517::
3509::
3486:.
3474::
3466::
3443:.
3439::
3431::
3408:.
3396::
3388::
3361:.
3357::
3349::
3216:/
3170:)
3166:r
3162:(
3157:a
3153:c
3128:0
3125:=
3119:)
3115:r
3111:(
3100:a
3096:T
3092:)
3088:r
3084:(
3079:a
3075:c
3051:)
3047:r
3043:(
3038:a
3034:c
3011:a
3007:T
2983:)
2979:r
2975:(
2911:2
2907:p
2900:E
2880:p
2874:E
2851:H
2844:→
2840:P
2838:⋅
2835:→
2831:α
2829:−
2825:−
2823:H
2808:0
2805:=
2802:p
2782:c
2779:p
2776:=
2773:E
2753:0
2750:=
2747:p
2725:n
2721:p
2714:E
2640:N
2624:Q
2618:Q
2600:=
2595:2
2584:0
2580:|
2576:Q
2538:0
2534:|
2530:Q
2527:)
2524:0
2521:(
2516:0
2512:j
2507:|
2503:0
2495:x
2490:3
2486:d
2479:=
2464:0
2460:|
2454:x
2451:P
2448:i
2441:e
2435:x
2432:P
2429:i
2425:e
2421:Q
2416:x
2413:P
2410:i
2403:e
2399:)
2396:0
2393:(
2388:0
2384:j
2378:x
2375:P
2372:i
2368:e
2363:|
2359:0
2351:x
2346:3
2342:d
2335:=
2320:0
2316:|
2312:Q
2307:x
2304:P
2301:i
2294:e
2290:)
2287:0
2284:(
2279:0
2275:j
2269:x
2266:P
2263:i
2259:e
2254:|
2250:0
2242:x
2237:3
2233:d
2226:=
2213:0
2209:|
2205:Q
2202:)
2199:x
2196:(
2191:0
2187:j
2182:|
2178:0
2172:x
2167:3
2163:d
2156:=
2146:0
2142:|
2138:Q
2135:Q
2131:|
2127:0
2098:P
2093:Q
2087:.
2083:Q
2078:Q
2064:0
2061:m
2057:A
2040:.
2027:|
2018:0
2014:m
1997:|
1993:H
1987:=
1972:|
1965:t
1962:d
1958:d
1937:0
1934:m
1928:0
1925:m
1907:.
1896:0
1892:|
1886:A
1882:Q
1871:A
1868:1
1852:0
1848:|
1842:A
1838:Q
1831:t
1828:d
1824:d
1791:,
1788:J
1781:)
1771:2
1767:A
1763:2
1757:2
1753:x
1743:e
1739:(
1730:x
1722:=
1719:J
1703:2
1699:A
1695:2
1689:2
1685:x
1675:e
1669:x
1658:=
1655:)
1652:x
1649:(
1644:0
1640:J
1629:2
1625:A
1621:2
1615:2
1611:x
1601:e
1595:x
1584:t
1581:d
1577:d
1572:=
1567:A
1563:Q
1556:t
1553:d
1549:d
1534:A
1522:k
1512:)
1510:k
1491:,
1488:0
1485:=
1479:t
1476:d
1466:g
1452:d
1432:〉
1430:g
1425:〈
1394:=
1391:)
1388:x
1385:(
1380:0
1376:J
1370:x
1359:t
1356:d
1352:d
1347:=
1344:Q
1338:t
1335:d
1331:d
1295:,
1289:0
1286:=
1278:j
1255:j
1242:i
1229:V
1224:2
1195:〉
1193:g
1188:〈
1170:i
1167:ϕ
1163:V
1154:g
1152:ϕ
1150:〈
1142:〉
1139:g
1135:〈
1123:g
1120:ϕ
1114:j
1111:ϕ
1101:θ
1097:0
1094:J
1087:ε
1083:θ
1081:〈
1071:θ
1069:〈
1064:θ
1059:Q
1042:.
1021:2
1017:v
1013:=
1004:J
990:U
971:2
967:v
961:2
957:m
931:.
923:2
919:v
913:2
909:m
902:)
886:(
883:)
867:(
862:2
857:2
853:v
847:=
819:,
814:2
810:v
804:2
800:m
793:)
772:i
768:e
764:v
761:i
758:(
755:)
734:i
727:e
723:v
720:i
714:(
709:2
706:1
701:=
683:2
679:m
659:)
636:(
631:2
628:1
623:=
618:L
572:,
563:=
543:θ
538:U
521:v
510:θ
487:i
483:e
479:v
476:=
459:θ
440:U
431:λ
413:,
405:2
401:)
395:2
391:v
371:(
335:2
331:v
327:=
303:ϕ
281:.
252:.
42:(
23:.
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