%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1401.5596v1 [hep-ph] 22 Jan 2014 %%CreationDate: Sun Jun 23 23:58:55 2019 %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman rtxr txsy Times-Italic Times-Bold rtxmi %%+ rtxi rtxbmi Times-BoldItalic txex txbsy rtxb txsya %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z manuscript_v2.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2019.06.23:2358 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. 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b(is)h(only)e(de\002ned)g(up)h(to)g(a)h(right)e(multiplica-)-287 5241 y(tion)23 b(by)f(an)g(element)g(of)i Fx(H)t FG(,)f(and)f(it)h(is)g (common)e(to)h(parametrize)f(it)i(as)-287 5340 y Fx(U)6 b FG(\()t Fx(x)p FG(\))23 b FF(=)g Fx(e)12 5310 y FE(i)p Fp(\031)58 5289 y Fm(a)83 5310 y FE(\()s Fo(x)p FE(\))p Fo(T)185 5319 y Fm(a)234 5340 y FG(in)d(terms)g(of)f(NG)h(\002elds)g Fr(\031)997 5310 y Fo(a)1030 5340 y FG(\()t Fx(x)p FG(\))g(and)f(the)g (brok)o(en)f(gen-)-287 5440 y(erators)27 b(of)e Fx(G)r FG(,)k Fx(T)224 5452 y Fo(a)256 5440 y FG(.)46 b(This)28 b(w)o(ay)-5 b(,)28 b(one)e(can)h(obtain)f(a)i Fx(one-to-one)c FG(\002eld)-287 5540 y(parametrization)e(for)i(an)o(y)g(uniform)e (internal)h(symmetry)-5 b(.)36 b(When)24 b(the)-287 5639 y(massi)n(v)o(e)h(modes)g(are)g(simply)f(dropped,)g Fr(\036)p FG(\()t Fx(x)p FG(\))31 b FF(=)h Fx(e)1240 5609 y FE(i)p Fp(\031)1286 5588 y Fm(a)1312 5609 y FE(\()s Fo(x)p FE(\))p Fo(T)1414 5618 y Fm(a)1443 5639 y Fr(\036)1489 5651 y FE(0)1522 5639 y FG(,)27 b(one)d(ar)n(-)-287 5739 y(ri)n(v)o(es)c(at)h (a)g(particular)d(EFT)j(kno)n(wn)e(as)h(the)h(nonlinear)d(sigma)i (model.)p Black Black 2080 160 a(F)o(or)i(nonuniform)e(symmetries,)j (on)f(the)h(contrary)-5 b(,)22 b(this)h(prescrip-)1955 259 y(tion)g(does)f(not)h(lead)g(to)g(a)g(one-to-one)d(\002eld)j (parametrization.)31 b(In)23 b(the)1955 359 y(e)o(xample)18 b(from)g(the)h(pre)n(vious)e(section,)i(the)g(\002eld)g Fr(\036)p FG(\()t Fx(x)p FG(\))g(can)g(ob)o(viously)1955 459 y(be)j(obtained)e(by)h(a)h(local)g(shift)g(generated)e(by)25 b Fx(M)s FG(,)e(acting)e(on)h(the)f(v)n(ac-)1955 558 y(uum,)g Fr(\036)2192 570 y FE(0)2251 558 y FF(=)k FG(0.)30 b(F)o(ormally)-5 b(,)20 b(this)i(can)g(be)f(written)h(as)g Fr(\036)p FG(\()t Fx(x)p FG(\))j FF(=)h Fx(e)3776 528 y FE(i)p Fp(\031)p FE(\()s Fo(x)p FE(\))s Fo(M)3948 558 y Fr(\036)3994 570 y FE(0)4026 558 y FG(.)1955 658 y(As)d(observ)o(ed)c (before,)i(local)g(\003uctuations)g Fr(\032)3281 670 y Fp(\026)3317 658 y FG(\()t Fx(x)p FG(\))g(induced)g(by)i Fx(N)3888 628 y Fp(\026)3946 658 y FG(are)1955 758 y(not)17 b(independent;)f(the)o(y)h(can)g(be)h(completely)e(remo)o(v)o(ed)f(by)i (a)h(rede\002ni-)1955 857 y(tion)24 b(of)h Fr(\031)p FG(\()t Fx(x)p FG(\).)37 b(As)26 b(another)d(e)o(xample,)h(consider)f (a)i(tw)o(o-dimensional)1955 957 y(crystal)16 b(lattice.)24 b(The)16 b(spontaneously)d(brok)o(en)h(generators)h(in)h(this)g(case) 1955 1056 y(include)24 b(translations)j Fx(P)2690 1068 y Fo(x)2719 1056 y Fr(;)16 b Fx(P)2807 1068 y Fo(y)2863 1056 y FG(and)24 b(the)h(rotation)j Fx(J)t FG(.)40 b(Using)25 b(the)h(well-)1955 1156 y(kno)n(wn)18 b(relation)23 b Fx(J)k FF(=)g Fx(x)r(P)2711 1168 y Fo(y)2757 1156 y Fl(\000)16 b Fx(y)r(P)2919 1168 y Fo(x)2949 1156 y FG(,)k(the)f(w)o(ould-be)f (rotation)h(NG)h(mode)1955 1256 y Fr(\022)q FG(\()t Fx(x)p FG(\))h(can)f(be)h(eliminated)e(in)i(f)o(a)n(v)n(or)f(of)g(the)g (translation)g(NG)h(modes,)e(as)1955 1355 y(follo)n(ws)25 b(from)f Fx(e)2452 1325 y FE(i)p Fp(\022)t Fo(J)2563 1355 y FF(=)32 b Fx(e)2685 1325 y FE(i[\()s Fo(x)p Fp(\022)q FE(\))q Fo(P)2851 1334 y Fm(y)2874 1325 y FC(\000)p FE(\()p Fo(y)p Fp(\022)q FE(\))q Fo(P)3041 1334 y Fm(x)3063 1325 y FE(])3087 1355 y FG(.)41 b(The)25 b(origin)f(and)h(importance)1955 1455 y(of)19 b(such)f(relations)h(among)e(symmetry)g(generators)h(will) h(be)g(discussed)1955 1555 y(at)i(length)e(in)h(Sec.)p 0 0 1 TeXcolorrgb 2511 1556 a SDict begin H.S end 2511 1556 a 0 0 1 TeXcolorrgb -1 x FG(5)p 0 0 1 TeXcolorrgb 2552 1498 a SDict begin H.R end 2552 1498 a 2552 1555 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 2552 1555 a Black FG(.)2080 1654 y(In)15 b(Ref.)h([)p 0 0 1 TeXcolorrgb 2350 1654 a SDict begin H.S end 2350 1654 a FG(12)2434 1654 y SDict begin 12 H.L end 2434 1654 a 2434 1654 a SDict begin [/Subtype /Link/Dest (cite.Nicolis:2013sga) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 2434 1654 a Black -1 w FG(],)g(this)g(redundanc)o(y)d(is)k(interpreted)c(as)k(a)f (sort)g(of)f(gauge)1955 1754 y(freedom:)23 b(one)d(starts)h(by)e (introducing)f(one)h(NG)i(\002eld)f(for)f(each)h(spon-)1955 1853 y(taneously)14 b(brok)o(en)g(generator)f(and)h(then)h(notices)g (that)g(the)h(correspond-)1955 1953 y(ing)g(\003uctuations)f(of)h(the)g (order)f(parameter)f(are)i(not)g(independent.)21 b(The)1955 2053 y(parametrization)d(of)h(the)h(microscopic)e(\002eld)i Fr(\036)p FG(\()t Fx(x)p FG(\))g(can)g(be)f(made)h(one-)1955 2152 y(to-one)e(by)h(\223\002xing)f(the)i(gauge\224,)d(that)j(is,)g(by) f(setting)g(some)g(of)g(the)g(NG)1955 2252 y(\002elds)e(to)h(zero.)23 b(From)16 b(the)h(point)f(of)h(vie)n(w)g(of)f(the)h(microscopic)f (theory)-5 b(,)1955 2352 y(ho)n(we)n(v)o(er)m(,)16 b(it)j(is)g(ob)o (viously)d(more)h(natural)g(to)h(w)o(ork)g(with)g(a)h(one-to-one)1955 2451 y(\002eld)h(parametrization)e(to)j(start)f(with.)1955 2567 y SDict begin H.S end 1955 2567 a 1955 2567 a SDict begin 12 H.A end 1955 2567 a 1955 2567 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.2) cvn /DEST pdfmark end 1955 2567 a 82 x Fx(4.2.)40 b(E)p Fw(\013)p Fx(ective)20 b(\002eld)g(theory)g(pictur)m(e)2080 2772 y FG(W)m(ithin)h(the)g(coset) h(construction,)d(one)i(b)n(uilds)g(a)g(model-indepen-)1955 2871 y(dent)j(lo)n(w-ener)o(gy)e(EFT)i(using)g(the)h(information)d (about)h(the)i(symme-)1955 2971 y(try)31 b(breaking)f(alone.)58 b(F)o(ormally)-5 b(,)32 b(this)g(means)f(that)h(the)f(dynamical)1955 3071 y(de)o(grees)c(of)h(freedom)f(are)h(gi)n(v)o(en)f(by)h(the)g (matrix)g Fx(U)6 b FG(\()t Fx(x)p FG(\))28 b(itself)h(rather)1955 3170 y(than)e(its)i(action)f(on)f(a)i(particular)d(order)h(parameter)f Fr(\036)3598 3182 y FE(0)3631 3170 y FG(.)49 b(Hence,)29 b(all)1955 3270 y(the)19 b(\002elds)h Fr(\031)2320 3240 y Fo(a)2352 3270 y FG(\()t Fx(x)p FG(\))f Fx(ar)m(e)h FG(independent)c(in)j(this)h(picture.)j(F)o(or)c(nonuniform)1955 3369 y(symmetries,)26 b(the)g(translations)f(ha)n(v)o(e)h(to)g(be)f (included)g(in)h(the)f(de\002ni-)1955 3469 y(tion)20 b(of)h(the)g(coset)g(element,)f(and)g(we)h(thus)f(set)i Fx(U)6 b FG(\()t Fx(x)p FG(\))23 b FF(=)h Fx(e)3648 3439 y FE(i)s Fo(x)3693 3418 y Fn(\026)3720 3439 y Fo(P)3756 3448 y Fn(\026)3787 3469 y Fx(e)3824 3439 y FE(i)p Fp(\031)3870 3418 y Fm(a)3895 3439 y FE(\()s Fo(x)p FE(\))p Fo(T)3997 3448 y Fm(a)4026 3469 y FG(.)1955 3569 y(The)h(action)f(of)h(a)g (transformation)d Fx(g)j FG(from)f(the)h(symmetry)e(group)e Fx(G)r FG(,)1955 3668 y(no)n(w)f(including)e(spacetime)i (transformations,)d(is)22 b(de\002ned)d(by)h([)p 0 0 1 TeXcolorrgb 3840 3668 a SDict begin H.S end 3840 3668 a FG(10)3924 3668 y SDict begin 12 H.L end 3924 3668 a 3924 3668 a SDict begin [/Subtype /Link/Dest (cite.Ogievetsky) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3924 3668 a Black -2 w FG(])2325 3848 y SDict begin H.S end 2325 3848 a 2325 3848 a SDict begin 12 H.A end 2325 3848 a 2325 3848 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 2325 3848 a Fx(ge)2404 3814 y FE(i)s Fo(x)2449 3793 y Fn(\026)2476 3814 y Fo(P)2512 3823 y Fn(\026)2542 3848 y Fx(e)2579 3814 y FE(i)p Fp(\031)2625 3793 y Fm(a)2651 3814 y FE(\()s Fo(x)p FE(\))p Fo(T)2753 3823 y Fm(a)2805 3848 y FF(=)j Fx(e)2918 3814 y FE(i)s Fo(x)2963 3793 y Fk(0)p Fn(\026)3004 3814 y Fo(P)3040 3823 y Fn(\026)3071 3848 y Fx(e)3108 3814 y FE(i)p Fp(\031)3154 3793 y Fk(0)p Fm(a)3194 3814 y FE(\()s Fo(x)3242 3793 y Fk(0)3260 3814 y FE(\))p Fo(T)3314 3823 y Fm(a)3344 3848 y Fx(h)p FG(\()p Fx(g)p Fr(;)14 b(\031)p FG(\()t Fx(x)p FG(\)\))p Fr(;)269 b FG(\(4\))1955 4028 y(where)24 b Fx(h)32 b Fl(2)j Fx(H)29 b FG(is)d(a)f(compensating)e(transformation)g(which)h(ensures)1955 4128 y(that)c Fx(U)27 b FG(remains)19 b(in)h(the)g(coset)h(space)d Fx(G)r Fr(=)r Fx(H)t FG(.)25 b(Denoting)18 b(the)i(unbrok)o(en)1955 4227 y(generators)f(of)j Fx(H)j FG(\(apart)20 b(from)f(the)i(momentum)d (operator)j Fx(P)3769 4239 y Fp(\026)3804 4227 y FG(\))g(as)g Fx(T)3993 4239 y Fp(\032)4026 4227 y FG(,)1955 4327 y(the)f(MC)h(form,) e(de\002ned)g(by)h Fr(!)j FF(=)g Fl(\000)p FG(i)p Fx(U)3097 4297 y FC(\000)p FE(1)3167 4327 y FG(d)p Fx(U)6 b FG(,)20 b(can)g(be)g(e)o(xpanded)d(as)2594 4507 y SDict begin H.S end 2594 4507 a 2594 4507 a SDict begin 12 H.A end 2594 4507 a 2594 4507 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 2594 4507 a Fr(!)23 b FF(=)g Fr(!)2808 4519 y Fo(P)2867 4507 y FF(+)18 b Fr(!)2995 4473 y Fo(a)2995 4527 y FC(?)3041 4507 y Fx(T)3091 4519 y Fo(a)3142 4507 y FF(+)h Fr(!)3271 4466 y Fp(\032)3271 4536 y FC(k)3305 4507 y Fx(T)3355 4519 y Fp(\032)3388 4507 y Fr(;)541 b FG(\(5\))1955 4701 y(where)24 b Fr(!)2241 4713 y Fo(P)2313 4701 y FF(=)31 b Fx(e)2460 4660 y Fp(\026)2434 4716 y(\027)2497 4701 y Fx(P)2548 4713 y Fp(\026)2584 4701 y FG(d)t Fx(x)2667 4671 y Fp(\027)2697 4701 y FG(.)39 b(The)24 b(indi)n(vidual)f(components)f(of)i(the)h(MC)1955 4801 y(form)19 b(then)h(transform)e(in)j(accord)e(with)h(Eq.)g(\()p 0 0 1 TeXcolorrgb 3302 4801 a SDict begin H.S end 3302 4801 a 0 0 1 TeXcolorrgb FG(4)p 0 0 1 TeXcolorrgb 3343 4745 a SDict begin H.R end 3343 4745 a 3343 4801 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 3343 4801 a Black FG(\))g(as)2225 4981 y SDict begin H.S end 2225 4981 a 2225 4981 a SDict begin 12 H.A end 2225 4981 a 2225 4981 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 2225 4981 a Fr(!)2283 4993 y Fo(P)-6 b Fp(;)p FC(?)2396 4981 y Fl(!)23 b Fx(h)p Fr(!)2599 4993 y Fo(P)-6 b Fp(;)p FC(?)2688 4981 y Fx(h)2730 4946 y FC(\000)p FE(1)2800 4981 y Fr(;)179 b(!)3057 4993 y FC(k)3108 4981 y Fl(!)23 b Fx(h)p Fr(!)3310 4993 y FC(k)3338 4981 y Fx(h)3380 4946 y FC(\000)p FE(1)3468 4981 y Fl(\000)18 b FG(i)p Fx(h)p FG(d)p Fx(h)3688 4946 y FC(\000)p FE(1)3757 4981 y Fr(:)172 b FG(\(6\))1955 5161 y(If)30 b(necessary)-5 b(,)32 b(the)e(EFT)h(can)f(be)h(augmented)d (with)j(other)m(,)g(non-NG)1955 5260 y(de)o(grees)f(of)g(freedom,)h (usually)g(called)f Fx(matter)j FG(\002elds.)57 b(Consider)30 b(a)1955 5360 y(\002eld)f Fr( )h FG(that)e(transforms)g(in)h(some)f (linear)h(representation)d Fx(R)j FG(of)g(the)1955 5459 y(unbrok)o(en)15 b(subgroup)i Fx(H)t FG(,)h Fr( )2769 5429 y FC(0)2793 5459 y FG(\()t Fx(x)2862 5429 y FC(0)2886 5459 y FG(\))23 b FF(=)g Fx(R)p FG(\()p Fx(h)p FG(\))p Fr( )p FG(\()t Fx(x)p FG(\).)g(This)17 b(can)g(be)h(e)o(xtended)1955 5559 y(using)i(Eq.)f(\()p 0 0 1 TeXcolorrgb 2317 5559 a SDict begin H.S end 2317 5559 a 0 0 1 TeXcolorrgb FG(4)p 0 0 1 TeXcolorrgb 2359 5503 a SDict begin H.R end 2359 5503 a 2359 5559 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 2359 5559 a Black FG(\))h(to)g(a)h(nonlinear)d(realization)h(of)h(the)h(whole)e(group)e Fx(G)r FG(,)2560 5739 y SDict begin H.S end 2560 5739 a 2560 5739 a SDict 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b(In)22 b(this)g(section,)g(we)g(will)-287 3805 y(sho)n(w)h(that)g (these)g(follo)n(w)f(directly)g(from)g(the)g(properties)g(of)g (classical)-287 3904 y(local)17 b(symmetry)f(transformations,)e(our)i (main)h(result)f(being)g(Eq.)h(\()p 0 0 1 TeXcolorrgb 1675 3905 a SDict begin H.S end 1675 3905 a 0 0 1 TeXcolorrgb -1 x FG(20)p 0 0 1 TeXcolorrgb 1757 3849 a SDict begin H.R end 1757 3849 a 1757 3904 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.20) cvn H.B /ANN pdfmark end 1757 3904 a Black FG(\).)-287 4022 y SDict begin H.S end -287 4022 a -287 4022 a SDict begin 12 H.A end -287 4022 a -287 4022 a SDict begin [/View [/XYZ H.V]/Dest (subsection.5.1) cvn /DEST pdfmark end -287 4022 a 81 x Fx(5.1.)40 b(De\002nition)19 b(of)i(a)f(Noether)g(curr)m(ent)-162 4228 y FG(Consider)h(a)i(theory)e(of)h(a)g(set)h(of)f(\(not)f (necessarily)h(scalar\))g(\002elds,)-287 4328 y(denoted)15 b(collecti)n(v)o(ely)g(as)i Fr(\036)p FG(\()t Fx(x)p FG(\),)g(de\002ned)e(by)h(the)h(action)f Fx(S)d FG([)p Fr(\036)p FG(\()t Fx(x)p FG(\)].)22 b(This)-287 4427 y(is)c(assumed)f(to)g(be)g(a)h(spacetime)e(inte)o(gral)g(of)h(a)g (local)g(Lagrangian)e(den-)-287 4527 y(sity)-5 b(,)31 b(which)d(is)i(a)f(function)e(of)h Fr(\036)p FG(\()t Fx(x)p FG(\))h(and)f(its)h(deri)n(v)n(ati)n(v)o(es.)49 b(Suppose)-287 4626 y(that)21 b(the)f(action)f(is)i(in)m(v)n(ariant)e (under)f(a)j(simultaneous)e(transformation)-287 4726 y(of)h(the)h(\002elds)f(and)g(the)g(spacetime)g(coordinates,)-91 4909 y SDict begin H.S end -91 4909 a -91 4909 a SDict begin 12 H.A end -91 4909 a -91 4909 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.16) cvn /DEST pdfmark end -91 4909 a Fr(\036)-45 4874 y FC(0)-21 4909 y FG(\()t Fx(x)48 4874 y FC(0)73 4909 y FG(\))i FF(=)h Fr(\036)p FG(\()t Fx(x)p FG(\))18 b FF(+)g Fr(\017)5 b(\020)g FG(\()p Fr(\036)p FG(\()t Fx(x)p FG(\))p Fr(;)18 b Fx(x)p FG(\))p Fr(;)182 b Fx(x)1023 4874 y FC(0)1071 4909 y FF(=)27 b Fx(x)19 b FF(+)f Fr(\017)5 b(\030)r FG(\()t Fx(x)p FG(\))p Fr(:)196 b FG(\(16\))-287 5091 y(Here)23 b Fr(\017)29 b FG(is)24 b(an)f(in\002nitesimal)f(parameter)g(of)g(the)h(transformation.)31 b(Let)-287 5191 y(us)25 b(no)n(w)g(perform)d(an)j(in\002nitesimal)f (transformation)e(with)j(a)g Fx(coor)m(di-)-287 5291 y(nate-dependent)d FG(parameter)h Fr(\017)5 b FG(\()t Fx(x)p FG(\))24 b(such)g(that)g(it)h(reduces)e(to)h(Eq.)f(\()p 0 0 1 TeXcolorrgb 1695 5292 a SDict begin H.S end 1695 5292 a 0 0 1 TeXcolorrgb -1 x FG(16)p 0 0 1 TeXcolorrgb 1778 5234 a SDict begin H.R end 1778 5234 a 1778 5291 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.16) cvn H.B /ANN pdfmark end 1778 5291 a Black FG(\))-287 5390 y(for)f(constant)f Fr(\017)5 b FG(\()t Fx(x)p FG(\))26 b FF(=)g Fr(\017)5 b FG(.)31 b(Owing)21 b(to)h(the)g(in)m(v)n(ariance)e(assumption,)h(the) -287 5490 y(v)n(ariation)e(of)h(the)g(action)g(has)g(to)h(tak)o(e)f (the)g(form)359 5695 y SDict begin H.S end 359 5695 a 359 5695 a SDict begin 12 H.A end 359 5695 a 359 5695 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.17) cvn /DEST pdfmark end 359 5695 a Fr(\016)p Fx(S)35 b FF(=)551 5582 y Ff(Z)655 5695 y FG(d)t Fx(x)17 b(J)796 5661 y Fp(\026)832 5695 y FG(\()t Fx(x)p FG(\))p Fr(@)972 5707 y Fp(\026)1006 5695 y Fr(\017)5 b FG(\()t Fx(x)p FG(\))p Fr(;)506 b FG(\(17\))p Black Black 1955 160 a(which)24 b(de\002nes)h(the)f(associated)h(Noether)f(current)j Fx(J)3535 130 y Fp(\026)3570 160 y FG(\()t Fx(x)p FG(\).)39 b(There)24 b(are)1955 259 y(tw)o(o)d(sources)f(of)h(ambiguity)e(in)i (this)g(de\002nition.)26 b(First,)21 b(the)g(e)o(xtension)1955 359 y(of)d(the)g(symmetry)f(transformation)f(to)i(local)g Fr(\017)5 b FG(\()t Fx(x)p FG(\))19 b(is)g(not)f(unique.)23 b(One)1955 459 y(could)29 b(nai)n(v)o(ely)f(tak)o(e)h(the)h(same)f (functional)f(form)g(\()p 0 0 1 TeXcolorrgb 3526 460 a SDict begin H.S end 3526 460 a 0 0 1 TeXcolorrgb -1 x FG(16)p 0 0 1 TeXcolorrgb 3609 402 a SDict begin H.R end 3609 402 a 3609 459 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.16) cvn H.B /ANN pdfmark end 3609 459 a Black FG(\))i(and)f(merely) 1955 558 y(replace)23 b Fr(\017)36 b Fl(!)30 b Fr(\017)5 b FG(\()t Fx(x)p FG(\),)25 b(b)n(ut)f(this)h(is)g(not)e(mandatory)-5 b(.)34 b(An)o(y)23 b(choice)h(com-)1955 658 y(patible)c(with)h(Eq.)f (\()p 0 0 1 TeXcolorrgb 2533 659 a SDict begin H.S end 2533 659 a 0 0 1 TeXcolorrgb -1 x FG(16)p 0 0 1 TeXcolorrgb 2615 601 a SDict begin H.R end 2615 601 a 2615 658 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.16) cvn H.B /ANN pdfmark end 2615 658 a Black FG(\))h(gi)n(v)o(es)f(a)h(current)e(that)i (is)g(conserv)o(ed)d(for)i(\002elds)1955 758 y(satisfying)j(the)h (equations)e(of)i(motion)e(\()p Fx(on-shell)p FG(\).)34 b(Second,)23 b(shifting)1955 857 y(the)i(current)f(by)g(a)i(v)o(ector)e (function)d Fx(G)3108 827 y Fp(\026)3143 857 y FG(\()t Fx(x)p FG(\))k(such)g(that)g Fr(@)3636 869 y Fp(\026)3669 857 y Fx(G)3731 827 y Fp(\026)3798 857 y FF(=)32 b FG(0)25 b(for)1955 957 y(all)d(\002eld)f(con\002gurations)d(\()p Fx(o)p Fw(\013)p Fx(-shell)p FG(\))i(does)h(not)f(a)p FF(\013)p FG(ect)h(Eq.)g(\()p 0 0 1 TeXcolorrgb 3741 958 a SDict begin H.S end 3741 958 a 0 0 1 TeXcolorrgb -1 x FG(17)p 0 0 1 TeXcolorrgb 3823 901 a SDict begin H.R end 3823 901 a 3823 957 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.17) cvn H.B /ANN pdfmark end 3823 957 a Black FG(\).)28 b(This)1955 1056 y(kind)19 b(of)h(ambiguity)e(gi)n(v) o(es)h(rise)h(to)g(an)g(equi)n(v)n(alence)e(relation)h(that)h(will)1955 1156 y(be)g(denoted)f(by)h(the)g(symbol)f Fb(,)p FG(,)h(that)h(is,)j Fx(J)3210 1126 y Fp(\026)3269 1156 y Fb(,)j Fx(J)3390 1126 y Fp(\026)3443 1156 y FF(+)16 b Fx(G)3574 1126 y Fp(\026)3610 1156 y FG(.)1955 1274 y SDict begin H.S end 1955 1274 a 1955 1274 a SDict begin 12 H.A end 1955 1274 a 1955 1274 a SDict begin [/View [/XYZ H.V]/Dest (subsection.5.2) cvn /DEST pdfmark end 1955 1274 a 81 x Fx(5.2.)40 b(Locally)20 b(identical)f(\002eld)h(tr)o(ansformations)2080 1480 y FG(Consider)f(tw)o(o)i(classes)h(of)e(symmetry)f (transformations,)f(charac-)1955 1580 y(terized)f(by)h(in\002nitesimal) f(parameters)g Fr(\017)3154 1549 y Fp(\013)3149 1608 y FE(1)3212 1580 y FG(and)g Fr(\017)3387 1549 y Fo(a)3382 1607 y FE(2)3420 1580 y FG(.)25 b(Assume)18 b(that)g(their)1955 1679 y(\(suitably)23 b(chosen\))f(local)h(forms)g(coincide,)g(that)g (is,)i(there)e(is)i(a)f(set)g(of)1955 1779 y(functions)34 b Fx(f)2335 1749 y Fo(a)2323 1799 y Fp(\013)2369 1779 y FG(\()t Fx(x)p FG(\))23 b(such)g(that)h(setting)f Fr(\017)3096 1749 y Fo(a)3091 1807 y FE(2)3130 1779 y FG(\()t Fx(x)p FG(\))28 b FF(=)41 b Fx(f)3384 1749 y Fo(a)3372 1799 y Fp(\013)3418 1779 y FG(\()t Fx(x)p FG(\))p Fr(\017)3552 1749 y Fp(\013)3547 1807 y FE(1)3590 1779 y FG(\()t Fx(x)p FG(\))24 b(mak)o(es)f(the)1955 1878 y(tw)o(o)e(local)g(transformations) e Fx(identical)p FG(.)26 b(The)21 b(v)n(ariation)f(of)g(the)h(action) 1955 1978 y(under)e(such)h(transformation)d(reads)2091 2183 y SDict begin H.S end 2091 2183 a 2091 2183 a SDict begin 12 H.A end 2091 2183 a 2091 2183 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.18) cvn /DEST pdfmark end 2091 2183 a Fr(\016)p Fx(S)35 b FF(=)2283 2070 y Ff(Z)2387 2183 y FG(d)t Fx(x)18 b(J)2529 2142 y Fp(\026)2525 2211 y FE(2)p Fo(a)2587 2183 y Fr(@)2630 2195 y Fp(\026)2665 2183 y Fr(\017)2702 2149 y Fo(a)2697 2207 y FE(2)2758 2183 y FF(=)2834 2070 y Ff(Z)2938 2183 y FG(d)t Fx(x)g(J)3080 2142 y Fp(\026)3076 2211 y FE(2)p Fo(a)3138 2183 y FG(\()12 b Fx(f)3213 2149 y Fo(a)3201 2204 y Fp(\013)3247 2183 y Fr(@)3290 2195 y Fp(\026)3324 2183 y Fr(\017)3361 2149 y Fp(\013)3356 2207 y FE(1)3419 2183 y FF(+)18 b Fr(\017)3527 2149 y Fp(\013)3522 2207 y FE(1)3566 2183 y Fr(@)3609 2195 y Fp(\026)3656 2183 y Fx(f)3691 2149 y Fo(a)3679 2204 y Fp(\013)3725 2183 y FG(\))p Fr(:)135 b FG(\(18\))1955 2401 y(Ho)n(we)n(v)o(er)m(,)20 b(since)i Fr(\017)2524 2371 y Fp(\013)2519 2430 y FE(1)2563 2401 y FG(\()t Fx(x)p FG(\))g(is)h(a)f(parameter)f(of)g(an)h(identical)f(local)h(trans-)1955 2511 y(formation,)h(this)i(should)f(be)h(equal)e(to)i Fr(\016)p Fx(S)43 b FF(=)3327 2438 y Ff(R)3396 2511 y FG(d)t Fx(x)18 b(J)3538 2470 y Fp(\026)3534 2540 y FE(1)p Fp(\013)3601 2511 y Fr(@)3644 2523 y Fp(\026)3679 2511 y Fr(\017)3716 2481 y Fp(\013)3711 2540 y FE(1)3755 2511 y FG(.)39 b(That)24 b(is)1955 2611 y(only)19 b(possible)h(pro)o(vided) 2733 2794 y SDict begin H.S end 2733 2794 a 2733 2794 a SDict begin 12 H.A end 2733 2794 a 2733 2794 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.19) cvn /DEST pdfmark end 2733 2794 a 4 w Fx(J)2778 2753 y Fp(\026)2774 2822 y FE(2)p Fo(a)2836 2794 y Fr(@)2879 2806 y Fp(\026)2926 2794 y Fx(f)2961 2759 y Fo(a)2949 2814 y Fp(\013)3018 2794 y FF(=)j Fr(@)3137 2806 y Fp(\026)3174 2794 y Fx(N)3234 2753 y Fp(\026)3229 2809 y(\013)3909 2794 y FG(\(19\))1955 2976 y(for)17 b(some)g(v)o(ector)f(function)i Fx(N)2844 2935 y Fp(\026)2839 2991 y(\013)2880 2976 y FG(\()t Fx(x)p FG(\).)24 b(By)18 b(comparison,)e(we)i(then)f(obtain)2701 3159 y SDict begin H.S end 2701 3159 a 2701 3159 a SDict begin 12 H.A end 2701 3159 a 2701 3159 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.20) cvn /DEST pdfmark end 2701 3159 a 4 w Fx(J)2746 3118 y Fp(\026)2742 3187 y FE(1)p Fp(\013)2833 3159 y Fb(,)35 b Fx(f)2956 3125 y Fo(a)2944 3180 y Fp(\013)2994 3159 y Fx(J)3035 3118 y Fp(\026)3031 3187 y FE(2)p Fo(a)3111 3159 y Fl(\000)21 b Fx(N)3245 3118 y Fp(\026)3240 3174 y(\013)3280 3159 y Fr(:)608 b FG(\(20\))1955 3342 y(This)26 b(alone)f(w)o(ould)g(be)g(a) i(v)n(oid)e(statement)g(\(an)o(y)g(tw)o(o)h(currents)e(di)p FF(\013)p FG(er)1955 3441 y(by)31 b(some)h(v)o(ector)e(function\))g(if) i(it)g(were)g(not)f(for)g(the)h(simultaneous)1955 3541 y(condition)19 b(\()p 0 0 1 TeXcolorrgb 2318 3543 a SDict begin H.S end 2318 3543 a 0 0 1 TeXcolorrgb -2 x FG(19)p 0 0 1 TeXcolorrgb 2401 3485 a SDict begin H.R end 2401 3485 a 2401 3541 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.19) cvn H.B /ANN pdfmark end 2401 3541 a Black FG(\))i(which)f(must)h(be)g(satis\002ed)h(o)p FF(\013)p FG(-shell.)k(Thanks)20 b(to)h(this)1955 3641 y(condition,)26 b(the)g(\(on-shell\))f(conserv)n(ation)f(of)30 b Fx(J)3386 3599 y Fp(\026)3382 3669 y FE(1)p Fp(\013)3477 3641 y FG(is)d(a)g(consequence)1955 3740 y(of)21 b(the)g(conserv)n(ation)d(of) 25 b Fx(J)2748 3699 y Fp(\026)2744 3768 y FE(2)p Fo(a)2806 3740 y FG(.)j(In)20 b(f)o(act,)h(a)h(stronger)d(conclusion)g(holds:) 1955 3840 y Fr(@)1998 3852 y Fp(\026)2037 3840 y Fx(J)2078 3799 y Fp(\026)2074 3868 y FE(1)p Fp(\013)2178 3840 y FF(=)48 b Fx(f)2314 3810 y Fo(a)2302 3860 y Fp(\013)2347 3840 y Fr(@)2390 3852 y Fp(\026)2429 3840 y Fx(J)2470 3799 y Fp(\026)2466 3868 y FE(2)p Fo(a)2556 3840 y FG(o)p FF(\013)p FG(-shell.)e(Note)27 b(that)g(the)g(ar)o(gument)e(leading)i (to)1955 3939 y(Eq.)f(\()p 0 0 1 TeXcolorrgb 2123 3940 a SDict begin H.S end 2123 3940 a 0 0 1 TeXcolorrgb -1 x FG(20)p 0 0 1 TeXcolorrgb 2205 3884 a SDict begin H.R end 2205 3884 a 2205 3939 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.20) cvn H.B /ANN pdfmark end 2205 3939 a Black FG(\))g(w)o(as)h(pre)n(viously)d(used)i(to)g(deri)n(v)o(e)f(the)h (conserv)o(ed)e(currents)1955 4039 y(associated)c(with)g(nonrelati)n (vistic)f(Galilei)i(in)m(v)n(ariance)d([)p 0 0 1 TeXcolorrgb 3615 4039 a SDict begin H.S end 3615 4039 a FG(26)3699 4039 y SDict begin 12 H.L end 3699 4039 a 3699 4039 a SDict begin [/Subtype /Link/Dest (cite.Nicolis:2010se) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3699 4039 a Black -1 w FG(].)2080 4139 y(Gi)n(v)o(en)g(the) h(freedom)e(that)i(one)f(has)h(in)g(the)g(de\002nition)f(of)g(the)h (Noe-)1955 4238 y(ther)30 b(currents,)i(one)e(may)h(w)o(onder)e(if)i (the)o(y)f(can)g(be)h(\223impro)o(v)o(ed\224)c(so)1955 4338 y(that)17 b(the)g(correction)e(term)k Fx(N)2802 4297 y Fp(\026)2797 4353 y(\013)2855 4338 y FG(v)n(anishes.)k(This)17 b(is)h(indeed)e(often)g(possi-)1955 4438 y(ble,)k(b)n(ut)g(cannot)f(be) h(achie)n(v)o(ed)e(in)i(general.)k(Recall)d(the)f(free)g(massless)1955 4537 y(scalar)i(\002eld)g(discussed)f(in)h(Sec.)p 0 0 1 TeXcolorrgb 2931 4538 a SDict begin H.S end 2931 4538 a 0 0 1 TeXcolorrgb -1 x FG(3)p 0 0 1 TeXcolorrgb 2973 4482 a SDict begin H.R end 2973 4482 a 2973 4537 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 2973 4537 a Black FG(.)30 b(The)22 b(Noether)e(currents)h (of)h(the)f Fr(\013)p FG(-)1955 4637 y(and)f Fr(\014)p FG(-shifts)g(are)g(easily)h(e)n(v)n(aluated)d(e)o(xplicitly)-5 b(,)2441 4820 y SDict begin H.S end 2441 4820 a 2441 4820 a SDict begin 12 H.A end 2441 4820 a 2441 4820 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.21) cvn /DEST pdfmark end 2441 4820 a 4 w Fx(J)2486 4785 y Fp(\026)2544 4820 y Fb(,)23 b Fr(@)2663 4785 y Fp(\026)2697 4820 y Fr(\036;)184 b Fx(J)2989 4778 y Fp(\026)2985 4834 y(\013)3047 4820 y Fb(,)27 b Fx(x)3164 4832 y Fp(\013)3202 4820 y Fr(@)3245 4785 y Fp(\026)3280 4820 y Fr(\036)18 b Fl(\000)h Fr(\021)3457 4778 y Fp(\026)3457 4834 y(\013)3495 4820 y Fr(\036:)347 b FG(\(21\))1955 5002 y(The)36 b(local)f Fr(\014)p FG(-shift)h(can)g(be)f(reproduced)e(by)j(a)g(local)g Fr(\013)p FG(-shift)g(with)1955 5102 y Fr(\013)p FG(\()t Fx(x)p FG(\))h FF(=)j Fx(x)2268 5114 y Fp(\013)2307 5102 y Fr(\014)2349 5072 y Fp(\013)2388 5102 y FG(\()t Fx(x)p FG(\),)29 b(that)f(is,)42 b Fx(f)2829 5114 y Fp(\013)2868 5102 y FG(\()t Fx(x)p FG(\))37 b FF(=)j Fx(x)3132 5114 y Fp(\013)3171 5102 y FG(.)48 b(W)-7 b(e)29 b(thus)e(ha)n(v)o(e)k Fx(J)3770 5072 y Fp(\026)3806 5102 y Fr(@)3849 5114 y Fp(\026)3895 5102 y Fx(f)3918 5114 y Fp(\013)3994 5102 y FF(=)1955 5201 y Fr(@)1998 5213 y Fp(\013)2036 5201 y Fr(\036)p FG(,)h(which)e(implies)h Fx(N)2705 5160 y Fp(\026)2700 5216 y(\013)2782 5201 y FF(=)41 b Fr(\021)2917 5160 y Fp(\026)2917 5216 y(\013)2955 5201 y Fr(\036)p FG(;)35 b(Eq.)30 b(\()p 0 0 1 TeXcolorrgb 3231 5202 a SDict begin H.S end 3231 5202 a 0 0 1 TeXcolorrgb -1 x FG(20)p 0 0 1 TeXcolorrgb 3313 5146 a SDict begin H.R end 3313 5146 a 3313 5201 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.20) cvn H.B /ANN pdfmark end 3313 5201 a Black FG(\))g(reco)o(v)o(ers)e(the)i(current)1955 5301 y(of)23 b(the)g Fr(\014)p FG(-shifts)g(gi)n(v)o(en)f(in)h(Eq.)f(\()p 0 0 1 TeXcolorrgb 2903 5301 a SDict begin H.S end 2903 5301 a 0 0 1 TeXcolorrgb FG(21)p 0 0 1 TeXcolorrgb 2986 5245 a SDict begin H.R end 2986 5245 a 2986 5301 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.21) cvn H.B /ANN pdfmark end 2986 5301 a Black FG(\))o(.)34 b(Ho)n(we)n(v)o(er)m(,)22 b(it)h(is)h(not)f(possible)g(to)1955 5401 y(impro)o(v)o(e)15 b(the)i(currents)f(so)h(that)2907 5384 y(\230)2888 5401 y Fx(J)2929 5359 y Fp(\026)2925 5416 y(\013)2987 5401 y Fb(,)27 b Fx(x)3104 5413 y Fp(\013)3166 5384 y FG(\230)3147 5401 y Fx(J)3188 5371 y Fp(\026)3240 5401 y FG(holds,)17 b(since)g(current)f(con-)1955 5500 y(serv)n(ation)e(w)o(ould)h(imply) 2730 5483 y(\230)2710 5500 y Fx(J)2747 5512 y Fp(\013)2809 5500 y FF(=)22 b FG(0)16 b(on-shell,)f(in)h(contradiction)d(with)j(the) 1955 5600 y(f)o(act)25 b(that)f(the)h(inte)o(gral)e(of)h(the)g (temporal)g(component)e(of)i(the)g(current)1955 5700 y(is)d(a)g(generator)d(of)i(the)g(associated)g(symmetry)f (transformation.)p Black 1860 5888 a(5)p Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 0 1 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Black Black 2080 160 a(The)24 b(w)o(ay)i(around)d (this)j(problem)d(is)j(to)g(couple)e(both)g(theories)h(to)1955 259 y(a)e(set)g(of)f(background)c(gauge)j(\002elds)k Fx(A)3113 271 y Fp(\026)3148 259 y FG(;)f(their)e(generating)e (function-)1955 359 y(als)26 b(in)e(terms)h(of)f(these)h(\002elds)g (then)g(ha)n(v)o(e)f(to)g(coincide.)38 b(T)-6 b(echnically)h(,)1955 459 y(we)23 b(replace)e(the)i(action)f Fx(S)13 b FG([)p Fr(\036)p FG(])21 b(with)i(a)g(ne)n(w)f(action)3497 442 y(\230)3482 459 y Fx(S)12 b FG([)p Fr(\036;)k Fx(A)p FG(])22 b(such)g(that)1970 541 y(\230)1955 558 y Fx(S)13 b FG([)p Fr(\036;)h FG(0])31 b FF(=)j Fx(S)13 b FG([)p Fr(\036)p FG(],)26 b(which)f(is)i(in)m(v)n(ariant)d(under)g(a)j (simultaneous)d Fx(local)1955 658 y FG(transformation)16 b(of)h Fr(\036)i FG(and)g Fx(A)2806 670 y Fp(\026)2841 658 y FG(.)25 b(If)18 b(we)h(manage)e(to)h(deri)n(v)o(e)f(the)h (Noether)1955 758 y(currents)23 b(based)h(on)h(the)f(transformation)e (rules)i(for)g(the)g(gauge)f(\002elds)1955 857 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Fx(S)57 b FF(=)45 b Fr(\016)2983 1467 y Fp(\036)3033 1438 y FG(\230)3018 1455 y Fx(S)40 b FF(+)27 b Fr(\016)3220 1467 y Fo(A)3275 1438 y FG(\230)3260 1455 y Fx(S)13 b FG(.)60 b(Using)32 b(the)g(assumed)1955 1555 y(gauge)21 b(in)m(v)n(ariance)g(and)i (setting)h Fx(A)k FF(=)g FG(0)23 b(so)h(that)f Fr(\016)3440 1567 y Fp(\036)3490 1538 y FG(\230)3475 1555 y Fx(S)36 b FG(becomes)22 b(equal)1955 1654 y(to)e(Eq.)g(\()p 0 0 1 TeXcolorrgb 2202 1655 a SDict begin H.S end 2202 1655 a 0 0 1 TeXcolorrgb -1 x FG(17)p 0 0 1 TeXcolorrgb 2285 1599 a SDict begin H.R end 2285 1599 a 2285 1654 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.17) cvn H.B /ANN pdfmark end 2285 1654 a Black FG(\))o(,)h(the)f(Noether)f(current)g (can)h(be)g(e)o(xtracted)f(from)2483 1877 y SDict begin H.S end 2483 1877 a 2483 1877 a SDict begin 12 H.A end 2483 1877 a 2483 1877 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.26) cvn /DEST pdfmark end 2483 1877 a 2497 1764 a Ff(Z)2601 1877 y FG(d)t Fx(x)f(J)2743 1842 y Fp(\026)2778 1877 y FG(\()t Fx(x)p FG(\))p Fr(@)2918 1889 y Fp(\026)2952 1877 y Fr(\017)5 b FG(\()t Fx(x)p FG(\))23 b FF(=)g Fl(\000)p Fr(\016)3278 1889 y Fo(A)3332 1860 y FG(\230)3317 1877 y Fx(S)3372 1784 y Ff(\014)3372 1819 y(\014)3372 1854 y(\014)3372 1888 y(\014)3392 1928 y Fo(A)p FD(=)p FE(0)3498 1877 y Fr(;)390 b FG(\(26\))1955 2107 y(that)20 b(is,)g(as)g(minus)g (the)f(coe)p FF(\016)p FG(cient)g(of)g Fr(@)3128 2119 y Fp(\026)3163 2107 y Fr(\017)25 b FG(in)20 b Fr(\016)3345 2119 y Fo(A)3400 2091 y FG(\230)3385 2107 y Fx(S)12 b FG(.)26 b(The)19 b(f)o(act)h(that)f(tw)o(o)1955 2207 y(di)p FF(\013)p FG(erent)31 b(global)g(symmetries)g(ha)n(v)o(e)h(the)g (same)g(local)g(form)f(means)1955 2307 y(that)15 b(the)o(y)g(can)g(be)g (gauged)e(simultaneously)h(by)g(adding)g(a)i(single)f(gauge)1955 2406 y(\002eld.)24 b(F)o(or)14 b(instance,)i(local)f(translations)g (and)g(local)g(rotations)f(are)i(both)1955 2506 y(included)22 b(in)i(the)g(group)f(of)g(local)h(di)p FF(\013)p FG(eomorphisms;)f(the) h(spacetime)1955 2606 y(metric)f(plays)h(the)g(role)f(of)h(the)f (background)e(gauge)h(\002eld)i(here.)35 b(This)1955 2705 y(naturally)18 b(leads)h(to)g(relations)f(of)h(the)f(type)h(\()p 0 0 1 TeXcolorrgb 3249 2706 a SDict begin H.S end 3249 2706 a 0 0 1 TeXcolorrgb -1 x FG(20)p 0 0 1 TeXcolorrgb 3331 2650 a SDict begin H.R end 3331 2650 a 3331 2705 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.20) cvn H.B /ANN pdfmark end 3331 2705 a Black FG(\),)h(for)e(the)h(tw)o(o)g(Noether) 1955 2805 y(currents)h(are)i(obtained)d(by)i(functional)f(di)p FF(\013)p FG(erentiation)f(of)i(the)g(action)1955 2904 y(with)f(respect)g(to)h(the)f(same)g(gauge)f(\002eld.)2080 3004 y(As)26 b(a)f(concrete)f(e)o(xample,)h(note)g(that)g(under)f (certain)g(conditions,)1955 3104 y(global)17 b(spatial)h(translations)g (in)g(a)g(nonrelati)n(vistic)f(theory)f(can)i(be)g(pro-)1955 3203 y(moted)c(to)i(nonrelati)n(vistic)d(general)i(coordinate)e(in)m(v) n(ariance)g([)p 0 0 1 TeXcolorrgb 3758 3203 a SDict begin H.S end 3758 3203 a FG(28)3842 3203 y SDict begin 12 H.L end 3842 3203 a 3842 3203 a SDict begin [/Subtype /Link/Dest (cite.Son:2005rv) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3842 3203 a Black -1 w FG(].)23 b(This)1955 3303 y(necessarily)k(also)h(includes) e(the)i(Galilei)g(boosts)f(\()p 0 0 1 TeXcolorrgb 3470 3304 a SDict begin H.S end 3470 3304 a 0 0 1 TeXcolorrgb -1 x FG(23)p 0 0 1 TeXcolorrgb 3553 3247 a SDict begin H.R end 3553 3247 a 3553 3303 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.23) cvn H.B /ANN pdfmark end 3553 3303 a Black FG(\))g(and)g(the)h(local)1955 3403 y(U\(1\))21 b(phase)h(transformations)d(of)j(the)f(\002eld)h Fr( )p FG(\()t Fx(x)p FG(\).)31 b(The)21 b(e)o(xtended)f(ac-)1955 3502 y(tion)2117 3485 y(\230)2102 3502 y Fx(S)13 b FG([)p Fr( ;)h Fx(g)p Fr(;)i Fx(A)p FG(])g(depends)h(on)g(the)h(matter)f (\002eld)h Fr( )p FG(\()t Fx(x)p FG(\),)g(the)g(spatial)g(met-)1955 3602 y(ric)31 b Fx(g)2116 3614 y Fo(i)9 b(j)2161 3602 y FG(\()t Fx(x)p FG(\))31 b(and)g(the)g(U\(1\))f(gauge)g(\002eld)j Fx(A)3224 3614 y Fp(\026)3259 3602 y FG(\()t Fx(x)p FG(\).)58 b(A)32 b(combined)d(local)1955 3701 y(U\(1\))d(rotation)f(with)h (parameter)f Fr(\022)q FG(\()t Fx(x)p FG(\))i(and)f(local)g(spatial)h (translation)1955 3801 y(with)20 b(parameter)f Fr(\030)2514 3771 y Fo(i)2535 3801 y FG(\()t Fx(x)p FG(\))h(tak)o(es)h(the)f(form)f ([)p 0 0 1 TeXcolorrgb 3175 3801 a SDict begin H.S end 3175 3801 a FG(28)3259 3801 y SDict begin 12 H.L end 3259 3801 a 3259 3801 a SDict begin [/Subtype /Link/Dest (cite.Son:2005rv) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 3259 3801 a Black -1 w FG(])1955 4131 y SDict begin H.S end 1955 4131 a 1955 4131 a SDict begin 12 H.A end 1955 4131 a 1955 4131 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.27) cvn /DEST pdfmark end 1955 4131 a 2336 3984 a Fr(\016)p Fx(g)2417 3996 y Fo(i)9 b(j)2485 3984 y FF(=)23 b Fl(\000)p Fr(\030)2652 3950 y Fo(k)2683 3984 y Fr(@)2726 3996 y Fo(k)2757 3984 y Fx(g)2799 3996 y Fo(i)9 b(j)2862 3984 y Fl(\000)18 b Fx(g)2975 3996 y Fo(ik)3022 3984 y Fr(@)3074 3996 y Fo(j)3093 3984 y Fr(\030)3131 3950 y Fo(k)3181 3984 y Fl(\000)g Fx(g)3294 3996 y Fo(k)10 b(j)3350 3984 y Fr(@)3393 3996 y Fo(i)3413 3984 y Fr(\030)3451 3950 y Fo(k)3482 3984 y Fr(;)2337 4119 y(\016)r Fx(A)2429 4131 y FE(0)2485 4119 y FF(=)23 b Fr(@)2604 4131 y FE(0)2636 4119 y Fr(\022)d Fl(\000)f Fr(\030)2811 4085 y Fo(j)2831 4119 y Fr(@)2883 4131 y Fo(j)2905 4119 y Fx(A)2956 4131 y FE(0)3007 4119 y Fl(\000)i Fx(A)3141 4131 y Fo(j)3161 4119 y Fr(@)3204 4131 y FE(0)3236 4119 y Fr(\030)3283 4085 y Fo(j)3304 4119 y Fr(;)2350 4255 y(\016)r Fx(A)2442 4267 y Fo(i)2485 4255 y FF(=)i Fr(@)2604 4267 y Fo(i)2624 4255 y Fr(\022)d Fl(\000)e Fr(\030)2798 4220 y Fo(j)2819 4255 y Fr(@)2871 4267 y Fo(j)2892 4255 y Fx(A)2943 4267 y Fo(i)2982 4255 y Fl(\000)i Fx(A)3115 4267 y Fo(j)3135 4255 y Fr(@)3178 4267 y Fo(i)3198 4255 y Fr(\030)3245 4220 y Fo(j)3284 4255 y FF(+)e Fx(mg)3457 4267 y Fo(i)9 b(j)3502 4255 y Fr(@)3545 4267 y FE(0)3577 4255 y Fr(\030)3624 4220 y Fo(j)3645 4255 y Fr(:)3909 4119 y FG(\(27\))1955 4444 y(The)28 b(local)g(phase)f(transformation)f(is)j(obtained)d(by)i (setting)g Fr(\030)3836 4414 y Fo(i)3895 4444 y FF(=)37 b FG(0,)1955 4544 y(i.e.)16 b Fr(\016)p Fx(g)2154 4556 y Fo(i)9 b(j)2221 4544 y FF(=)23 b FG(0)16 b(and)g Fr(\016)r Fx(A)2584 4556 y Fp(\026)2642 4544 y FF(=)23 b Fr(@)2761 4556 y Fp(\026)2795 4544 y Fr(\022)q FG(.)i(The)16 b(v)n(ariation)e(of) i(the)g(action)g(becomes)2552 4785 y SDict begin H.S end 2552 4785 a 2552 4785 a SDict begin 12 H.A end 2552 4785 a 2552 4785 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.28) cvn /DEST pdfmark end 2552 4785 a Fr(\016)2605 4768 y FG(\230)2591 4785 y Fx(S)35 b FF(=)23 b Fr(\016)2783 4797 y Fp( )2838 4768 y FG(\230)2823 4785 y Fx(S)31 b FF(+)2967 4672 y Ff(Z)3071 4785 y FG(d)t Fx(x)3194 4729 y Fr(\016)3248 4712 y FG(\230)3233 4729 y Fx(S)p 3178 4766 127 5 v 3178 4842 a Fr(\016)r Fx(A)3270 4854 y Fp(\026)3314 4785 y Fr(@)3357 4797 y Fp(\026)3392 4785 y Fr(\022)q(:)459 b FG(\(28\))1955 5014 y(From)15 b(here)h(we)g(immediately)e(obtain)h (the)h(e)o(xpression)e(for)h(the)h(particle)1955 5114 y(number)23 b(current,)36 b Fx(j)2551 5084 y Fp(\026)2617 5114 y Fb(,)31 b Fl(\000)p Fr(\016)2808 5097 y FG(\230)2793 5114 y Fx(S)12 b Fr(=\016)r Fx(A)2976 5126 y Fp(\026)3010 5039 y Ff(\014)3010 5073 y(\014)3010 5108 y(\014)3031 5147 y Fo(A)p FD(=)p FE(0)3137 5114 y FG(.)38 b(\(W)-7 b(e)25 b(denote)f(by)i Fx(A)k FF(=)h FG(0)25 b(the)1955 5213 y(limit)d(of)g(both)f Fx(g)2438 5225 y Fo(i)9 b(j)2509 5213 y Fl(!)26 b(\000)p Fr(\016)2707 5225 y Fo(i)9 b(j)2774 5213 y FG(and)24 b Fx(A)2970 5225 y Fp(\026)3031 5213 y Fl(!)i FG(0.\))k(Analogously)-5 b(,)19 b(we)k(obtain)1955 5313 y(the)28 b(v)n(ariation)e(of)h(the)h(action)f(under)g(local)g (translations)g(and)h(thence)1955 5413 y(the)20 b(momentum)e(current,)h (de\002ned)g(by)h Fr(\016)p Fx(S)35 b FF(=)3314 5339 y Ff(R)3383 5413 y FG(d)t Fx(x)13 b(T)3535 5371 y Fp(\026)3560 5440 y Fo(i)3580 5413 y Fr(@)3623 5425 y Fp(\026)3658 5413 y Fr(\030)3696 5383 y Fo(i)3717 5413 y FG(,)2346 5669 y SDict begin H.S end 2346 5669 a 2346 5669 a SDict begin 12 H.A end 2346 5669 a 2346 5669 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.29) cvn /DEST pdfmark end 2346 5669 a Fx(T)2402 5634 y FE(0)p Fo(i)2475 5669 y Fb(,)23 b Fl(\000)p Fx(m)2683 5613 y Fr(\016)2736 5596 y FG(\230)2722 5613 y Fx(S)p 2674 5650 112 5 v 2674 5726 a Fr(\016)r Fx(A)2766 5738 y Fo(i)2795 5559 y Ff(\014)2795 5594 y(\014)2795 5628 y(\014)2795 5663 y(\014)2795 5698 y(\014)2815 5737 y Fo(A)p FD(=)p FE(0)2921 5669 y Fr(;)180 b Fx(T)3178 5634 y Fo(i)9 b(j)3246 5669 y Fb(,)23 b FG(2)3390 5613 y Fr(\016)3443 5596 y FG(\230)3429 5613 y Fx(S)p 3374 5650 126 5 v 3374 5726 a Fr(\016)p Fx(g)3455 5738 y Fo(i)9 b(j)3509 5559 y Ff(\014)3509 5594 y(\014)3509 5628 y(\014)3509 5663 y(\014)3509 5698 y(\014)3529 5737 y Fo(A)p FD(=)p FE(0)3635 5669 y Fr(:)253 b FG(\(29\))p Black 1860 5888 a(6)p Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 0 1 TeXcolorrgb -295 -446 a SDict begin H.S end -295 -446 a -295 -446 a SDict begin H.R end -295 -446 a -295 -446 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -295 -446 a Black Black Black -287 160 a FG(Finally)-5 b(,)26 b(by)e(setting)h Fr(\022)q FG(\()t Fx(x)p FG(\))31 b FF(=)h Fl(\000)p Fx(m)t(x)750 172 y Fo(i)770 160 y Fx(v)807 130 y Fo(i)827 160 y FG(\()t Fx(x)p FG(\))25 b(and)f Fr(\030)1132 130 y Fo(i)1153 160 y 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180 1170 a 180 1170 a SDict begin 12 H.A end 180 1170 a 180 1170 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.31) cvn /DEST pdfmark end 180 1170 a Fx(T)236 1135 y FE(0)p Fo(i)309 1170 y Fb(,)23 b Fx(m)12 b(j)480 1135 y Fo(i)500 1170 y Fr(;)183 b Fx(B)755 1135 y Fp(\026)p Fo(i)828 1170 y Fb(,)23 b Fx(t)q(T)984 1135 y Fp(\026)p Fo(i)1054 1170 y Fl(\000)c Fx(m)t(x)1227 1135 y Fo(i)1259 1170 y Fx(j)1282 1135 y Fp(\026)1318 1170 y Fr(:)328 b FG(\(31\))-287 1320 y(The)17 b(reason)f(why)f(we)i(obtained)e (stronger)h(relations)g(than)g(in)h(Eqs.)f(\()p 0 0 1 TeXcolorrgb 1695 1320 a SDict begin H.S end 1695 1320 a 0 0 1 TeXcolorrgb FG(24)p 0 0 1 TeXcolorrgb 1778 1264 a SDict begin H.R end 1778 1264 a 1778 1320 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.24) cvn H.B /ANN pdfmark end 1778 1320 a Black FG(\))-287 1420 y(and)g(\()p 0 0 1 TeXcolorrgb -122 1421 a SDict begin H.S end -122 1421 a 0 0 1 TeXcolorrgb -1 x FG(25)p 0 0 1 TeXcolorrgb -40 1363 a SDict begin H.R end -40 1363 a -40 1420 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.25) cvn H.B /ANN pdfmark end -40 1420 a Black FG(\))g(is)h(the)e Fx(assumption)g FG(that)h(the)f(theory)g(can)g(be)h(made)f(generally)-287 1519 y(coordinate)h(in)m(v)n(ariant)h(by)g(introducing)f(background)e (gauge)i(\002elds)j Fx(g)1761 1531 y Fo(i)9 b(j)-287 1619 y FG(and)20 b Fx(A)-95 1631 y Fp(\026)-60 1619 y FG(.)25 b(This)19 b(is)g(a)g(nontri)n(vial)e(assumption)g(which)h (imposes)h(stronger)-287 1718 y(constraints)29 b(on)g(the)g(action)g (than)f(mere)h(global)g(U\(1\),)h(translational)-287 1818 y(and)20 b(Galilei)h(in)m(v)n(ariance)d([)p 0 0 1 TeXcolorrgb 490 1818 a SDict begin H.S end 490 1818 a FG(28)574 1818 y SDict begin 12 H.L end 574 1818 a 574 1818 a SDict begin [/Subtype /Link/Dest (cite.Son:2005rv) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 574 1818 a Black -1 w FG(].)-287 1961 y SDict begin H.S end -287 1961 a -287 1961 a SDict begin 12 H.A end -287 1961 a -287 1961 a SDict begin [/View [/XYZ H.V]/Dest (section.6) cvn /DEST pdfmark end -287 1961 a 87 x Fz(6.)41 b(Conclusions)-162 2203 y FG(In)29 b(this)h(paper)m(,)g(we)f(ga)n(v)o(e)f(a)i(ne)n(w)f(interpretation)e (of)i(the)g(in)m(v)o(erse)-287 2302 y(Higgs)23 b(constraints,)f (commonly)f(emplo)o(yed)f(in)j(constructions)e(of)h(ef-)-287 2402 y(fecti)n(v)o(e)k(Lagrangians)e(for)h(spontaneously)f(brok)o(en)g (spacetime)i(sym-)-287 2501 y(metries.)31 b(W)-7 b(e)23 b(sho)n(wed)f(that)g(this)g(procedure)e(is)j(less)g(arbitrary)e(than)g (it)-287 2601 y(seems:)28 b(the)21 b(number)f(of)g(true)h(massless)i (NG)e(bosons)f(depends)g(solely)-287 2701 y(on)i(the)f (symmetry-breaking)c(pattern;)22 b(all)g(nonlinear)e(realizations)g(of) -287 2800 y(the)25 b(brok)o(en)e(symmetry)g(can)h(be)g(obtained)f(by)h (augmenting)f(the)h(EFT)-287 2900 y(for)35 b(these)h(NG)g(modes)f(with) g(matter)g(\002elds.)71 b(The)36 b(spontaneously)-287 3000 y(brok)o(en)22 b(\223redundant\224)f(symmetries)j(can)f(in)h(some) f(systems)i(manifest)-287 3099 y(themselv)o(es)20 b(by)g(the)g (presence)f(of)h(massi)n(v)o(e)g(modes)f(in)i(the)f(spectrum.)-162 3199 y(T)-7 b(ogether)28 b(with)h(recent)g(progress)f(in)i (understanding)c(of)j(brok)o(en)-287 3298 y(symmetries)j(in)g(the)g (nonrelati)n(vistic)f(conte)o(xt)g([)p 0 0 1 TeXcolorrgb 1170 3298 a SDict begin H.S end 1170 3298 a FG(1)1212 3298 y SDict begin 12 H.L end 1212 3298 a 1212 3298 a SDict begin [/Subtype /Link/Dest (cite.Watanabe:2011ec) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 1212 3298 a Black -1 w FG(\226)p 0 0 1 TeXcolorrgb 1253 3298 a SDict begin H.S end 1253 3298 a FG(3)1295 3298 y SDict begin 12 H.L end 1295 3298 a 1295 3298 a SDict begin [/Subtype /Link/Dest (cite.Hidaka:2012ym) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 1295 3298 a Black -1 w FG(,)p 0 0 1 TeXcolorrgb 1348 3298 a SDict begin H.S end 1348 3298 a FG(15)1432 3298 y SDict begin 12 H.L end 1432 3298 a 1432 3298 a SDict begin [/Subtype /Link/Dest (cite.Watanabe:2013iia) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 1432 3298 a Black -1 w FG(],)k(this)d(pro-)-287 3398 y(vides)f(a)h(f)o(airly)f(complete)f(picture)g(of)h(NG)h(bosons)e (in)i(an)f(arbitrary)-287 3498 y(quantum)22 b(man)o(y-body)e(system,)k (e)o(xcept)e(for)h(the)h(cases)g(where)f(trans-)-287 3597 y(lational)18 b(in)m(v)n(ariance)d(is)k(spontaneously)c(brok)o (en.)23 b(T)-7 b(o)17 b(\002ll)i(this)f(gap)f(rep-)-287 3697 y(resents)k(one)e(of)h(the)h(main)e(goals)h(of)g(our)g(future)f(w) o(ork.)-287 3926 y SDict begin H.S end -287 3926 a -287 3926 a SDict begin 12 H.A end -287 3926 a -287 3926 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end -287 3926 a Fz(Ackno)o(wledgments)-162 4081 y FG(W)-7 b(e)32 b(are)f(indebted)f(to)i(Hitoshi)f(Murayama)e(for) i(numerous)e(dis-)-287 4181 y(cussions)k(of)g(the)g(topic.)63 b(The)33 b(w)o(ork)g(of)f(T)-6 b(.B.)33 b(w)o(as)h(funded)d(by)i(the) -287 4281 y(Academy)27 b(of)g(Finland,)i(grant)e(No.)g(273545.)46 b(H.W)-8 b(.)28 b(appreciates)f(\002-)-287 4380 y(nancial)16 b(support)f(from)h(the)g(Honjo)g(International)e(Scholarship)i(F)o (oun-)-287 4480 y(dation.)60 b(W)-7 b(e)33 b(further)d(ackno)n(wledge)f (the)j(hospitality)f(of)h(the)g(Ka)n(vli)-287 4579 y(IPMU,)21 b(and)f(of)g(IFT)g(U)m(AM-CSIC)h(through)d(the)i(Centro)g(de)h (Excelen-)-287 4679 y(cia)g(Se)n(v)o(ero)e(Ochoa)h(Program)f(under)g (grant)g(SEV)-8 b(-2012-0249,)16 b(where)-287 4779 y(parts)21 b(of)f(the)g(w)o(ork)f(were)h(carried)g(out.)-287 4942 y SDict begin H.S end -287 4942 a -287 4942 a SDict begin 12 H.A end -287 4942 a -287 4942 a SDict begin [/View [/XYZ H.V]/Dest (section*.2) cvn /DEST pdfmark end -287 4942 a Fx(Note)f(added.)48 b FG(While)19 b(this)g(paper)f(w)o(as)h (being)f(\002nished,)g(Ref.)h([)p 0 0 1 TeXcolorrgb 1570 4942 a SDict begin H.S end 1570 4942 a FG(29)1654 4942 y SDict begin 12 H.L end 1654 4942 a 1654 4942 a SDict begin [/Subtype /Link/Dest (cite.Endlich:2013vfa) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 1654 4942 a Black -1 w FG(])g(ap-)-287 5042 y(peared)h(which)g (mak)o(es)g(the)h(same)g(k)o(e)o(y)e(observ)n(ation)g(that)i(some)f(of) g(the)-287 5141 y(NG)k(\002elds)g(of)f(brok)o(en)f(spacetime)h (symmetries)g(can)g(acquire)g(a)h(mass)-287 5241 y(term.)36 b(Our)24 b(\002eld)g(rede\002nition)e(based)i(on)g(Eq.)f(\()p 0 0 1 TeXcolorrgb 1143 5243 a SDict begin H.S end 1143 5243 a 0 0 1 TeXcolorrgb -2 x FG(9)p 0 0 1 TeXcolorrgb 1184 5185 a SDict begin H.R end 1184 5185 a 1184 5241 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.9) cvn H.B /ANN pdfmark end 1184 5241 a Black FG(\))h(mak)o(es)g(the)g(nature)-287 5340 y(of)30 b(these)h(modes)e(more)g(transparent)g(as)i(it)g (demonstrates)e(that)h(the)o(y)-287 5440 y(ha)n(v)o(e)e(no)h(attrib)n (utes)f(of)g(NG)h(bosons.)50 b(On)28 b(the)h(other)f(hand,)h(the)g(au-) -287 5540 y(thors)23 b(of)g(Ref.)h([)p 0 0 1 TeXcolorrgb 189 5540 a SDict begin H.S end 189 5540 a FG(29)273 5540 y SDict begin 12 H.L end 273 5540 a 273 5540 a SDict begin [/Subtype /Link/Dest (cite.Endlich:2013vfa) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [0 1 0] H.B /ANN pdfmark end 273 5540 a Black -2 w FG(])g(observ)o(e)d(that)j(in)f(some)g (cases,)h(the)g(de\002cit)f(in)g(the)-287 5639 y(number)d(of)i(true)f (NG)h(bosons)f(cannot)f(be)i(e)o(xplained)d(by)j(redundanc)o(y)-287 5739 y(in)f(\003uctuations)e(of)h(the)g(order)f(parameter)g(as)i(in)f (Sec.)p 0 0 1 TeXcolorrgb 1304 5740 a SDict begin H.S end 1304 5740 a 0 0 1 TeXcolorrgb -1 x FG(4.1)p 0 0 1 TeXcolorrgb 1408 5683 a SDict begin H.R end 1408 5683 a 1408 5739 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (subsection.4.1) cvn H.B /ANN pdfmark end 1408 5739 a Black FG(.)p Black Black 1955 160 a SDict begin H.S end 1955 160 a 1955 160 a SDict begin 12 H.A end 1955 160 a 1955 160 a SDict begin [/View [/XYZ H.V]/Dest (section*.3) cvn /DEST pdfmark end 1955 160 a Fz(Refer)o(ences)p Black 1988 313 a SDict begin H.S end 1988 313 a Fu([1])2065 313 y SDict begin 9.5 H.A end 2065 313 a 2065 313 a SDict begin [/View [/XYZ H.V]/Dest (cite.Watanabe:2011ec) cvn /DEST pdfmark end 2065 313 a Black 42 w Fu(H.)i(W)-5 b(atanabe,)26 b(T)-5 b(.)21 b(Brauner)m(,)46 b(On)23 b(the)g(number)g(of)g(Namb)o(u-Goldstone)i (bosons)2107 392 y(and)18 b(its)f(relation)j(to)d(char)o(ge)h (densities,)26 b(Phys.)16 b(Re)n(v)l(.)i(D84)f(\(2011\))g(125013.)p Black 1988 471 a SDict begin H.S end 1988 471 a Fu([2])2065 471 y SDict begin 9.5 H.A end 2065 471 a 2065 471 a SDict begin [/View [/XYZ H.V]/Dest (cite.Watanabe:2012hr) cvn /DEST pdfmark end 2065 471 a Black 42 w Fu(H.)22 b(W)-5 b(atanabe,)25 b(H.)c(Murayama,)44 b(Uni\002ed)24 b(Description)g(of)f(Namb)o (u-Goldstone)2107 550 y(Bosons)17 b(without)i(Lorentz)f(In)m(v)n (ariance,)28 b(Phys.)16 b(Re)n(v)l(.)h(Lett.)g(108)g(\(2012\))h (251602.)p Black 1988 629 a SDict begin H.S end 1988 629 a Fu([3])2065 629 y SDict begin 9.5 H.A end 2065 629 a 2065 629 a SDict begin [/View [/XYZ H.V]/Dest (cite.Hidaka:2012ym) cvn /DEST pdfmark end 2065 629 a Black 42 w Fu(Y)-9 b(.)18 b(Hidaka,)30 b(Counting)21 b(rule)e(for)f(Namb)o(u-Goldstone)j (modes)d(in)h(nonrelati)n(vistic)2107 708 y(systems,)24 b(Phys.)16 b(Re)n(v)l(.)h(Lett.)h(110)f(\(2013\))g(091601.)p Black 1988 787 a SDict begin H.S end 1988 787 a Fu([4])2065 787 y SDict begin 9.5 H.A end 2065 787 a 2065 787 a SDict begin [/View [/XYZ H.V]/Dest (cite.Coleman:1969sm) cvn /DEST pdfmark end 2065 787 a Black 42 w Fu(S.)h(R.)h(Coleman,)h(J.)e(W)-5 b(ess,)18 b(B.)h(Zumino,)30 b(Structure)21 b(of)e(Phenomenological)k (La-)2107 866 y(grangians.)c(I,)k(Phys.)16 b(Re)n(v)l(.)i(177)f (\(1969\))g(2239\2262247.)p Black 1988 944 a SDict begin H.S end 1988 944 a Fu([5])2065 944 y SDict begin 9.5 H.A end 2065 944 a 2065 944 a SDict begin [/View [/XYZ H.V]/Dest (cite.Callan:1969sn) cvn /DEST pdfmark end 2065 944 a Black 42 w Fu(C.)26 b(G.)f(Callan,)31 b(S.)25 b(R.)h(Coleman,)k(J.)25 b(W)-5 b(ess,)27 b(B.)f(Zumino,)56 b(Structure)28 b(of)e(Phe-)2107 1023 y(nomenological)21 b(Lagrangians.)d(II,)24 b(Phys.)16 b(Re)n(v)l(.)h(177)g(\(1969\))h(2247\2262250.)p Black 1988 1102 a SDict begin H.S end 1988 1102 a Fu([6])2065 1102 y SDict begin 9.5 H.A end 2065 1102 a 2065 1102 a SDict begin [/View [/XYZ H.V]/Dest (cite.Leutwyler:1993iq) cvn /DEST pdfmark end 2065 1102 a Black 42 w Fu(H.)i(Leutwyler)m(,)37 b(On)20 b(the)h(F)o(oundations)h(of)e(Chiral)i(Perturbation)i(Theory,) 35 b(Ann.)2107 1181 y(Phys.)16 b(235)h(\(1994\))h(165\226203.)p Black 1988 1260 a SDict begin H.S end 1988 1260 a Fu([7])2065 1260 y SDict begin 9.5 H.A end 2065 1260 a 2065 1260 a SDict begin [/View [/XYZ H.V]/Dest (cite.Leutwyler:1993gf) cvn /DEST pdfmark end 2065 1260 a Black 42 w Fu(H.)26 b(Leutwyler)m(,)60 b(Nonrelati)n(vistic)31 b(e)p Ft(\013)p Fu(ecti)n(v)o(e)f(Lagrangians,) 60 b(Phys.)26 b(Re)n(v)l(.)h(D49)2107 1339 y(\(1994\))18 b(3033\2263043.)p Black 1988 1418 a SDict begin H.S end 1988 1418 a Fu([8])2065 1418 y SDict begin 9.5 H.A end 2065 1418 a 2065 1418 a SDict begin [/View [/XYZ H.V]/Dest (cite.Watanabe:2013uya) cvn /DEST pdfmark end 2065 1418 a Black 42 w Fu(H.)28 b(W)-5 b(atanabe,)33 b(T)-5 b(.)28 b(Brauner)m(,)33 b(H.)c(Murayama,)66 b(Massi)n(v)o(e)30 b(Namb)o(u-Goldstone)2107 1497 y(Bosons,)24 b(Phys.)16 b(Re)n(v)l(.)i(Lett.)f(111)g(\(2013\))h (021601.)p Black 1988 1575 a SDict begin H.S end 1988 1575 a Fu([9])2065 1575 y SDict begin 9.5 H.A end 2065 1575 a 2065 1575 a SDict begin [/View [/XYZ H.V]/Dest (cite.Volkov:1973vd) cvn /DEST pdfmark end 2065 1575 a Black 42 w Fu(D.)h(V)-9 b(.)20 b(V)-9 b(olk)o(o)o(v)l(,)35 b(Phenomenological)24 b(Lagrangians,)36 b(Fiz.)20 b(Elem.)f(Chast.)h(Atom.)2107 1654 y(Y)-7 b(adra)18 b(4)f(\(1973\))h(3\22641.)p Black 1955 1733 a SDict begin H.S end 1955 1733 a Fu([10])2065 1733 y SDict begin 9.5 H.A end 2065 1733 a 2065 1733 a SDict begin [/View [/XYZ H.V]/Dest (cite.Ogievetsky) cvn /DEST pdfmark end 2065 1733 a Black 42 w Fu(V)-9 b(.)19 b(Ogie)n(v)o(etsk)o (y)l(,)35 b(Nonlinear)22 b(realizations)i(of)19 b(internal)j(and)e (spacetime)i(symme-)2107 1812 y(tries,)f(Acta)16 b(Uni)n(v)o(ersitatis) i(Wratisla)o(viensis)g(207)d(\(1974\))h(117\226141.)f(In)g(Proceed-) 2107 1891 y(ings)j(of)e(10th)i(W)m(inter)h(School)f(of)f(Theoretical)j (Physics)d(in)g(Karpacz.)p Black 1955 1970 a SDict begin H.S end 1955 1970 a Fu([11])2065 1970 y SDict begin 9.5 H.A end 2065 1970 a 2065 1970 a SDict begin [/View [/XYZ H.V]/Dest (cite.Ivanov:1975zq) cvn /DEST pdfmark end 2065 1970 a Black 42 w Fu(E.)e(Iv)n(ano)o(v)l(,)i(V)-9 b(.)16 b(I.)g(Ogie)n(v)o (etsk)o(y)l(,)24 b(The)16 b(In)m(v)o(erse)i(Higgs)e(Phenomenon)i(in)f (Nonlinear)2107 2049 y(Realizations,)28 b(T)-5 b(eor)l(.)17 b(Mat.)h(Fiz.)f(25)g(\(1975\))g(164\226177.)p Black 1955 2127 a SDict begin H.S end 1955 2127 a Fu([12])2065 2127 y SDict begin 9.5 H.A end 2065 2127 a 2065 2127 a SDict begin [/View [/XYZ H.V]/Dest (cite.Nicolis:2013sga) cvn /DEST pdfmark end 2065 2127 a Black 42 w Fu(A.)d(Nicolis,)j(R.)d(Penco,)i(F)-5 b(.)14 b(Piazza,)i(R.)f(A.)f(Rosen,)20 b(More)15 b(on)g(gapped)h (Goldstones)2107 2206 y(at)i(\002nite)g(density:)23 b(more)17 b(gapped)h(Goldstones,)25 b(JHEP)16 b(1311)i(\(2013\))f(055.)p Black 1955 2285 a SDict begin H.S end 1955 2285 a Fu([13])2065 2285 y SDict begin 9.5 H.A end 2065 2285 a 2065 2285 a SDict begin [/View [/XYZ H.V]/Dest (cite.McArthur:2010zm) cvn /DEST pdfmark end 2065 2285 a Black 42 w Fu(I.)26 b(N.)g(McArthur)m(,) 58 b(Nonlinear)29 b(realizations)i(of)26 b(symmetries)i(and)f (unphysical)2107 2364 y(Goldstone)19 b(bosons,)24 b(JHEP)16 b(1011)h(\(2010\))h(140.)p Black 1955 2443 a SDict begin H.S end 1955 2443 a Fu([14])2065 2443 y SDict begin 9.5 H.A end 2065 2443 a 2065 2443 a SDict begin [/View [/XYZ H.V]/Dest (cite.Low:2001bw) cvn /DEST pdfmark end 2065 2443 a Black 42 w Fu(I.)23 b(Lo)n(w)l(,)i(A.)e(V)-9 b(.)24 b(Manohar)m(,)49 b(Spontaneously)27 b(Brok)o(en)f(Spacetime)g (Symmetries)2107 2522 y(and)18 b(Goldstone')l(s)h(Theorem,)24 b(Phys.)16 b(Re)n(v)l(.)h(Lett.)g(88)g(\(2002\))h(101602.)p Black 1955 2601 a SDict begin H.S end 1955 2601 a Fu([15])2065 2601 y SDict begin 9.5 H.A end 2065 2601 a 2065 2601 a SDict begin [/View [/XYZ H.V]/Dest (cite.Watanabe:2013iia) cvn /DEST pdfmark end 2065 2601 a Black 42 w Fu(H.)c(W)-5 b(atanabe,)16 b(H.)e(Murayama,)20 b(Redundancies)e(in)d(Namb)o (u-Goldstone)h(Bosons,)2107 2680 y(Phys.)g(Re)n(v)l(.)h(Lett.)h(110)f (\(2013\))h(181601.)p Black 1955 2758 a SDict begin H.S end 1955 2758 a Fu([16])2065 2758 y SDict begin 9.5 H.A end 2065 2758 a 2065 2758 a SDict begin [/View [/XYZ H.V]/Dest (cite.Hayata:2013vfa) cvn /DEST pdfmark end 2065 2758 a Black 42 w Fu(T)-5 b(.)15 b(Hayata,)j(Y)-9 b(.)16 b(Hidaka,)23 b(Brok)o(en)18 b(spacetime)g(symmetries)f(and)g(elastic)h(v)n(ariables)2107 2837 y(\(2013\).)p Black 1955 2916 a SDict begin H.S end 1955 2916 a Fu([17])2065 2916 y SDict begin 9.5 H.A end 2065 2916 a 2065 2916 a SDict begin [/View [/XYZ H.V]/Dest (cite.Goon:2012dy) cvn /DEST pdfmark end 2065 2916 a Black 42 w Fu(G.)23 b(Goon,)i(K.)e(Hinterbichler)m(,)29 b(A.)23 b(Jo)o(yce,)j(M.)d(T)n(rodden,)47 b(Galileons)26 b(as)e(W)-5 b(ess-)2107 2995 y(Zumino)17 b(T)-5 b(erms,)24 b(JHEP)16 b(1206)h(\(2012\))h(004.)p Black 1955 3074 a SDict begin H.S end 1955 3074 a Fu([18])2065 3074 y SDict begin 9.5 H.A end 2065 3074 a 2065 3074 a SDict begin [/View [/XYZ H.V]/Dest (cite.Weinberg:1996v2) cvn /DEST pdfmark end 2065 3074 a Black 42 w Fu(S.)e(W)-5 b(einber)o(g,)17 b(The)g(Quantum)g(Theory)g (of)g(Fields,)g(v)o(olume)g(II,)f(Cambridge)i(Uni-)2107 3153 y(v)o(ersity)h(Press,)d(Cambridge,)i(UK,)e(1996.)p Black 1955 3232 a SDict begin H.S end 1955 3232 a Fu([19])2065 3232 y SDict begin 9.5 H.A end 2065 3232 a 2065 3232 a SDict begin [/View [/XYZ H.V]/Dest (cite.Endlich:2013spa) cvn /DEST pdfmark end 2065 3232 a Black 42 w Fu(S.)g(Endlich,)i(A.)e (Nicolis,)i(R.)f(Penco,)25 b(Spontaneously)20 b(brok)o(en)e(mass)e (\(2013\).)p Black 1955 3311 a SDict begin H.S end 1955 3311 a Fu([20])2065 3311 y SDict begin 9.5 H.A end 2065 3311 a 2065 3311 a SDict begin [/View [/XYZ H.V]/Dest (cite.Altman) cvn /DEST pdfmark end 2065 3311 a Black 42 w Fu(S.)21 b(D.)f(Huber)m(,)j(B.)e(Theiler)m(,)i(E.)d(Altman,)j(G.)d (Blatter)m(,)42 b(Amplitude)23 b(Mode)f(in)f(the)2107 3389 y(Quantum)d(Phase)g(Model,)24 b(Phys.)17 b(Re)n(v)l(.)g(Lett.)g (100)g(\(2008\))h(050404.)p Black 1955 3468 a SDict begin H.S end 1955 3468 a Fu([21])2065 3468 y SDict begin 9.5 H.A end 2065 3468 a 2065 3468 a SDict begin [/View [/XYZ H.V]/Dest (cite.Son) cvn /DEST pdfmark end 2065 3468 a Black 42 w Fu(D.)e(T)-5 b(.)15 b(Son,)23 b(Lo)n(w-Ener)o(gy)17 b(Quantum)g(E)p Ft(\013)p Fu(ecti)n(v)o(e)i(Action)f(for)e(Relati)n (vistic)21 b(Super)o(-)2107 3547 y(\003uids)c(\(2002\).)p Black 1955 3626 a SDict begin H.S end 1955 3626 a Fu([22])2065 3626 y SDict begin 9.5 H.A end 2065 3626 a 2065 3626 a SDict begin [/View [/XYZ H.V]/Dest (cite.Son:supersolid) cvn /DEST pdfmark end 2065 3626 a Black 42 w Fu(D.)24 b(T)-5 b(.)23 b(Son,)50 b(E)p Ft(\013)p Fu(ecti)n(v)o(e)27 b(Lagrangian)g(and) e(topological)j(interactions)g(in)d(super)o(-)2107 3705 y(solids,)g(Phys.)16 b(Re)n(v)l(.)h(Lett.)g(94)g(\(2005\))h(175301.)p Black 1955 3784 a SDict begin H.S end 1955 3784 a Fu([23])2065 3784 y SDict begin 9.5 H.A end 2065 3784 a 2065 3784 a SDict begin [/View [/XYZ H.V]/Dest (cite.Kohn) cvn /DEST pdfmark end 2065 3784 a Black 42 w Fu(W)-6 b(.)18 b(K)n(ohn,)29 b(Cyclotron)21 b(Resonance)g(and)e(de)g(Haas-v)n(an)h (Alphen)g(Oscillations)i(of)2107 3863 y(an)c(Interacting)i(Electron)e (Gas,)24 b(Phys.)16 b(Re)n(v)l(.)i(123)f(\(1961\))g(1242\2261244.)p Black 1955 3941 a SDict begin H.S end 1955 3941 a Fu([24])2065 3941 y SDict begin 9.5 H.A end 2065 3941 a 2065 3941 a SDict begin [/View [/XYZ H.V]/Dest (cite.Horava) cvn /DEST pdfmark end 2065 3941 a Black 42 w Fu(T)-5 b(.)22 b(Gri)p Ft(\016)p Fu(n,)j(K.)d(T)-5 b(.)22 b(Grosv)o(enor)m(,)j(P)-7 b(.)22 b(Ho\020)-22 b(ra)o(v)n(a,)25 b(Z.)d(Y)-7 b(an,)46 b(Multicritical)28 b(symmetry)2107 4020 y(breaking)23 b(and)e(naturalness)j(of)c(slo)n(w)i(Namb)o(u-Goldstone)g(bosons,)38 b(Phys.)20 b(Re)n(v)l(.)2107 4099 y(D88)d(\(2013\))h(101701.)p Black 1955 4178 a SDict begin H.S end 1955 4178 a Fu([25])2065 4178 y SDict begin 9.5 H.A end 2065 4178 a 2065 4178 a SDict begin [/View [/XYZ H.V]/Dest (cite.Sinova) cvn /DEST pdfmark end 2065 4178 a Black 42 w Fu(J.)12 b(Sino)o(v)n(a,)j (C.)e(B.)g(Hanna,)h(A.)f(H.)f(MacDonald,)19 b(Quantum)14 b(Melting)h(and)f(Absence)2107 4257 y(of)h(Bose-Einstein)h (Condensation)i(in)d(T)-5 b(w)o(o-Dimensional)16 b(V)-9 b(orte)o(x)16 b(Matter,)21 b(Phys.)2107 4336 y(Re)n(v)l(.)c(Lett.)h(89) f(\(2002\))g(030403.)p Black 1955 4415 a SDict begin H.S end 1955 4415 a Fu([26])2065 4415 y SDict begin 9.5 H.A end 2065 4415 a 2065 4415 a SDict begin [/View [/XYZ H.V]/Dest (cite.Nicolis:2010se) cvn /DEST pdfmark end 2065 4415 a Black 42 w Fu(A.)c(Nicolis,)20 b(Galilean)c(currents)f(and)g(char)o (ges,)k(Phys.)13 b(Re)n(v)l(.)h(D85)g(\(2012\))g(085026.)p Black 1955 4494 a SDict begin H.S end 1955 4494 a Fu([27])2065 4494 y SDict begin 9.5 H.A end 2065 4494 a 2065 4494 a SDict begin [/View [/XYZ H.V]/Dest (cite.Higashijima:1994zg) cvn /DEST pdfmark end 2065 4494 a Black 42 w Fu(K.)29 b(Higashijima,)68 b(Namb)o(u-Goldstone)32 b(theorem)e(for)g(conformal)h(symmetry)2107 4572 y(\(1994\).)p Black 1955 4651 a SDict begin H.S end 1955 4651 a Fu([28])2065 4651 y SDict begin 9.5 H.A end 2065 4651 a 2065 4651 a SDict begin [/View [/XYZ H.V]/Dest (cite.Son:2005rv) cvn /DEST pdfmark end 2065 4651 a Black 42 w Fu(D.)25 b(T)-5 b(.)24 b(Son,)k(M.)d(W)m(ingate,)55 b(General)27 b(coordinate)i(in)m(v)n(ariance)g(and)d(conformal)2107 4730 y(in)m(v)n(ariance)c(in)d(nonrelati)n(vistic)k(physics:)i(Unitary) 20 b(Fermi)f(gas,)28 b(Ann.)18 b(Phys.)g(321)2107 4809 y(\(2006\))g(197\226224.)p Black 1955 4888 a SDict begin H.S end 1955 4888 a Fu([29])2065 4888 y SDict begin 9.5 H.A end 2065 4888 a 2065 4888 a SDict begin [/View [/XYZ H.V]/Dest (cite.Endlich:2013vfa) cvn /DEST pdfmark end 2065 4888 a Black 42 w Fu(S.)27 b(Endlich,)32 b(A.)27 b(Nicolis,)k(R.)d(Penco,)62 b(UV)28 b(completion)i(without)g(symmetry)2107 4967 y(restoration)20 b(\(2013\).)p Black 1860 5888 a FG(7)p Black eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF