15:50-16:10, 9th Feb., GC2018
On fermionic ghosts and the removal from scalar-fermion systems
Yuki Sakakihara (Osaka City University)
ref. Rampei Kimura, YS, Masahide Yamaguchi Phys. Rev. D96 (2017) 044015, another paper in prep. WHY FERMIONS?
How are inflaton and SM particles coupled with each other? Interactions between inflation and SM particles … unknown e.g. Reheating process depends on the details of the interactions Generalization of interactions including only bosonic fields has been more frequently discussed. e.g. Horndeski, Beyond Horndeski,Vector-tensor theories, … How can fermions be coupled to gravity? Examination of the interaction between scalar fields and fermions can be a first step to investigate gravity-fermion system
Can we have quadratic terms of derivatives of fermion? Usually it is difficult because of the appearance of ghosts. It may be possible by the introduction of scalar fields. WHAT KINDS OF INTERACTIONS ARE ALLOWED IN PRINCIPLE?
Avoidance of negative norm states is crucial Fermions can easily have negative norm states (because of extra dofs) Let us consider systems both with bosons and fermions and how we can avoid such the negative norm states
In the sense that we should eliminate “extra degrees of freedom”, the analysis is technically similar to that of higher derivative theories of scalar fields ref. H. Motohashi, et.al., JCAP 1607 (2016) 033.
Extra dofs become explicit in Hamiltonian formulation 2 2 2 fermionic systemfermionic and explain system that the and absence explain of that negative the absence norm states of negative requires norm the reduction states requires in the the dimension reduction of in the dimension of phase space. In Sec.phaseIVfermionic, space. we give Insystem our Sec. setupIV and, of we explain coexistence give our that setup thesystem absence of coexistencewith of bosonic negative system and norm fermionic with states bosonic point requires particles and the fermionic reduction and derive point in particles the dimension and derive of the condition forthe avoidingphase condition fermionic space. for In avoiding Sec. ghosts,IV, we which fermionic give we our call ghosts,setup the of maximally-degenerate whichcoexistence we call system the with maximally-degenerate condition. bosonic and We fermionic then condition. perform point particles We then and derive perform Hamiltonian analysisHamiltonianthe of the condition system analysis for satisfying avoiding of the the system fermionic condition satisfying ghosts, and find the which another condition we call condition andthe maximally-degenerate find guaranteeing another condition that secondarycondition. guaranteeing We then that perform secondary constraints are notconstraints producedHamiltonian and are notall analysis the produced Lagrange of the and system multipliers all the satisfying Lagrange are uniquely the multipliers condition determined. and are find uniquely In another the last determined. condition part3 of Sec. guaranteeing InIV the,we last part that of secondary Sec. IV,we also show how thesealso conditionsconstraints show how can these are be not understood conditions produced in can and Lagrangian be all understood the Lagrange formulation. in multipliers Lagrangian In Sec. are formulation.V uniquely, we provide determined. In concrete Sec. V, In examples, we the provide last part concrete of Sec. examples,IV,we Derivative:which areleft free derivatives fromwhich thealso with are fermionic show respect free how from to ghosts, these Grassmann-odd the conditions and fermionic explicitly variables can ghosts, be show understood are and defined the explicitly consistency in by Lagrangian removing show with the theformulation. our variable consistency analysis. from In the Sec. The withV final, our we provide section analysis. concrete The examples, final section isleft, devoted to summary.is devotedwhich In are Appendix to free summary. fromA, the we In fermionic mention Appendix how ghosts,A to, we produce and mention explicitly primary how show to constraints produce the consistency primary properly with constraints even our when analysis. the properly The evenfinal section when the maximally-degeneratemaximally-degenerateis condition devoted to is not summary. satisfied. condition In In Appendix is Appendix not satisfied.A,B we, we mention In discuss Appendix how Hamiltonian toB produce, we discuss analysis primary Hamiltonian including constraints fermions analysis properly including even when fermions the whenmaximally-degenerate we possibly have also condition secondary@ isLf not constraints. satisfied. In In Appendix AppendixB,C we, we discuss explicitly Hamiltonian prove that analysis the maximally-degenerate including fermions when we possibly have also secondary constraints. f = ✓ In↵ Appendix. C, we explicitly prove that the maximally-degenerate(8) condition is equivalentconditionwhen to the we is possibly presenceequivalent have of toN also theprimary secondary presence@✓ constraints.↵ constraints. of N primary In Appendix In Appendix constraints.D,C we, we In calculate explicitly Appendix the prove DiracD, that we brackets calculate the maximally-degenerate the Dirac brackets condition is equivalent to the presence of N primary constraints. In Appendix D, we calculate the Dirac brackets between canonicalbetween variables canonical in the maximally-degenerate variables in the maximally-degenerate case. In Appendix E case., we Ingive Appendix a simple↵E extension, we give to a ghost simple extension to ghost Throughout this paper,between we use left canonical derivatives variables and omit in the superscript maximally-degenerateL for the derivative case. In operator, Appendix@/@✓E, we give a simple extension to ghost freeL boson-fermion↵ free co-existence boson-fermion field theory. co-existence field theory. ⌘ @ /@✓ . free boson-fermion co-existence field theory. Complex conjugate: let us define complex conjugate in Grassmann algebra. In order to be consistent with Hermi- tian conjugation of operators, the complex conjugation is required to have the following properties: II. GRASSMANNII. ALGEBRA GRASSMANN AND ALGEBRA CANONICAL AND FORMALISM CANONICAL FORMALISM II. GRASSMANN ALGEBRA AND CANONICAL FORMALISM ↵ ↵ (✓ ✓ )⇤ = ✓ ⇤ ✓ ⇤, (9) In quantum fieldIn theories, quantum fermionic field theories, fields↵ obey fermionic↵ canonical fields anti-commutation obey canonical relations, anti-commutation a(t, x), ⇡ relations,b(t, y) + = a(t, x), ⇡b(t, y) + = In quantum field theories,(✓ ⇤)⇤ = fermionic✓ , fields obey canonical anti-commutation{ (10) relations,} a(t, x), ⇡b(t, y) + = i ab(x y), where a is a fermion and ⇡a is↵ its conjugate↵ momentum. For the purpose of constructing general{ { } } i abi( x (xy),y where), where a is(ais a✓ afermion)⇤ fermion= a⇤ ✓ and⇤ and, ⇡⇡a isis its its conjugate conjugate momentum. For For(11) the the purpose purpose of of constructing constructing general general action with bosons andab fermions, we woulda like to start witha classical (or “pseudo-classical”) treatment of them. actionaction with with bosons bosons and and fermions, fermions, we we would would like like to to start start with with classical classical (or (or “pseudo-classical”) “pseudo-classical”) treatment treatment of them.of them. To deal with fermions in the context of classical mechanics, one needs to reformulate canonical formalism such that where a is a complexTo number. dealTo deal with with fermions fermions in in the the context context of of classical classical mechanics, mechanics, one one needs needs to to reformulate reformulate canonical canonical formalism formalism such such that that classical analysis isclassical consistent analysis with anti-commutation is consistent with relations anti-commutation in quantum relations theory. inIn quantum the first part theory. of this In section, the first we part of this section, we Inverse matrix: whether aclassical matrix is analysis invertible is or consistent not plays withan important anti-commutation role in degenerate relations theories in quantum as we will theory. see in In the first part of this section, we briefly provide anbriefly overviewbriefly provide of provide the an basics an overview overview of Grassmann of of the the basics basics algebra. of of Grassmann Grassmann Then, we focus algebra. algebra. on Hamiltonian Then, Then, we we focus focus formulation on on Hamiltonian Hamiltonian including formulation formulation including including fermionicSec. IV. The degrees condition of freedom. for the existence (All the of materials the inverse described matrix of in a Grassmann this section valued and Sec. squareIII matrixare based is obtained on [44].) as follows. We introducefermionicfermionic two degrees square degrees matrices of freedom.of freedom. that are (All (All functions the the materials materials of the variables described describedqi in inand this✓↵, section section which can and and be Sec. Sec. in IIIIIIareare based based on on [44 [].)44].) general written as
i ↵ i A. Grassmanni ↵ i algebra↵ A(q , ✓ )=A0(q )+A↵(q )✓ + A↵ (q )✓ A.✓ + Grassmann, algebra (12) A. Grassmann··· algebra i ↵ i i ↵ i ↵ B(q , ✓ )=B0(q )+B↵(q )✓ + B↵ (q )✓ ✓ + , (13) A ··· A Grassmann algebraA is Grassmann formed by algebra generators is formed⇠ with by generatorsA =1, 2,...,M⇠AA withsatisfyingA =1, the2,...,M anti-symmetricsatisfying the relations, anti-symmetric relations, A B B A A Grassmann algebra is formed by generators ⇠ i with A =1, 2,...,M satisfyingA A the anti-symmetric relations, ⇠where⇠ +A⇠0,A⇠↵,...,B= 0.0,B From↵A,...Bare this fullyB definition,A anti-symmetric it is clear matrices that depending each generator on q . The squared condition should that beB zero,is the ⇠ ⇠ =0(no A A ⇠A⇠B⇠ +⇠ ⇠+B⇠⇠A ⇠== 0. 0. From From this this definition, definition, it it is is clear clear that that each each generator generator squared squared should should be be zero, zero,⇠ ⇠⇠A⇠=0(noA =0(no summation),inverse of A is which given by suggestssummation),AB = I the,where Pauli whichI is exclusion the suggests identity principle the matrix, Pauli at which exclusion the levelleads principle to of the classical following at the mechanics. level equations, of classical In terms mechanics. of generators In terms of generators ⇠A, an arbitrary functionsummation),A g can which be expressed suggests as the Pauli exclusion principle at the level of classical mechanics. In terms of generators A ⇠ , an arbitrary function g can be expressed as ⇠ , an arbitrary function g can be expressedA0B0 as= I, (14)
A A B A A AB1 AM A1 AM g = g0 + gA⇠ + gABg⇠A=0⇠Bg↵0+++AgA↵B⇠+0 g=0+A1g...AAB, M⇠ ⇠⇠ ...+⇠ +,gA ...A ⇠ ...(15)⇠ , (1) (1) ···A A B 1 M A1 AM 1 g = g0 + gA⇠ + gAB⇠ ⇠ + ···+ gA1...AM ⇠ ... ⇠ , (1) where the coe cients g areA0B completely↵ + (A↵B anti-symmetric. A B↵)+A↵ B The0 =0 terms, made··· of an even (odd)(16) number of ⇠A are A whereA1...A then coe cients2 gA1...A n are completely anti-symmetric. The terms made of an even (odd) number of ⇠ are i i A called “Grassmann-even”wherecalled the (“Grassmann-odd”). “Grassmann-even” coe cients gA1...A (“Grassmann-odd”).n Noware we completely introduce anti-symmetric. even. Now dynamical we introduce variables The even terms dynamicalq (t made)(i =1 of variables, an2, evenn)q and( (odd)t)(i =1 number, 2, n of) and⇠ are . ··· i odd ones ✓↵(t)(↵called=1odd, 2, “Grassmann-even” onesN)✓ as↵(t follows,)(↵ =1, 2 (“Grassmann-odd”)., N) as follows, . Now. we introduce even dynamical variables q (t)(i =1,···2, n) and odd ones···✓↵(t)(↵ =1, 2, ···N) as follows, ··· You will find that if and only if A0 has thei inverse,···i the equationsi i A canB bei solved successivelyi A B as q (t)=q0(t)+qAB(tq)⇠(t)=⇠ +q0(t)+, qAB(t)⇠ ⇠ + , (2) (2) i i i A B ↵ ↵ A ↵ ↵ A ↵···B CA ↵ A ···B C 1 ✓ (t)=✓ (t)⇠ + ✓q✓(t()=t()=t)⇠q✓0⇠(t()+t⇠)⇠ +qAB+ ✓(t),⇠ (⇠t)⇠ +⇠ ⇠ ,+ , (3) (3) (2) B = A , A ABC A ABC (17) 0 0 ↵ ↵ A ···↵ A ···B C ··· 1 1 ✓ (t)=✓A(t)⇠ + ✓ABC (t)⇠ ⇠ ⇠ + , (3) where the coe cientswhereqBi ↵ = theCLASSICALAand0 coeA ↵✓A↵cients0 , qarei completelyand ✓SETUP↵ anti-symmetricare completely and anti-symmetric time-dependent.··· and(18) (Since time-dependent. we do not (Since we do not A1...A n A1...An A1...An A1...An 1 i ↵ ↵ ↵ require the covariance,whererequire superscripts the the coe covariance,1 cients and subscripts1q superscripts1 and are just✓1 and labels subscripts1 are of completelythe1 are variables, just1 labels anti-symmetric e.g., of✓ the= variables,✓↵, except and e.g., time-dependent. for✓ Appendix= ✓↵, except (Since for Appendix we do not B↵ = A0 (A↵A0 AA 1A...A0 n A A0 AA1...A↵A0 n ) A0 A↵ A0 , (19) E.) These variablesrequire satisfyE.) These the the2 covariance, following variables (anti-)commutation satisfy superscripts the following and subscripts relations: (anti-)commutation are just labels relations: of the variables, e.g., ✓↵ = ✓ , except for Appendix . ↵ E.) These. variables satisfy the following (anti-)commutation relations: bosons: commuting(Grassmann-evenqiqj qj)q propertyi =0, qiqj qjqi =0, (4) (4) ↵ i i ↵ i↵j i ji i↵ which also satisfy BAfermions= I. Therefore,: anti-commuting we conclude (Grassmann-odd) that✓ q amatrixq ✓ =0A has property, theq✓q inverseq qq q✓ if=0 and=0, only, if A0 has the (5) (5) (4) ↵ ↵ inverse, i.e., ✓↵✓ + ✓ ✓↵ =0 ,. ✓✓↵q✓i +q✓i✓↵✓ =0=0,. (6) (6) (5) ↵ ↵ i det(A ) =0, i where A = A↵ .✓ ✓ + ✓ ✓ =0↵ . (20) (6) From the above relations,LetFrom us considerq the(t) above can simplest be relations,0 regarded6 modelsq as(t) bosons can be and0 regarded✓|✓(=0t) as as bosons fermions. and ✓ (t) as fermions. i ↵ From the above relations, q (t) can be regarded as bosons and ✓ (t) as fermions.i ↵ A Function:Grassmann-evenbecause of Grassmann real Lagrangian nature, an arbitraryi (super)function↵ f(q , ✓ ), whichA depends on ⇠ only through Function: because of GrassmannB. nature, Hamiltonian an arbitrary formulation (super)function f(q , ✓ ), which depends on ⇠ only through i ↵ Noqi andspatial✓↵, derivatives can be expanded (not field in powers theories) of↵ the odd variables ✓↵, q and ✓ , canFunction: be expandedbecause in powers of Grassmann of the odd nature, variables an✓ arbitrary, (super)function f(qi, ✓↵), which depends on ⇠A only through i Up to↵ first time derivatives (no higher derivatives)i ↵ Now we move on to Hamiltonianq and formulation✓ , can be both expanded with n Grassmann-even in powersi ↵ of the variables oddi variablesq (t) andi ↵N✓ Grassmann-odd, i ↵ ones ✓↵(t). In the present paper, weReal consider variablesf( theqi, Lagrangian✓(complex↵)=f (q variables containingi)+ff(q(q, ican)✓ up✓↵)= to+be thef fdecomposed)0( firstq(q)+i)✓ derivatives,↵f✓↵ (q+)✓ + namely,, f↵ (q )✓ ✓ + , (7) (7) 0 ↵ ↵ ··· ··· t2 i ↵ i i ↵ i ↵ i i i ↵i f(↵q , ✓ )=f0(q )+f↵(q )✓ + f↵ (q )✓ ✓ + , (7) i wherei f0(Sq =) and fL↵(1q...,↵q˙k,(✓q ), ✓˙ are)dt Grassmann-even . functions with fully anti-symmetric(21) indices. where f0(q ) and f↵1...↵k (q ) are Grassmann-even functions with fully anti-symmetric indices. ··· Zt1 i i where f0(q ) and f↵1...↵k (q ) are Grassmann-even functions with fully anti-symmetric indices. 2
Inverse matrix: whether a matrix can be invertible or not plays an important role in degenerate theories as we will see in Sec. IV. The condition for the existence of an inverse matrix of a Grassmann valued square matrix is obtained as follows. We introduce two square matrices that are functions of the variables qi and ✓↵, which can be in general written as
A(qi, ✓↵)=A (qi)+A (qi)✓↵ + A (qi)✓↵✓ + , (12) 0 ↵ ↵ ··· B(qi, ✓↵)=B (qi)+B (qi)✓↵ + B (qi)✓↵✓ + , (13) 0 ↵ ↵ ··· i where A0,A↵,...,B0,B↵,... are matrices depending on q . The condition that B is the inverse of A is given by AB = I,whereI is the identity matrix, leads to the following equations,
A0B0 =0, (14)
A0B↵ + A↵B0 =0, (15) 1 A B + (A B A B )+A B =0, (16) 0 ↵ 2 ↵ ↵ ↵ 0 . .
1 1 1 1 A =A0 A0 A↵A0 5 1 1 1 1 1 1 1 1 5 + A0 (A↵A0 A A0 A A0 A↵A0 ) A0 A↵ A0 + (17) 2 III. NECESSITY OF DEGENERACY - EXAMPLE: PURELY FERMIONIC··· SYSTEM III. NECESSITY OF DEGENERACY - EXAMPLE: PURELY FERMIONIC SYSTEM 5 Contrary to purely bosonic system, N-fermionic system needs constraints eliminating N/2 ghostly degrees1 of freedom The above equations can be solved in succession if and only if A0 has an inverse. Then, assuming A0 3exists, to realize healthy system.Contrary We will see to purely theIII. appearance bosonic NECESSITY system, of negative OFN DEGENERACY-fermionic norm states system for - the EXAMPLE: needs fermionic constraints PURELY system eliminating without FERMIONIC anyN/2 ghostlySYSTEM degrees of freedom we get constraints, i.e., in non-degenerateto realize healthy theories. system. We alsoWe will show see how the the appearance negative norm of negative states are norm avoided states for for usual the fermionic system without any Contrary to purely bosonic system, N-fermionic system needs constraints eliminating N/2 ghostly degrees of freedom where we require theDirac-type variation fermions. of zA to Weconstraints, vanish omit at bosonic the i.e., endpoints, degrees in non-degenerate of freedomzA(t1)= here theories.zA for(t2) simplicity, = 0. We The also but canonical show the essence how momenta the does negative not are change norm for states are avoided for usual defined by boson-fermion1 co-existence systemto realize as we healthy will see system. in the We next will section. see the appearance of negative norm states for the fermionic system without any B0 = A0 , Dirac-typeconstraints, fermions. i.e., We in non-degenerate omit bosonic theories. degrees of We freedom also show here how for the simplicity, negative norm(18) but states the essence are avoided does for not usual change for 1 1 boson-fermionDirac-type@L co-existence fermions.@ WeL system omit bosonic as we will degrees see in of freedomthe next here section. for simplicity, but the essence does not change for B↵ = A0 A↵A0 , pi = , ⇡↵ = . (25) (19) boson-fermion@q˙iA. Non-degenerate co-existence@✓˙↵ system fermionic as we will system see in the next section. 5 1 1 1 1 1 1 1 1 5 Note that pBi are↵ even= andA0 real(A↵ variablesIII.A0 NECESSITYA asA usual0 whileA OF A DEGENERACY⇡0 ↵ areA↵ oddA0 and) - imaginary EXAMPLE:A0 A.A Non-degenerate↵ variablesA PURELY0 , from FERMIONIC the fermionic reality systemof SYSTEM the (20) Lagrangian. Then, the2In Hamiltonian this subsection, is we given begin by with the action given by A. Non-degenerate fermionic system III. NECESSITYt2 OF DEGENERACY - EXAMPLE: PURELY FERMIONIC SYSTEM . In thisi subsection,˙↵ wei begini ↵ with˙↵↵ the˙↵ action given by . Contrary to purelyH bosonic=˙q piIn system,+ this✓ ⇡ subsection,↵N-fermionicL(qS ,=q we˙ , ✓ begin system,L✓(✓ with),, needs✓ the)dt constraints action . given eliminating by N/2 ghostly(26) degrees of(34) freedom . t to realize healthy system. We will see the appearanceZ 1 of negative norm statest2 for the fermionic system without any t2 ↵ ↵ and the variationalWe principleconstraints, assume the that action i.e., the Lagrangian in yields non-degenerate Hamilton’s is non-degenerate,Contrary theories. equations, to purely We bosonic also show system, howN the-fermionicS = negativeL system( norm✓↵ ,˙✓↵˙ needs states)dt . constraints are avoided eliminating for usualN/2 ghostly degrees of freedom(34) to realize healthy system. We will see theS appearance= L(✓ , of✓ ) negativedt . norm states for the fermionic system(34) without any Dirac-type fermions. We omit bosonic degrees of freedom here for simplicity,tt11 but the essence does not change for which also satisfy BA = I. Therefore, we@ concludeH that@H (0)amatrix@AH has the inverse@H(0) ifZZ and only if A0 has the boson-fermionq˙i = co-existence, p˙ = @ system2constraints,L , as we✓˙↵ will i.e.,= see in in non-degenerate, the⇡˙ next=@2 section.L theories.. @2 WeL also show how(27) the negative norm states are avoided for usual Wedeti assumeWe assume thati the that=0 Lagrangian, the Lagrangianwhere is non-degenerate,↵ is non-degenerate,↵ = , (35) inverse, i.e., @pi ˙ @Dirac-typeq˙↵ fermions. @⇡↵ We omit˙ bosonic ˙@✓↵ degrees˙ of˙↵ freedom here for simplicity, but the essence does not change for @✓ @✓ 6 @✓ @✓ @✓ @✓ ✓,✓˙=0 ✓ boson-fermion◆ co-existence✓2 system(0)(0) as◆ we will see in the next2 section.(0)(0) 2 The time-evolution of a function F (qi, ✓↵) can be expressed as F˙ = @@F/2L@@tL+ F, H , where the Poisson@@L2L bracket of@ L@2L and the Euler-Lagrange equations (22) thenA. contain Non-degeneratedet det the second timefermionic=0=0 derivatives, , systemwherewhere of ✓ ↵. In other words,= = this system , , (35) (35) @✓˙ @✓˙↵{ 6 } @✓˙ @✓˙↵ @✓˙ @✓˙↵ ˙ arbitrary variables F and G isdet( definedA0) as=0 [38], where A0 = A ✓@=0✓˙✓ @.✓˙↵ ◆ 6 ✓ @✓˙ @✓˙◆↵ @✓(21)˙ @ ✓✓˙,✓↵=0 ˙ does not have any constraints, and the total number✓ of degrees◆ of freedom is the same with✓ the number◆ of the original ✓,✓=0 6 | A. Non-degenerate fermionic system ↵ variablesIn thisN. (The subsection, phase@F @ space weG beginand is@F spanned the with@G Euler-Lagrange the by action 2N canonical given@ equationsF by@G variables.) (22@)F then@G contain the second time derivatives of ✓ . In ↵ other words, this system Now we would like toand show thedoes that Euler-Lagrange not non-degenerateFERMIONIC have any constraints,"F equations fermionic and (22 system the) then total inevitably contain number GHOST of the gives degrees second negative of freedom time norm derivatives is states. the same Similar of with✓ the. In number other of words, the original this system F, G = i i +( ) ↵ + ↵ . (28) situations{ are} known@q todoes@ bepi found not variables@ havepi in@q the anyN non-degenerate. constraints, (The phase@✓ space and Lagrangian@⇡t2 is the↵ spanned total@⇡↵ with number by@✓ 2 higherN canonical of derivatives degrees variables.) of of freedom bosonic is variables. the same In with the number of the original ✓ In this◆ subsection,✓S = we beginL(✓↵ with, ✓˙↵) thedt . action◆ given by (34) bosonic case,B. after Hamiltonian thevariables replacementNowN. of (The formalismwe the would phase higher like space derivative to show is spanned that terms non-degenerate with by 2 newlyN canonical fermionic defined variables.) variables, system inevitably one finds gives that negative the norm states. Similar t1 t2 Hamiltonian is linear in momentumNowsituations we would and@ are notF like known@ boundedG to show to@ beF from thatZ found@G below, non-degenerate in the which non-degenerate leads fermionic to the Lagrangian Ostrogradsky’s system↵ with↵ inevitably higher ghost derivatives instabil- gives negative of bosonic norm variables. states. In Similar "F S = L(✓ , ✓˙ )dt . (34) We assume that theF, G LagrangianPurely=(bosonic fermionic) is case, non-degenerate, after non-degenerate the+ replacement. of system the higher derivative terms with newly(29) defined variables, one finds that the ity [46, 47]. This ghostsituations can be interpreted are known↵ as to the be appearance found in↵ the of negative non-degeneratei norm states Lagrangiant in1 the quantized with higher theory derivatives [48]. of bosonic variables. In { } Hamiltonian @✓ is@⇡ linear↵ in@⇡ momentum↵ @✓ and not bounded fromZ below, which leads to the Ostrogradsky’s ghost instabil- Now we move on to HamiltonianIn formalism fermionic system, with thebosonic both positivity case,n Grassmann of2 ✓ afterHamiltonian(0) the replacement is even not guaranteed variables◆ of the in higher2 theq classical((0)t derivative) and level,2 N terms andGrassmann we with should newly discuss odd defined the variables, one finds that the ↵ ityWe [46@, assumeL47]. This that ghost the can Lagrangian be interpreted is@ non-degenerate,L as the appearance@ L of negative norm states in the quantized theory [48]. ones ✓ (t). In the presentHere, "F paper,represents westability the consider Grassmann after the the quantization. parity LagrangianHamiltonian ofdetF ,i.e., Based containing is"F on linear=0if the=0 canonical inF momentum,is up even,where quantization to and first and"F not derivatives,=1if (32 bounded), weF is obtain= odd. from namely, anti-commutation As below, a consequence, which, leads relations, to the Ostrogradsky’s(35) ghost instabil- ˙ ˙↵ ˙ ˙↵ ˙ ˙↵ In fermionic@✓ @✓ system,6 the positivity of2 Hamiltonian@✓ @✓(0) is not@✓ guaranteed@✓ ✓,✓˙=0 in the2 classical(0) level,2 and we should discuss the the Poisson brackets of the canonical variablesity [46, are47✓ ]. found This◆ to ghost be↵ can be interpreted↵ ✓@ L as the◆ appearance of negative@ L norm states@ L in the quantized theory [48]. stability after the✓ˆ quantization., ⇡ˆ + = i Based, on the canonical quantization (32), we obtain anti-commutation relations, det =0, where↵ = , (35) i and thei Euler-Lagrange↵ In fermionic equations˙ and↵ π system, have (i22)j the then{ the one-to-one contain positivity} the ofcorrespondence.↵ second@ Hamiltonian✓˙ @✓˙↵ time derivatives6 is (No not guaranteedprimary of ✓ . In constraints)@ other in✓˙ @ the✓˙↵ words, classical this@(36)✓˙ level, system@✓˙↵ and˙ we should discuss the q ,pj = j , ✓ , ⇡ = ✓⇡ , q ,q =ˆ↵ ˆp i,pj = ✓✓ , ✓ =◆⇡ˆ↵↵, ⇡ =0. ↵ ✓ (30)◆ (22) ✓,✓=0 { does} not have{ any constraints,stability} after and{ the the quantization. total} ✓ ,{ number✓ + =} of Based⇡ˆ degrees{↵, ⇡ˆ on+} of=0 the freedom{. canonical✓ , ⇡ˆ is}+ the= quantization samei with, the (32 number), we obtain of the original anti-commutation relations, { } { } { } ↵ (36) variables N. (The phase spaceand is spanned the Euler-Lagrange by 2N canonicalquantization equations variables.) (22)ˆ then↵ ˆ contain the second time derivatives of ✓ . In other words, this system Here, the canonical operators ✓ˆ↵↵ and⇡ ˆ are↵ now Hermitian and anti-Hermitianˆ↵✓ , ✓ + = operators,⇡ˆ↵,↵⇡ˆ + respectively.=0. Then, we ✓ ,does⇡ not=↵ have , any constraints, and the✓{ total, ⇡ˆ number}+ = { i of degrees}, of freedom is the same with the number of the original Now we would liket2 to show that non-degenerate fermionic system inevitably gives negative norm states. Similar introduce orthogonal Hermitian{ operators,variables} N. (The phase spaceˆ↵ is spanned{ by} 2N canonical variables.) (36) situations are known to beHere,i↵ foundi the in↵ canonical the↵ non-degenerate operators ✓ Lagrangianand⇡ ˆ↵ areˆ↵ with nowˆ Hermitian higher derivatives and anti-Hermitian of bosonic operators, variables. respectively. In Then, we S = L(q✓ ,,q˙✓ , ✓=, ✓⇡˙↵,)⇡dt. =0. ✓ , ✓ + = ⇡ˆ↵, ⇡ˆ + =0. (31) (23) bosonic case, after the replacement{ introduceNowˆ} orthogonal of{ we1 the wouldˆ higher} Hermitian like derivative to showˆ operators, that terms1 non-degenerateˆ{ with newly} { defined fermionic} variables, system one inevitably finds that gives the negative norm states. Similar t A↵ = (✓↵ i⇡ˆ↵) , B↵ = (✓↵ + i⇡ˆ↵) , (37) Hamiltonian is linear1 in momentumsituations andp2 are not known bounded to↵ be from found below,p2 in the which non-degenerate leads to the Lagrangian Ostrogradsky’s with ghost higher instabil- derivatives of bosonic variables. In The Poisson bracket satisfies the followingZ Here, identities, the canonical operators ✓ˆ and⇡ ˆ are1 nowˆ Hermitian and1 anti-Hermitianˆ operators, respectively. Then, we orthogonalbosonic Hermitian case, after operators the replacementAˆ↵↵= of( the✓↵ higheri⇡ˆ↵) , derivativeBˆ↵ = terms(✓↵ + i with⇡ˆ↵) , newly defined variables, one(37) finds that the andity the [46 anti-commutation, 47]. This ghostintroduce relations can be orthogonal interpretedbetween them Hermitian as are the given appearance operators, by ofp negativei 2 norm↵ states inp2 the quantized theory1 [48]. The Lagrangian should be an even andIn fermionic real function, system,F, G the and=( positivityHamiltonian the)"F "G of dynamical+1 HamiltonianG, is F linear, variablesis in not momentum guaranteedq andand in not the bounded✓ classicalare from level, taken below, and to we which(32) be should real leads discuss to the the Ostrogradsky’s ghost instabil- and the anti-commutation relations between them are given by { }AAˆ↵, Aityˆ i [+46=,↵47{ ].↵ This, }" ghostAˆ"↵, Bˆ can+ =0 be interpreted, 1 Bˆ↵, Bˆ as+ the= ↵ appearance. 1 of negative norm(38) states in the quantized theory [48]. throughout this paper. The variation withstability respect afterF, the G to quantization.Gz ==(F,q G Based, ✓G) on+( yields the) canonicalF theG1 G Euler-LagrangeF, quantizationˆ G , ˆ (32), we equations, obtainˆ anti-commutationˆ (33) relations, 1 2 { In} fermionic1 2 system,{ the1ˆ positivity} Aˆ↵ =2 of{ ( Hamiltonian✓↵ i}⇡ˆˆ↵) ,ˆ isB not↵ = guaranteed(ˆ✓↵ +ˆ ini⇡ˆ the↵) , classical level, and we should discuss(37) the { } { } " " A{↵, A + }=p ↵ , A↵, B + =0,p B↵, B + = ↵ . (38) One immediately notices that all eigenvalues ofˆ theF↵2 G first{ anti-commutator}↵ 2 have{ the} negative2 sign,{ leading} to the F1F2,G = F1stabilityF2,G after+( the)✓ quantization., ⇡ˆ F+1=,Gi F Based2,, on the canonical quantization(34) (32), we obtain anti-commutation relations, negative norm states,{d whileand@L} the thoseOne anti-commutation{ of immediately@ theL third} anti-commutator notices{ relations that{} all between} eigenvalues have the them correct of are the givenfirst sign anti-commutator guaranteeing by the have positivity the negative of (36) sign, leading to the norm of states. This fact tells us that each=0 fermionic, ✓ˆNegative↵, ✓ˆ degree= norm⇡ˆ of, freedom⇡ˆ states=0 in. physical(ghosts)↵ spacePositive should↵ carry norm 1 degree(24) of which are easily proved from the definitionA of thenegative PoissonA norm bracket states, ( while28).+ those of↵ the third+ anti-commutator✓ˆ , ⇡ˆ + = havei the, correct sign guaranteeing the positivity of dt @z˙ 2@z { ˆ} ˆ { } ˆ ˆ ˆ ˆ If the system containsfreedom (second in phase class) space, constraints otherwisenorm negative, of one states. should norm This states rather factA inevitably↵ tells, useA us the+ that= appear, Diraceach ↵ bracket fermionicwhich, {A implies↵ instead, degreeB } + that of=0 of freedomN/ the, 2 physical PoissonB in↵, physicalB degrees+ = space ↵ of should. carry 1 degree of (38)(36) ✓ ◆ ˆ↵ { } { ˆ↵ ˆ } { } bracket, defined byfreedomHere, (N thedegrees canonical of freedom operatorsfreedom in phase✓ and in space) phase⇡ ˆ↵ are space, are now extra otherwise Hermitian degrees negative of and freedom norm anti-Hermitian corresponding states✓ , inevitably✓ + operators,= to⇡ˆ appear, fermionic↵, ⇡ˆ respectively.+which=0 ghosts. implies. Since Then, that N/ we2 physical degrees of One immediately notices that all eigenvalues of the{ first} anti-commutator↵ { } have the negative sign, leading to the thisintroduce is a direct orthogonal consequence Hermitian offreedom the canonical operators, (N degrees quantization of freedom of in the phase canonical space) are variables extra degrees✓ and of⇡ freedom↵ (36), corresponding any fermionic to fermionic ghosts. Since negative normHere, states, the canonical while those1 operatorsab of the✓ˆ↵ thirdand⇡ ˆ anti-commutatorare now Hermitian have and the anti-Hermitian correct sign↵ guaranteeing operators, respectively. the positivity Then, of we non-degenerate theoriesA, B alwaysD = this suA,↵ is Ber a from directA, negative consequence 1a (C norm ) of states theb,B canonical even, if1 we quantization↵ have bosonic of variablesthe canonical in addition.(35) variables ✓ and ⇡↵ (36), any fermionic { norm} ofnon-degenerate{ states.introduce} Aˆ This{ = orthogonal theories fact}(✓ˆ tells always Hermitiani⇡ˆ us{) su that, ↵erBˆ}each fromoperators,= negative fermionic(✓ˆ + normi⇡ˆ degree) , states of even freedom if we have in physical bosonic variables space(37) should in addition. carry 1 degree of 1 ↵ p ↵ ↵ ↵ p ↵ ↵ Since complex variableswhere can bea are decomposed second class into constraints real and and imaginaryfreedomCab = in phase partsa, b . space, and expressed2 otherwise negative in terms norm of2 a states set of inevitably two real variables,appear, which we implies can that N/2 physical degrees of { } ˆ 1 ˆ ˆ 1 ˆ always identify complexThe variables prescription with doubled of the canonical real variables quantization without is simply lossB. of replacing generality. Degenerate the fermionic Poisson brackets systemA↵ = of(✓↵ canonicali⇡ˆ↵) , variablesB↵ = (or(✓↵ + i⇡ˆ↵) , (37) and the anti-commutationfreedom relations (N degrees between of freedom them are in given phase by space) arep extra degrees of freedomp corresponding to fermionic ghosts. Since B. Degenerate2 fermionic system2 ↵ the Dirac brackets if secondary constraintsthis exist) is a by direct commutation consequence relations of the for canonical bosons andquantization anti-commutation of the canonical relations variables ✓ and ⇡↵ (36), any fermionic Aˆ↵, Aˆ + = ↵ , Aˆ↵, Bˆ + =0, Bˆ↵, Bˆ + = ↵ . (38) for fermions, Although the appearancenon-degenerate of negative{ and the norm theories} anti-commutation states always seems{ su to↵ relationser be} from a generic negative between feature{ them norm of} non-degenerate are states given even by if fermionic we have system bosonic variables in addition. as weOne saw immediately in the previous notices subsection, thatAlthough all we eigenvalues the have appearance already of the known of first negativeˆ that anti-commutatorˆ norm a Dirac states field, seems have for to example,ˆ the beˆ a negative generic does feature sign,not suˆ leading of↵er non-degenerateˆ from to the fermionic system 1 ˆ ˆ A↵, A + = ↵ , A↵, B + =0, B↵, B + = ↵ . (38) suchnegative a problem.(i~ norm) Here,A, states,B we review whileasif thoseA we whyand saw such ofB in the aarethe theory third previous bosons anti-commutator can, subsection, avoid{ negative we have} have norm the already states correct known by{ sign illustrating that guaranteeing} a Dirac a simplest field,{ the for positivity model, example,} of does not su↵er from 1{ ˆ ˆ} suchOne a problem. immediately Here, noticeswe review that why all such eigenvalues a theory can of avoid the first negative anti-commutator norm states by have illustrating the negative a simplest sign, model, leading to the A, B norm( ofi~ states.) A, ThisB factif tellsA is us a that bosoneach (fermion) fermionici and degreeB isB. of a freedom fermion Degenerate in (boson) physical fermionic, space systemshould(36) carry 1 degree of { } ! 8 { } negative norm states, while˙↵ those of the third anti-commutator have the correct sign guaranteeing the positivity of freedom in phase1 ˆ ˆ space, otherwise negative normL = states✓ inevitably✓↵ . appear, which impliesi ˙↵ that N/2 physical degrees(39) of <>(i~) A, B + if A and B are fermions ,2 L = ✓ ✓↵ . (39) freedom (N degrees{ } of freedomnorm in phase of states. space) This are extra fact tells degrees us that of freedomeach fermionic corresponding 2 degree to of fermionic freedom ghosts. in physical Since space should carry 1 degree of Althoughfreedom the appearance in phase space, of negative otherwise norm negative states norm seems states to be inevitably↵ a generic appear, featurewhich of non-degenerate implies that N/ fermionic2 physical system degrees of this> is a direct consequence of the canonical quantizationˆ ˆ of theˆ canonicalˆ ˆ ˆ variablesˆ ˆ ✓ andˆ ˆ⇡↵ (ˆ36ˆ), any fermionic where the commutator and: anti-commutatoras we are saw respectively in the previous defined subsection, as A, B we= A haveB B alreadyA and knownA, B + that= A aB Dirac+BA. field, for example, does not su↵er from non-degenerate theories alwaysfreedom su↵er from(N degrees negative of{ freedom norm} states in phase even i space) if we have are{ ↵ extra bosonic} degrees variables of freedom in addition. corresponding to fermionic ghosts. Since Hereafter, we set = 1 in this paper. One should note that real variables such as q ,pi, and ✓ will be promoted ↵ ~ such a problem.this is a Here, direct we consequence review why of such the canonical a theory can quantization avoid negative of the normcanonical states variables by illustrating✓ and ⇡↵ a simplest(36), any model, fermionic to Hermitian operators, and then the imaginary variables ⇡ become anti-Hermitian operators through canonical non-degenerate↵ theories always su↵er from negativei norm states even if we have bosonic variables in addition. quantization. B. Degenerate fermionic systemL = ✓˙↵✓ . (39) 2 ↵ Although the appearance of negative norm states seems to be a genericB. feature Degenerate of non-degenerate fermionic system fermionic system 2 as we saw in the previous subsection, we have already known that a Dirac field, for example, does not su↵er from When the system contains first class constraints (sometimesAlthough in addition the to appearance second class of constraints), negative norm we add states gauge seems fixing to conditions be a generic to feature of non-degenerate fermionic system the set of the constraints,such which a problem. e↵ectively Here, leads towe a review system why only such with asecond theory class can constraints. avoid negative norm states by illustrating a simplest model, as we saw in the previous subsection, we have already known that a Dirac field, for example, does not su↵er from i such a problem. Here,L we= review✓˙↵✓ why. such a theory can avoid negative norm states(39) by illustrating a simplest model, 2 ↵ i L = ✓˙↵✓ . (39) 2 ↵ 5 5
B. Non-degenerate fermionic system B. Non-degenerate fermionic system In this subsection, we begin with the action given by In this subsection, we begin with the action given by t2 ↵ ↵ t2 S = L(✓ , ✓˙ )dt. (46) ↵ ˙↵ S = L(✓ , ✓ )dt. Zt1 (46) t We assume that the LagrangianZ is1 non-degenerate, We assume that the Lagrangian is non-degenerate, @2L 2 det =0, (47) @ L ˙ ˙↵ ↵ ↵ 6 det ✓=0@✓ @,✓ ◆ ✓ =✓˙ =0 (47) ↵ @✓˙ @✓˙ ↵ ˙↵ 6 ✓ ◆ ✓ =✓ =0 ↵ and the Euler-Lagrange equations (24) then contain the second time derivatives of ✓ . In other words, this system does not have any constraints, and the total number of degrees of↵ freedom are the same with the number of the and the Euler-Lagrange equations (24) then contain the second time derivatives of ✓ . In other words, this system does not have any constraints,original variables and theN. total (The number phase space of degrees is spanned of freedom by 2N canonical are the variables.) same with the number of the original variables N. (TheNow phase we would space like is spanned to show by that 2N non-degeneratecanonical variables.) fermionic systems inevitably give negative norm states. Similar situations are well-known to be found in a non-degenerate Lagrangian with higher derivatives of bosonic variables. Now we would like to show that non-degenerate fermionic systems inevitably give negative norm states. Similar After the replacement of the higher derivative terms with newly defined variables, one find that the Hamiltonian is situations are well-knownlinear to in be momentum found in a and non-degenerate is not bounded Lagrangian from below, with which higher leads derivatives to the Ostrogradsky’s of bosonic variables. ghost instability [39, 40]. After the replacementThis of the ghost higher can derivativebe interpreted terms as the with appearance newly defined of negative variables, norm one states find in that the the quantized Hamiltonian theory [ is41]. In fermionic linear in momentum andsystems, is not the bounded positivity from of Hamiltonian below, which are leads not guaranteed to the Ostrogradsky’s in the classical ghost level, instability and we should [39, discuss40]. the stability 3 This ghost can be interpretedafter the quantization. as the appearance Based of on negative the canonical norm quantization states in the (38 quantized), we obtain theory anti-commutation [41]. In fermionic relations, systems, the positivity of Hamiltonian are not guaranteedwhere we require in the the classical variation level, of z andto vanish we should at the discuss endpoints, the z stability(t )= z (t ) = 0. The canonical momenta are ↵ ↵ A A 1 A 2 after the quantization. Based on the canonical quantization (38),✓ˆ we, ⇡ˆ obtain+ = anti-commutationi , relations, (48) defined by { } 3 ↵ ↵ ↵ ✓ˆ , ✓ˆ + = ⇡ˆ↵, ⇡ˆ + =0. @L @L (49) ˆ pi = , ⇡↵ = . (25) ✓ , ⇡ˆ + = i ,{ } { } i ˙↵ (48) where we require the variation of zA to vanish{ at} the endpoints, zA(t1)= zA(t2) = 0. The canonical@q˙ momenta@ are✓ Here, the canonical operators↵ ✓ˆ↵ and⇡ ˆ are now Hermitian and anti-Hermitian, respectively. Then, we introduce defined by ✓ˆ , ✓ˆNote+ = that⇡ˆp↵↵i,are⇡ˆ even+ =0 and. real variables as usual while ⇡↵ are odd and(49) imaginary variables from the reality of the orthogonal operators, { Lagrangian.} @L { Then,} @ theL Hamiltonian is given by p = , ⇡ = . (25) ˆ↵ i i ↵ i ˙↵ i i ↵ ˙↵ Here, the canonical operators ✓ and⇡ ˆ↵ are now Hermitian@q˙ and@✓˙ anti-Hermitian,↵ respectively.H =˙q pi + ✓ Then,⇡↵ L we(q , introduceq˙ , ✓ , ✓ ), (26) ˆ 1 ˆ ˆ 1 ˆ orthogonal operators, A↵ = (✓↵ i⇡ˆ↵), B↵ = (✓↵ + i⇡ˆ↵), (50) Note that pi are even and real variables as usualand while the variational⇡↵ arep2 odd principle and imaginary the action variablesp yields2 Hamilton’s from the reality equations, of the Lagrangian. Then, the Hamiltonian is given by 1 1 i @H @H ˙↵ @H @H ˆ ˆ ˆ ˆ i ˙↵ ˆ i i ↵ˆ˙↵ q˙ = , p˙i = , ✓ = , ⇡˙ ↵ = . (27) where both A↵ Aand↵ =B↵ Hare(=˙✓↵ Hermitian,q pi i+⇡ˆ↵✓),⇡↵ andBL↵( the=q , q˙ anti-commutation, ✓(✓, ↵✓ +), i⇡ˆ↵), @p relations between@qi these operators(26)@⇡ (50) are given@✓↵ by p2 p2 i ↵ and the variational principle the action yields Hamilton’sTheAˆ , time-evolutionAˆ equations,= , of aA functionˆ , Bˆ F=0(qi,,✓↵) canBˆ , beBˆ expressed= as. F˙ = @F/@t + F, H , where(51) the Poisson bracket of ˆ ˆ { ↵ }+ ↵ { ↵ }+ { ↵ }+ ↵ { } where both A↵ and B↵ are Hermitian,i and@H the anti-commutationarbitrary@H variables↵ relationsF@Hand G is between defined@H as these [38] operators are given by q˙ = , p˙ = , ✓˙ = , ⇡˙ = . (27) ↵ (Since we do not require thei covariance,i superscripts and↵ subscripts↵ are just labels of the variables, e.g., ✓ = ✓ .) @pi @q @⇡↵ @✓ @F @G @F @G @F @G @F @G ↵ ˆ ˆ ˆ ˆ ˆ ˆ "F One immediatelyA↵, A + notice=i ↵ that↵ , allA eigenvalues↵, B + =0 of, the firstB↵F,, B commutator G =+ = ↵ i . have the negativei +( ) sign, leading↵(51) + to the negative↵ . (28) The time-evolution of a function F (q , ✓ ) can be expressed as F˙ = @F/{@t +} F, H@,q where@pi the@pi Poisson@q bracket of@✓ @⇡↵ @⇡↵ @✓ norm states,{ while} those of the{ third commutator} have{ the correct{} ✓} sign guaranteeing◆ the positivity✓ of the norm.◆ This arbitrary variables F and G is defined as [38] ↵ (Since we do not requirefact the tells covariance, us that each superscripts fermionic degree and subscripts of freedom arein physical just labels space of should the variables, carry 1@F degree e.g.,@G ✓ of@ freedom=F ✓@↵G.) in phase space, @F @G @F @G @F @G @F @GF, G =( )"F + . (29) One immediately notice that all eigenvalues of the first commutator"F have the negative sign, leading↵ to the negative↵ otherwise negativeF, G = norm states inevitably+( appear,) which+ implies{ that.}N/2 degrees@✓ of@⇡ freedom↵ (28)@⇡↵ are@✓ extra degrees of { } @qi @p @p @qi @✓↵ @⇡ @⇡ @✓↵ ✓ ◆ norm states, while thosefreedom of the corresponding third commutator✓ to Ostrogradsky’si Here, havei " the◆represents correct ghosts.✓ sign the Since Grassmann guaranteeing↵ this is↵ a parity direct the◆ of positivity consequenceF ,i.e.," of=0if the of canonicalF norm.is even, This quantization and " =1if ofF theis odd. As a consequence, F ↵ F F fact tells us that eachPoisson fermionic brackets degree (49 of) freedom of the canonicalthe in Poisson physical variables brackets space✓ of shouldand the canonical⇡ carry↵, any 1 fermionic variables degree of are non-degenerate freedom found to in be phase theories space, always su↵er from "F @F @G @F @G otherwise negative normnegative states norm inevitably states evenF, appear, G if=( wewhich have) both implies↵ bosonic+ thati andN/↵ fermionic2i degrees. ↵ variables of freedom↵ in the arei Lagrangian. extraj degrees(29) of↵ { } @✓ @⇡↵ q @⇡,p↵ @✓= , ✓ , ⇡ = , q ,q = p ,p = ✓ , ✓ = ⇡ , ⇡ =0. (30) freedom corresponding to Ostrogradsky’s ghosts. Since✓ this is a{ directj} consequence◆j { of} canonical { quantization} { i j} of the{ } { ↵ } 6 Here, "F represents the Grassmann parity of F↵,i.e.,"F =0ifF is even, and "F =1ifF is odd.↵ As a consequence,↵ Poisson bracketsthe Poisson (49 brackets) of the of canonical the canonical variables variables✓ areand found⇡↵, to any be fermionic non-degenerate theories✓ , ⇡ = always , su↵er from C.where Degeneratemeans fermionic the weak system equality.{ Since} the Hamiltonian vanishes H = 0, the total Hamiltonian is simply given by negative norm states eveni if we havei both↵ bosonic↵ and fermionici j variables↵ in the Lagrangian.↵ q ,pj = , ✓ , ⇡ = , q ,q = pi,pj ⇡= ✓ , ✓ = ⇡↵, ⇡ =0✓ , ✓. = ⇡↵, ⇡ (30)=0. (31) j { } { } ↵ { } { } { } { } { } { } HT = ↵µ 6 (54) Although the appearance of↵ negative norm↵ states looks generic fori fermionic system as we saw in the previous ✓ , ⇡ WEYL= , FERMION¯↵˙ µ ↵ ¯↵˙ µ ↵ subsection, we have already{ known} that a Dirac field,↵ for example,L = does( not ↵↵˙ su@3µ↵ er from@µ such ↵↵˙ a problem.) Here, we C. Degenerate↵ fermionicwhere systemµ is the Lagrange multiplier2 . The Poisson brackets of the primary constraints are where means✓ , the✓ weak= ⇡↵ equality., ⇡ =0 Since. the Hamiltonian vanishes H = 0, the(31) total Hamiltonian is simply given by review why such a theory⇡ can{ avoid} negative{ } norm states by illustrating= i 1 a ˙ simplest1 + i 1 ˙ model,1 + i 2 ˙ 2 + i 2 ˙ 2 (32) R R ↵ I I R R ↵, I I= i ↵ (55) i i HT = ↵µ { } (54) Although the appearance of negative normL = states( ¯↵˙ µ looks@ ↵ generic@ ¯↵˙ µ for ↵ fermionic) ˙↵ system as we↵ saw↵ in the↵ previous ↵↵˙ µ µ which↵↵˙ meansL = that✓↵✓ all. ↵ are the second-class= R + i constraints,I and no further(52) constraints are not(33) added. Then the time subsection, we have already known thatwhere aµ Dirac↵ is2 the field, Lagrange for example, multiplier does3. The not2 Poisson su↵er brackets from such of the a problem. primary constraints Here, we are 6 6 = i 1The ˙ 1 + Poissoni 1 ˙ 1 + bracketi evolution2 ˙ 2 satisfies+ i of2 ˙ the2 the constraintsfollowing identities, (53) determines the(32) Lagrange multipliers as review why such a theoryObviously, can avoid the Euler-Lagrange negative normR equationsR statesI byI are illustrating first-orderR R aI di simplestI↵erential+ (without model, equations, time derivatives) and this could be regarded as a classical F,↵, G =(= )i" F↵ "G+1 G,˙ F , (55) (34) counterpartwhere of ameans Dirac fermion. the weak↵ The equality. canonical↵ Since↵ momenta the Hamiltonian are{ given vanishes by} ⇡↵ H== 0,i✓↵ the/ 2,↵ total which ↵ Hamiltonian, lead µ to= theiµ is primary↵ simply0 . given by (56) where means =i theR + i weak I equality. Since{ the} Hamiltonian { vanishes}(33)" " H = 0, the total Hamiltonian is simply given by ˙↵ ⇡ {F G1 } ⇡ constraints, ⇡which↵ means that⇡↵ L↵ all= ↵✓↵are✓ the.↵ second-classα = 1, …,3 N constraints,F, G1G2 = and3F, no G1 furtherG2 +( constraints) G(52)1 areF, G not2 , added. Then the time (35) HT = ↵µ ,whereHT =µ ↵isµ the,where Lagrangeµ is multiplier the Lagrange. The{ multiplier Poisson} brackets{ . The} Poisson of the primary brackets constraints{ of the} primary are constraints are The Poisson bracket satisfies the following identities, 2 The dimension of the phase space spanned by the"F "G canonical variables is 2N.SincewehaveN primary constraints, evolution of the constraints (53) determines theF Lagrange1F2,G = multipliersF1 F2,G as+( ) 2 F1,GF2, 6 (36) "F "G+1 the number of physicali { degrees} of{ freedom} is (2N N{ )/2=}N/2 as it should be. F, G =( ) G, F , , = i (34) (54) Obviously, the Euler-Lagrange equations are{ N first-order primary} which constraints di are↵erential easily{ proved}For equations, ↵ confirmation, from⇡↵ + the˙ and↵ definition✓↵ this=0 we, couldnow of the check↵ be Poisson↵ regarded the, absence bracket= asi a of↵ ( classical28 negative). norm states(53) in this system. Since we have(54) second class "F "G ⌘ {↵2 ↵}, µ ={ 2 iµ↵ } 0 . (56) counterpart of a Dirac fermion.where TheF, canonicalmeans G1G2 the= momenta weakF,If the G equality.1 systemG2 are+( contains givenconstraints, Since) 1 thebyG (second1 ⇡F, Hamiltonian↵ we G= evaluate2⇡ class), {i✓↵ constraints/ the2, vanishes} which Dirac , bracketsH lead one=⇡ should0, to the of the all total rather(35) primarycanonical Hamiltonian use the variables, Dirac is bracket simply instead given by of the Poisson which means{ that} all { are} the second-class " " π {are constraints, determined} and by no constraints further constraints are not added. Then the time constraints, The⇡ F dimensionF which,G = meansbracket,F of↵ F the,G that defined phase+( all space by) F↵2 areG spannedF the,G second-class byF , the canonical constraints, variables isand 2N no.Sincewehave(36) further constraintsN primary are constraints, not added. Then the time 1 2 1 2 1 2 ↵ 1 i evolution{ of theHamiltonian: constraints} { ( H53=0,}) determines Total Hamiltonian:{ the Lagrange} H = multipliers µ as 1 ab ˙ (54) the numberevolution of physical of the degrees constraints of freedom (53) determines is (2N A,T N B) the/✓2=↵↵=, Lagrange✓ N/A,D2 B= as iti multipliers A, should↵ , (C be.✓↵,) as⇡ D↵,B= , ↵↵ , , µ ⇡↵=, ⇡ iµD↵= 0. ↵ The, (37) dimension (57) which are easily proved from the definition of the Poisson bracket (28). D a b ⇡ {2 } ⇡4 For confirmation,of the phase2 we space nowi spanned check the by absence the canonical of{ negative}{ variables{ norm} } states is{ 2N in.Sincewehave} this{ system.{} Since } N weprimary have{ second constraints,} class the number of If the system contains (second class)↵ constraints ↵where⇡,↵ one +are should✓ second↵ =03 rather, class˙ use constraints the Dirac and bracket C = instead , of. the Poisson (53) where constraints,µ is thephysicalNo Lagrange secondary we evaluate⌘ degrees multipliera constraints2 the of freedom Diracand. The the brackets is canonical↵ Poisson (2N of ↵ all,N brackets quantization canonical)/µ2==abN/ ofiµ variables, the2 leads{↵ asa primary it0 tob should}. the constraints following be. anti-commutation are relation,(55) bracket, defined by The prescription of the canonical⇡ { quantization} is⇡ simply replacing the Poisson brackets of canonical variables (or For confirmation, we now1 ab check the absence of negative1 norm statesi in thisi system. Since we1 have second class The dimensionA, B ofD the=the phaseA, Dirac B space bracketsA, spanneda (C if secondary) by b the,B constraints canonical, ↵, exist)= variablesˆ iˆ by↵ commutation is 2N.Sincewehaveˆ(37) relations forN bosonsprimary and anti-commutation constraints,(55) relations { Dirac} brackets{ } { ✓↵, ✓} D = {i ↵ },{ ✓↵, ⇡} D✓↵ =, ✓ + ↵ = ,↵ , ⇡↵,✓⇡↵ , ⇡ˆD =+ = ↵ , ↵ , ⇡ˆ↵, ⇡ˆ + = (57) ↵ . (58) the number ofconstraints, physicalfor degrees fermions, we evaluate of{ freedom} the is Dirac (2N bracketsN{)/2= of}N/{ all2 as canonical}2 it should{ variables, be.{ } } 4 2 { } 4 where a are second classwhich constraints means and thatCab all= a,are b . the second-class constraints, and no further constraints are not added. Then the time { ↵ } One should note 1 thatˆ ˆ these anti-commutationPositive norm relations between the canonical variables are consistent with the pri- The prescription of the canonicalFor confirmation,and quantization the canonical we is now quantization simply check replacing the leads absence the to Poisson the(i of~) following negative bracketsA, B anti-commutation norm of canonicalif statesA and inB variables thisare relation, bosons1 system. (or , Since we havei second class evolution of the constraints (53) determines the Lagrange{ } multipliers as ˆ the Dirac brackets if secondaryconstraints, constraints we evaluate exist) by the commutation Dirac bracketsmary relations✓ constraints,↵ of, ✓ all forD canonical bosons=1 i.e.,ˆi and ˆ↵ plugging variables, anti-commutation, ✓↵ in, ⇡ ⇡ ˆ↵D== relationsi✓↵ /↵ 2 for, the⇡↵ second, ⇡ D and= third↵ , expressions in (58)recoversthefirst(55) ˆ A,ˆ B{ 8(}i~) ˆ A, B {ifiA is a} boson (fermion)2 and1{ B is a} fermion4 (boson), (38) ✓↵{,anti-commutation✓ +}=! ↵ , ✓{↵ relations., ⇡ˆ } + = It is ↵ clear, that⇡ˆ↵, ⇡ˆ that + the= negative ↵ . norm state does not(58) appear in this system since the for fermions, ˙>(i ) 1 A,,ˆ Bˆ µ = if iµA and 0B .are fermions, (56) and the canonical{ quantization} <↵ˆ leads~ˆ { ↵ to the }+ following1 2 ↵ anti-commutation4 { } i 4 relation, 1 ˆ ˆ ✓ , ✓ =relationsi , ✓↵⇡,✓✓{ , ⇡+{ is positive}}= definite, ⇡⇡. , ⇡ = , (56) (i~) A, B if A and↵ B areD bosons↵ , ↵ D ↵ ↵ D ↵ One should notewhere that{ the these commutator} anti-commutation and{ anti-commutator{ } relations} between are2 respectively the{ canonical defined} as variables4 A,ˆ Bˆ are= A consistentˆBˆ BˆAˆ and withA,ˆ theBˆ pri-= AˆBˆ+BˆAˆ. The dimension1{ ˆ ˆ} of the phase space spanned:> by the canonical variables is 2Ni.SincewehaveN primary1 constraints,+ A, B (i~) maryA, B constraints,if A is i.e., a boson plugging (fermion) inˆ⇡ ˆ andˆ= B iis✓ˆ a/2 fermion for the (boson)ˆ second, and third(38) expressions{ } in (58 i)recoversthefirst{ ↵ } { } !the8 number{ of} physicalHereafter, degrees we of set freedom~ = 1✓↵↵ in is, ✓ this (2N+ paper.=↵N ↵ )/ One2=, N/ should✓↵2, ⇡ˆ as note it+ should= that real be.↵ variables, ⇡ˆ↵, ⇡ suchˆ + as=q ,pi, ↵ and. ✓ will be promoted (56) and> the1 canonicalˆ ˆ quantization leads to the{ following} anti-commutation{ } relation, 2 { } 4 <(i~) anti-commutationA, B + toif A Hermitianand relations.B are operators, fermions It is clear, and thatIII. then that DEGENERATE the the imaginary negative variables norm THEORIES state⇡↵ doesbecome IN not BOSON-FERMION anti-Hermitian appear in this operators system CO-EXISTENCE since through the canonical SYSTEM For confirmation,{ } ˆ ˆ we now check the absence4 of negative norm states in this system. Since we have second class relationsOne✓↵ should, ✓quantization. + is noteˆ positiveˆ that definite these anti-commutationˆ. ˆˆ ˆ ˆ ˆ ˆi relationsˆ ˆ betweenˆ ˆ ˆ theˆ1 canonical variables are consistent with the pri- where the commutator andconstraints,:> anti-commutator we{ evaluate are respectively} the✓↵, Dirac✓ defined+ = brackets ↵ as , A, ofB✓↵ all, ⇡ˆ= canonical A+B=BA variables,and ↵ ,A, B⇡ˆ↵+,=⇡ˆ A+B+=BA. ↵ . (57) Hereafter, we set = 1 in this paper.mary One should constraints,{ note that} i.e., realAs plugging variables seen{ { in} such the in⇡ ˆ previous} as q =i,p ,2 section,i and✓ˆ /{2✓↵ for thewill{ } the unique be promoted} second solution 4 and to third avoid negative expressions norm in states (56)recoversthefirst in fermionic system is to have a ~ ↵ i ↵1 i to Hermitian operators, and then theanti-commutation imaginary variables ⇡ relations.↵ subecome cient anti-Hermitian It number is clear of constraints that operators that eliminating the through negative canonicalN/2 norm ghostly state degrees does of freedom not appear in phase in this space. system In this since section, the we provide a One should note thatIII.2 theseWhen DEGENERATE✓ the anti-commutation↵, system✓ D = containsi THEORIES↵ first, relations class✓↵ constraints, ⇡ IN betweenD BOSON-FERMION= (sometimes the ↵ canonical, in addition⇡↵, ⇡ variables CO-EXISTENCE to secondD = class are ↵ constraints), consistent, SYSTEM we with add the gauge pri- fixing(57) conditions to { } general approach{ to4 constructing} 2 degenerate{ Lagrangian} 4 of boson-fermion co-existence system, whose physical degrees quantization. mary constraints,relations i.e.,the plugging✓ˆ set, ✓ ofˆ the in constraints,is⇡ ˆ positive= whichi definite✓ˆ / e↵2ectively for. the leads second to a system and only third with expressions second class constraints. in (57)recoversthefirst ↵ + of↵ freedom is↵ n+N/2 with n being the number of bosonic variables and N that of fermionic ones. We focus on the most anti-commutationand the canonical relations. quantization{ It} is leads clear to that the that following the negative anti-commutation norm state relation, does not appear in this system since the As seen in the previous section,general the Lagrangian unique solution containing to avoid up negative to first derivatives norm states of bosonsin fermionic and fermions system is (23 to). have In the a first part of this section, we ˆ ˆ 4 i 1 2 When the system containsrelations first classsu constraints ✓cient↵, ✓ number+ (sometimesis positive of constraints inˆ addition definiteˆ derive to eliminating second. (su class cient) constraints),N/ˆ 2 conditions,ghostly we degrees add which gauge of fixing yieldfreedom conditionsN constraints in phase to space. to eliminate In this section, fermionic we Ostrogradsky’s provide a ghosts, in Hamiltonian { } III.✓↵, ✓ DEGENERATE + = ↵ , ✓↵, THEORIES⇡ˆ + = ↵ IN, BOSON-FERMION⇡ˆ↵, ⇡ˆ + = ↵ . CO-EXISTENCE SYSTEM(58) the set of the constraints, which e↵ectivelygeneral leads approach to a system to constructing{ only with} secondformulation. degenerate class constraints.{ In Lagrangian the} latter of2 part, boson-fermion we show{ that co-existence} the condition4 system, imposed whose physical in Hamiltonian degrees formulation is equivalent to One shouldof freedom note is thatn+N/ these2 with anti-commutationn beingrequiring the number that relations the of equations bosonic between variables of motion the and canonical ofN fermionsthat variables of fermionic are first-order are ones. consistent di We↵erential focus with on equations. the the most pri- generalIII. Lagrangian DEGENERATE containing THEORIES up to first derivatives IN BOSON-FERMION of bosons and fermions CO-EXISTENCE (23). In the first SYSTEM part of this section, we mary constraints,As seen i.e., inplugging the previous in⇡ ˆ section,= i✓ˆ / the2 for unique the second solution and to avoid third negative expressions norm in states(58)recoversthefirst in fermionic system is to have a derive (su cient) conditions, which↵ yield N↵constraints to eliminate fermionic Ostrogradsky’s ghosts, in Hamiltonian anti-commutationsu cient relations. number It of is constraints clear that thateliminating the negativeN/2 ghostly norm state degrees does of not freedom appear in phase in this space. system In since this section, the we provide a formulation. In the latter part, we show that the condition imposed inA. Hamiltonian Degeneracy formulation conditions is equivalent to As seen inˆ thegeneralˆ previous approach section, to the constructing unique4 solution degenerate to avoid Lagrangian negative norm of boson-fermion states in fermionic co-existence system system, is to have whose a physical degrees relationsrequiring✓↵, ✓ that+ is the positive equations definite of motion. of fermions are first-order di↵erential equations. su cient number{ of freedom} of constraints is n+N/ eliminating2 with nN/being2 ghostly the number degrees of of bosonic freedom variables in phase space. and N Inthat this of section, fermionic we provide ones. We a focus on the most i ↵ i i ↵ ↵ general approachgeneral to constructing Lagrangian degenerate containingIf the time Lagrangian up derivatives to first of derivatives boson-fermion of q and ✓ ofare bosons co-existence expressed and fermions in system, terms ofwhose (23 the). canonical physical In the first degrees variables part of (q this,p , ✓ section,, ⇡ ), we we do not have of freedom is n+N/2 with n being theany number primary of bosonic constraints. variables Then, and weN lookthat for of the fermionic conditions ones. where We the focus time on derivatives the most of qi and ✓↵ are not written in III.derive DEGENERATE (su cient) conditions, THEORIES whichA. IN yield Degeneracy BOSON-FERMIONN constraints conditions to eliminate CO-EXISTENCE fermionic Ostrogradsky’s SYSTEM ghosts, in Hamiltonian terms of the canonical variables. general Lagrangianformulation. containing In up the to latter first derivatives part, we of show bosons that and the fermions condition (23). imposed In the first in Hamiltonian part of this section, formulation we is equivalent to derive (su cient) conditions, which yieldLetN usconstraints consider the to infinitesimal eliminate fermionic variation Ostrogradsky’s of the canonical ghosts, momenta in Hamiltonianwith respect to all variables, As seenIf in therequiring the time previous derivatives that section, the of equationsqi theand unique✓↵ are of motion expressed solution of to in fermions avoid terms negative of are the first-order canonical norm states variables di↵erential in fermionic (qi,p equations.i, ✓↵, system⇡↵), we is do to not have have a formulation. In the latter part, we show that the condition imposed in Hamiltonianj formulation is equivalentj to su cientany number primary of constraints. constraints Then, eliminating we lookN/ for2 ghostly the conditions degrees where of freedom thepi time in phase derivatives q˙ space. of InLqqi˙i thisandqj section,✓L↵q˙arei✓ not we q written provide in a requiring that the equations of motion of fermions are first-order di↵erential= equations.K ˙ + , (59) terms of the canonical variables. ⇡↵ ✓ L✓˙↵qj L✓˙↵✓ ✓ general approach to constructing degenerate Lagrangian of boson-fermion✓ ◆ co-existence✓ ◆ system,✓ whose physical◆✓ ◆ degrees of freedomLet is usn+ considerN/2 with then infinitesimalbeing the number variation of bosonic of the canonical variablesA. Degeneracy momenta and N that with conditions of respect fermionic to all ones. variables, We focus on the most j j j pi Lq˙iq˙j L ij˙ q˙ Lq˙iqj Lq˙i✓ q general Lagrangian containing up to first p derivativesA. Degeneracy ofq˙ bosons conditionsL andq˙iqj fermionsL i (23 q˙).q✓ In the first part of this section, we i = K + = q˙ ✓ , ˙ + (59) , (60) derive (su cient) conditions, which yield Ni constraints↵ ˙ to eliminate ⇡↵ fermionicL ˙↵ ˙j L Ostrogradsky’s˙↵ ˙ ✓ ghosts,L✓˙↵qj inL Hamiltoniani✓˙↵✓i ↵ ✓↵ If the time derivatives of ⇡q↵ and ✓ are ✓ expressed✓ L✓˙◆↵qj in✓ termsL✓˙✓↵✓✓ of the✓ ✓✓ canonical◆✓ ◆ variables✓ (q ,p , ✓◆✓, ⇡ ),◆ we do not have ✓ ◆ ✓ ◆ ✓ ◆✓ ◆ i ↵ formulation.If the time derivativesany In the primary latter of q constraints.i part,and ✓ we↵ are show Then, expressed that we the look in terms condition for the of the conditions imposed canonical in where variables Hamiltonian the ( timeqi,p formulationi, derivatives✓↵, ⇡↵), we is of do equivalentq notand have✓ toare not written in requiring that the equations of motion of fermions are first-orderj di↵erential equations.i j ↵ any primary constraints.terms of the Then, canonical we p look variables. forL thei j conditionsL i ˙ where q˙ the timeL i derivativesj L i of qq and ✓ are not written in i = q˙ q˙ q˙ ✓ + q˙ q q˙ ✓ , (60) Let us consider the3 infinitesimal variation of˙ the canonical momenta with ↵ respect to all variables, ↵ terms of the canonical variables. ⇡↵ The orderL✓˙↵✓˙ ofj theL✓˙ constraints↵✓˙ ✓ ↵ and theL Lagrange✓˙↵qj L✓˙ multipliers↵✓ ✓µ in the total Hamiltonian should be like ↵µ in the left derivative ✓ ◆notation.✓ We also note◆✓ that µ↵ are◆ Grassmann✓ odd numbers.◆✓ ◆ Let us consider the infinitesimal variation of the canonical momentaj with respect to all variables,j 4 i j i Here, weA. adopt Degeneracy✓ˆ ↵pias independent conditions q variables˙ sinceLq˙ theyq areL Hermitianq˙ ✓ operators.q If one would like to adopt⇡ ˆ↵ instead, they should be j = K ˙ + j , (57) pimultiplied by iq˙to ⇡ be↵ Hermitian.Lq˙iqj ✓Lq˙i✓ L ✓q˙↵qj L✓˙↵✓ ✓ 3 i ↵ = K ✓˙ +◆ ✓↵ ◆ ✓ , ◆✓i i ↵◆ ↵↵ (58) If the timeThe order derivatives of the constraints of q and ⇡↵✓and↵ are the expressed Lagrange ✓ multipliers in termsL✓˙↵µq ofj in theL the✓˙↵ canonical✓ total Hamiltonian ✓ variables should (q be,p like, ✓ ,↵⇡µ ),in we the do left not derivative have notation. We also note that µ✓↵ are◆ Grassmann✓ odd numbers.◆ ✓ ◆✓ ◆ i ↵ any primary4 constraints.ˆ Then, we look for the conditions where the timej derivatives of q and ✓ arej not written in Here, we adopt ✓↵ as independent variables pi since theyL areq˙iq˙ Hermitianj Lq˙i✓˙ operators. q˙ If one wouldLq˙i likeqj to adoptLq˙i✓ ⇡ ˆ↵ instead, q they should be terms ofmultiplied the canonical by i to variables. be Hermitian. = j ˙ + j , (58) pi Lq˙iq˙j ⇡↵Lq˙i✓˙ L✓˙ ↵q✓˙˙j L✓˙↵✓˙ Lq˙iqj ✓Lq˙i✓ L q✓˙↵qj L✓˙↵✓ ✓ Let us consider the infinitesimal= variation✓ of◆ the✓ canonical˙ + momenta◆✓ with◆ respect✓ to all, variables,◆✓ ◆ (59) ⇡↵ L✓˙↵✓˙j L✓˙↵✓˙ ✓ L✓˙↵qj L✓˙↵✓ ✓ ✓ ◆ ✓ ◆✓j ◆ ✓ ◆✓j ◆ where K is the kinetic p matrix,i q˙ Lq˙iqj Lq˙i✓ q = K + , (59) ˙ L ↵ j L ↵ ⇡↵ ✓ ✓˙ q ✓˙ ✓A ✓ ✓ ◆ ✓ ◆ ✓ K = ij◆✓Bi ◆ , (59) ↵j D↵ 3 The order of the constraints and the Lagrange multipliers µ↵ inj the total Hamiltonian should bej like µ↵ in the left derivative ↵ L i j L i q˙ L✓i jC L i ◆ ↵ ↵ pi q˙ q˙ q˙ ✓˙ q˙ q q˙ ✓ q notation. We also note that µ are Grassmann= odd numbers. ˙ + , (60) 4 ˆ ⇡↵ L✓˙↵✓˙j L✓˙↵✓˙ ✓ L✓˙↵qj L✓˙↵✓ ✓ Here, we adopt ✓↵ as independent✓ variables◆ ✓ since they are Hermitian◆✓ operators.◆ ✓ If one would like◆✓ to adopt◆ ⇡ ˆ↵ instead, they should be multiplied by i to be Hermitian. 3 ↵ ↵ The order of the constraints ↵ and the Lagrange multipliers µ in the total Hamiltonian should be like ↵µ in the left derivative notation. We also note that µ↵ are Grassmann odd numbers. 3 4 ˆ ↵ ↵ The order of theHere, constraints we adopt ↵✓↵andas independent the Lagrange variables multipliers sinceµ theyin the are total Hermitian Hamiltonian operators. should If be one like would ↵µ likein the to adopt left derivative⇡ ˆ↵ instead, they should be notation. We alsomultiplied note that byµi↵toare be Grassmann Hermitian. odd numbers. 4 Here, we adopt ✓ˆ↵ as independent variables since they are Hermitian operators. If one would like to adopt⇡ ˆ↵ instead, they should be multiplied by i to be Hermitian. AVOIDANCE OF NEGATIVE NORM STATES
A fermionic variable should carry 1 dof in phase space (1/2 dof in physical sp.) to avoid the fermionic ghost
In the system with m bosons and N fermions, we need N constraints for eliminating N dofs of fermions in phase sp.
In addition, another condition, the constraint matrix is invertible, should be satisfied in order that Hamiltonian analysis becomes closed. (This condition is actually important since it guarantees definite time evolution of the system. Here we will not discuss this point.) 6
Obviously, the Euler-Lagrange equations are first-order di↵erential equations, and this model could be regarded as 6 a classical counterpart of a Dirac fermion. The canonical momenta are given by ⇡↵ = i✓↵/2, which lead to the primary constraints, Obviously, the Euler-Lagrange equations are first-order di↵erential equations, and this model could be regarded as i a classical counterpart of a Dirac fermion. The ↵ canonical⇡↵ + momenta✓↵ =0. are given by ⇡↵ = i✓↵/2, which lead to the(40) primary constraints, ⌘ 2 ↵ ↵ Since the Hamiltonian vanishes, H = 0, the total Hamiltoniani is simply given by HT = ↵µ ,whereµ are the 4 Lagrange multipliers . The Poisson brackets between ↵ ⇡↵ the+ primary✓↵ =0. constraints are , = i , which means(40) ⌘ 2 { ↵ } ↵ that all ↵ are second class constraints, and no further constraints are added. Then the↵ time evolution↵ of the Since the Hamiltonian vanishes, H = 0, the total Hamiltonian˙ is simply given by HT = ↵µ ,whereµ are the constraints (40) determines4 the Lagrange multipliers as ↵ = ↵, µ = iµ↵ 0, where means the weak equality.Lagrange The multipliers dimension. of The the Poisson phase space brackets spanned between by the primarycanonical{ constraints variables} is are 2N .Sincewehave⇡↵, = i ⇡↵ N, which(second means class) that all are second class constraints, and no further constraints are added. Then{ the} time evolution of the primary constraints,↵ the number of physical degrees of freedom is (2N N)/2=N/2 as it should be. constraints (40) determines the Lagrange multipliers as ˙ = , µ = iµ 0, where means the weak For confirmation, we now check the absence of negative↵ norm↵ states in this system.↵ Since we have second class equality. The dimension of the phase space spanned by the canonical{ variables} is 2N.Sincewehave⇡ ⇡ N (second class) constraints, we evaluate the Dirac brackets between all canonical variables, primary constraints, the number of physical degrees of freedom is (2N N)/2=N/2 as it should be. For confirmation, we now check the absence of negative norm1 states in this system.i Since we have second class ✓ , ✓ = i , ✓ , ⇡ = , ⇡ , ⇡ = , (41) constraints, we evaluate{ the↵ Dirac }D brackets ↵ between{ ↵ all } canonicalD 2 ↵ variables,{ ↵ }D 4 ↵ and the canonical quantization leads to the following anti-commutation1 relations, i ✓↵, ✓ D = i ↵ , ✓↵, ⇡ D = ↵ , ⇡↵, ⇡ D = ↵ , (41) { } { } i 2 { } 41 ✓ˆ , ✓ˆ = , ✓ˆ , ⇡ˆ = , ⇡ˆ , ⇡ˆ = . (42) and the canonical quantization{ ↵ leads}+ to↵ the following{ ↵ } anti-commutation+ 2 ↵ { relations,↵ }+ 4 ↵ One should note that these anti-commutation relations betweeni the canonical variables1 are consistent with the primary ✓ˆ↵, ✓ˆ + = ↵ , ✓ˆ↵, ⇡ˆ + = ↵ , ⇡ˆ↵, ⇡ˆ + = ↵ . (42) constraints, i.e., plugging⇡ ˆ{ = }i✓ˆ /2 into the{ second} and 2 the third{ expressions} in4 (42) recovers the first one. It is ↵ ↵ clearOne that should negative note that norm these states anti-commutation do not appear relations in this system between since the canonical the relations variables✓ˆ , are✓ˆ consistentare positive with definite. the primary5 { ↵ }+ constraints, i.e., plugging⇡ ˆ↵ = i✓ˆ↵/2 into the second and the third expressions in (42) recovers the first one. It is 5 clear that negative norm states do not appear in this system since the relations ✓ˆ↵, ✓ˆ + are positive definite. IV. DEGENERATE THEORIES IN BOSON-FERMION CO-EXISTENCE{ } SYSTEM
As seen in theIV. previous DEGENERATE section, the THEORIES unique solution IN BOSON-FERMION to avoid negative norm CO-EXISTENCE states in N-fermionic SYSTEM system is to have asu cient number of constraints eliminating N/2 ghostly degrees of freedom. In this section, we provide a general approachAs seen to in constructing the previous degenerate section, the Lagrangian unique solution of boson-fermion to avoid negative co-existence norm states system, in N-fermionic whose physical system is degrees to have of freedomasu cient is n+ numberN/2 with of constraintsn being the eliminating number of bosonicN/2 ghostly variables. degrees We of focus freedom. on the In most this section, general we Lagrangian provide a containing general upapproach to first time to constructing derivatives degenerate of bosons and Lagrangian fermions of (21 boson-fermion). In the former co-existence part of thissystem, section, whose we physical derive a degrees (su cient) of 7 conditionfreedom which is n+N/ yields2 withN nconstraintsbeing the number to eliminate of bosonic fermionic variables. ghosts We in focus Hamiltonian on the most formulation. general Lagrangian In the latter containing part, we showup to that first the time condition, derivatives imposed of bosons in Hamiltonian and fermionswhere K formulation, (is21 the). kineticIn the is equivalentformer matrix, part to of requiring this section, that we the derive equations a (su ofcient) motion ofcondition fermions which are first-order yields N diconstraints↵erential to equations. eliminate We fermionic also introduce ghosts in another Hamiltonian condition, formulation. which In we the call latter the uniqueness part, we condition,show that to the have condition, no more imposed constraints in Hamiltonian in Hamiltonian formulation, formulation is equivalent and show to that requiring it is responsible that the equationsAij for thei unique of motion time of fermions are first-order di↵erential equations. We also introduce another condition, whichK = we call theB uniqueness, (61) evolution of the system in Lagrangian formulation. ↵j D↵ condition, to have no more constraints in Hamiltonian formulation and show that it is responsible✓ C for the unique◆ time evolution of the system in Lagrangian formulation.which components are defined by
A. Degeneracy condition @pi @pi i j Aij = j = Lq˙ q˙ , i = = Lq˙i✓˙ , A. Degeneracy condition @q˙ B @✓˙ i ↵ i i ↵ ↵ If the time derivatives of q and ✓COEXISTINGare expressed in terms of the canonical SYSTEM@⇡ variables↵ (q ,p , ✓ , ⇡ @⇡),↵ we do not have ↵j = j = L✓˙↵q˙j ,D↵ = i = L✓˙↵↵✓˙ = L✓˙ ✓˙↵ . (62) anyIf primary the time constraints. derivatives Therefore, of qi and ✓↵ weare need expressed to lookn bosons in for terms the condition ofand the N canonical wherefermionsC variables the@q˙ time ( derivativesqi,pi, ✓↵, ⇡↵ of),@q we✓˙ and do not✓ are have7 not written in terms of the canonical variables. Let us then consider the infinitesimal variations of the canonicali ↵ momenta any primary constraints. Therefore, we needHere to look for the condition where the time derivatives of q and ✓ are not whosewithwritten respect components in terms to all ofare variables, the defined canonical by variables. Let us then consider the infinitesimal variations of the canonical momenta with respect to all variables,Variations of momentaj j @ @L @p i pi q˙ L@q˙piiqj Lq˙i✓ q LXY = . (63) =i Kj + , (43) Aij = j = Lq˙ q˙ , ˙ j i = = Lq˙i✓˙ , j @Y @X @q˙ ⇡ p↵i ✓q˙ Lq˙i˙↵q jj LL˙q˙↵i✓ ✓q = K B + @✓✓ q ✓ ✓ , ＊ ⇣ ⌘ (43) ✓ ◆ ✓ ˙ ◆ ✓ ◆✓ ◆ ― i i ↵ ⇡↵ It should ✓ be noticeL✓˙↵ thatqj L all✓˙↵ sub-matrices✓ ✓ depend on (q , q˙ , ✓ , ✓˙ ) in general. By construction, the symmetric matrix where K is the kinetic matrix, @⇡✓↵ ◆ ✓ ◆ ✓ @⇡↵ ◆✓ ◆ ↵j = = L ˙↵ j ,D↵ = = L ˙↵ ˙ = L ˙ ˙↵ . (45) 5 j ✓ Aq˙ ij and the anti-symmetric˙ ✓ matrix✓ D↵ ✓ are✓ Grassmann-even and Hermitian ,7 and i and ↵j are Grassmann-odd where K is the kinetic matrix,C @q˙ T@✓ B C and related asAij =i . In the present paper, we assume that the purely bosonic submatrix of the kinetic matrix Here we have introduced an abbreviatedKinetic matrix notation,K = C B B , (44) Aij is non-degenerate,A↵ijj Di↵ i.e., invertible. This assumption is equivalent to requiring whose components are defined by K = ✓ C B ◆ , (44) @2L ↵j D@↵ @L @pi ✓ C ◆@pi (0) (0) LXY = = . det A =0, where A = Aij(46)˙ , (64) Aij = = Lq˙i@q˙jY,@X i@ Y= @X = L i ˙ , ij ij ✓,✓=0 @q˙j B @✓˙ q˙ ✓ 6 | 4 i ↵ i ↵⇣˙ ⌘ ↵ It shouldThe order be noticed of the constraints that all the ↵ and sub-matrices the Lagrangeas depend discussed multipliers on ( inqµ, Sec.q˙in, ✓ the II., ✓ A.total) in Hamiltonian general. By should construction, be like ↵µAijinis the a Hermitian left derivative 7 4 ↵ @⇡↵ ↵ @⇡↵ ↵ notation.The order We of also the note constraints that µ are↵ and Grassmann-odd the↵j = Lagrange= numbers. multipliersL ˙↵ j ,Dµ in↵ the= total Hamiltonian= L ˙↵ ˙ ij = shouldL ˙ be ˙↵ like. ↵µ in the left derivative6 (45) i symmetric matrix, while D↵ is an anti-Hermitianj Multiplied✓ anti-symmetricq˙ by the inverse matrix, of Aij both✓, ✓A of, the which first✓ ✓ are line Grassmann-even of (59) can be solvedi for q˙ as 5 ↵ C @q˙ @✓˙ Here,notation. we adopt We also✓ˆ↵ as note independent that µ are variables Grassmann-odd sinceT they numbers. are Hermitian operators. If one would like to adopt⇡ ˆ↵ instead, they shouldB be and5 ↵j are Grassmann-oddˆ and related as = . In the present paper, we assume that the bosonic submatrix of multipliedHere, we by adopti to✓ be↵ as Hermitian. independent variables since they are Hermitian operators. If onei wouldij like toij adopt ⇡ ˆ↵↵instead,ij they shouldk beij ↵ C Here we have introduced an abbreviatedC notation, B q˙ = A p A ✓˙ A L j k q + A L j ↵ ✓ , (65) the kineticmultiplied matrix by i toA beij Hermitian.is non-degenerate,Assumption: i.e., No invertible. asdegeneracy discussed This in assumptionbosonic in Sec. sector II. is equivalent A. j to requiringBj↵ q˙ q q˙ ✓ @2L @ @L ij i (0) MultipliedL = (0) by= the inverse. of Aij, A , the first line(46) of (57) can be solved for q˙ as det A =0, XYwhere A = A ˙ , (47) ij 6 @Y @Xij @Yi ij@|✓X,✓=0 1 1 ˙ 1 1 q˙ ⇣=(L⌘q˙q˙) p +(Lq˙q˙) Lq˙✓˙ ✓ (Lq˙q˙) Lqq˙ q +(Lq˙q˙) Lq˙✓ ✓ , (66) as discussed in Sec. II. A. i i ↵ ˙ i ij ij ↵ ij k ij ↵ It should be noticed that all the sub-matricesThe first depend line on of (q ＊, q ˙can, ✓ ,be✓ )solved in general. for By construction, Aij is a Hermitian˙ j k j ↵ ij i q˙ = A pj A j↵6 ✓ A Lq˙ q q + A Lq˙ ✓ ✓ , (62) Multipliedsymmetric by the matrix, inverse of whileAijD, A↵ ,is the an firstanti-Hermitianand line plugging of (43 in) anti-symmetric this can for be solvedthe second matrix, for lineq˙ bothas of (59 of) which gives are Grassmann-even B i T B and ↵j are Grassmann-oddi andij relatedij as ˙=↵ .ij In the presentk ij paper, we↵ assume that the bosonic submatrix of C q˙ = A pj A andj↵C ✓ plugging BA Lq˙ijj q thisk q +˙ into A L theq˙j ✓↵ ✓ second,ij line ofij (57) gives(48) k ij the kinetic matrix Aij is non-degenerate, B i.e.,(D invertible.↵ ↵iA Thisj assumption) ✓ = ⇡↵ is equivalent↵iA pj + to requiring↵iA Lq˙j qk L✓˙↵qk q ↵iA Lq˙j ✓ + L✓˙↵✓ ✓ . (67) and plugging this into the second line of (43) gives C B C C C det A(0) =0, where A(0) = A , ⇣ ⌘ (47) ij ij ij ✓,✓˙=0 ij ˙ ij ij k ij ˙ ij 6 ij (kD| ↵ ij ↵iA j ) ✓ = ⇡↵ ↵iA pj + ↵iA L j k L ˙↵ k q (D↵ ↵iA j ) ✓ = ⇡↵ ↵iA pj + ↵iA Lq˙j qk1 L✓˙↵qk q ↵iA 1Lq˙j ✓ + L✓˙↵✓ ✓1 . (49) 1 q˙ q ✓ q as discussed C inB Sec. II. A. C (L ˙ ˙ +CL ˙ (Lq˙q˙) L ˙) ✓˙ = ⇡ L ˙ C(Lq ˙q˙) C p + LB˙ (Lq˙q˙) Lqq˙ L ˙ qC L ˙ (Lq˙q˙) Lq˙✓ C+ L ˙ ✓ . (68) ij ✓✓ ✓q˙ q˙✓ ✓q˙ i ✓q˙ ✓q ✓q˙ ✓✓ Multiplied by the inverse of Aij, A , the first⇣ line of (43) can˙↵ be⌘ solved for q˙ as ij ⇣ ⌘ Now we would like to consider the situation such that the velocities ✓ cannot be expressed in terms⇣ of other canonical⌘ ↵iA⇣ Lq˙j ✓ + L ˙↵ ⌘ ✓ (63) i ij ↵ ij ˙↵ ij k ij ↵ ✓ ✓ variables, that is, the coe cient matrix q˙ = ofA ✓˙ pjdoesA notj↵ have ✓ theA L inverse,j k q + equivalentA Lq˙j ✓↵ ✓ to imposing, the degeneracy C(48) B q˙ q 1 condition, L✓˙✓˙ + L✓˙q˙(Lq˙q˙) Lq˙✓˙ = 0 (69) and plugging this into the second line of (43Now) gives we would like to consider the situation such that the velocities ✓˙↵ cannot be expressed in terms of other (0) (0) ij ˙ ij ij k ij ˙↵ ↵ (D A ) det✓ =D ⇡↵ =0Now, weA would wherep + likeD toA↵ considerL=j Dk ↵ theL✓,✓↵˙ situation=0k . q such thatA L thej velocities+ L ↵ ✓ ✓cannot(50). (49) be expressed˙ in terms of other canonical ↵ ↵i j ↵ canonical↵i j variables,↵i q˙ q | that✓˙ q is, the↵i coeq˙ ✓ cient✓˙ ✓ matrix of ✓ does not have the inverse, equivalent to imposing the C B variables, C that is,C the coe cient matrix of ✓˙↵ Cdoes not have the inverse. According to the property of the Grassmann We consider the cases of N = 1, N = 2, and N 3 separately.⇣ ⌘ Now we would like to consider the situationdegeneracy such that the condition, velocities ✓˙↵ cannot be expressed in terms of(0) other canonical(0) valued matrix (21), this condition is equivalent to impose det D↵ = 0, where D↵ = D↵ ✓,✓˙=0 . N =1variables,case: that is, the coe cient matrix of ✓˙↵ does not have the inverse, equivalent to imposing the degeneracy | • (0) (0) Let uscondition, start with a single fermionic variable,WeN consider= 1. In the this cases case, of bothN =1D,Nand= 2,A and1 Nare3 zero separately. due to the Grass- det D↵ =0, where D↵ = D↵ ✓,✓˙=0 . (64) mann property, and the degeneracy condition(0) of the kinetic matrix(0) is automaticallyC B satisfied. More importantly, | det D =0, where D = D ˙ . (50) 1 ↵ N =1case: ↵ ↵ |✓,✓=0 the coe cient matrix D A always vanishes,• and we have a primary constraint 1 = ⇡1 f1(q, p, ✓), which We consider the cases ofC N =B 1, N = 2,We and N consider3 separately. the cases of N = 1, N = 2, and N 3 separately. 1 will remove the fermionic ghost properly. Let us start with a single fermionic variable, N = 1. In this case, both D and A are zero due to the N =1case: Grassmann property, and then, the degeneracy of the kinetic matrix is automaticallyC satisfied.B More importantly, N =2case• : N =1case: 1 the coe cient matrix D A always vanishes,1 and we have a primary constraint = ⇡ f (q, p, ✓), which • Let us start with a single fermionic variable,• N = 1. In this case, both D and A are zero due to the Grass- 1 1 1 When N = 2, the matrix D is no longer zero,removes which, Ostrogradsky’s in general, has ghost theC following properly.B form,C B 1 mann property, and the degeneracy conditionLet of us the start kinetic with matrix a is single automatically fermionic satisfied. variable, More importantly,N = 1. In this case, both D and A are zero due to the Grass- 1 the coe cient matrix D A always vanishes,0 andD12 we have a primary constraint 1 = ⇡1 f1(q, p, ✓), which C B will remove the fermionic C ghostB properly.D = mann property,. and the degeneracy condition (51) of the kinetic matrix is automatically satisfied. More importantly, D12 0 1 the✓ coe cient◆ matrix D A always vanishes, and we have a primary constraint 1 = ⇡1 f1(q, p, ✓), which 5 ˙ ˙ N =2case: Since ✓↵ is real and Lagrangian should be Hermite, C✓↵✓ is alwaysB accompanied by the imaginary unit i in Lagrangian, which can be Let us explicitly• write the Lagrangian for this case.will remove the fermionic ghost properly. When N = 2, the matrix D is noeasily longer checked zero, by which, referring in ( general,11). Then, has the the anti-symmetric following form, matrix D↵ is a pure imaginary matrix, guaranteeing its Hermitian property. 1 0 D ND=2= case: 12 i✓.1✓2 (51) • D12 00 ˙ 1 ✓ ◆i✓1✓1 When N = 2, the matrix˙ D is no longer zero, which, in general, has the following form, Let us explicitly write the Lagrangiani i for this case. B i✓2✓2 C L = GI (q , q˙ )xI , where x = B ˙ C , (52) B i✓1✓2 C B C1 B i✓ ✓˙ C 0 D12 B 2 1 i✓C1✓2 D = . (65) B i✓˙ 0✓˙ C ˙ 1 D 0 B 1 2 i✓C1✓1 12 B✓ ✓ ✓˙ ✓˙i✓C✓˙ ✓ ◆ i i B 1 2B1 2C2 2 C L = GI (q , q˙ )xI , where x = B ˙ C , (52) @ B i✓A1✓2 C Let us explicitly writeB theC Lagrangian for this case. B i✓ ✓˙ C B 2 1 C 6 B i✓˙ ✓˙ C The product of two real fermionic variables is not real but imaginary because of the GrassmannB 1 2 propertyC (11)andshouldbealways 1 accompanied by the imaginary unit i in (Grassmann-even real) Lagrangian. Then, the matrixB✓1✓D2✓↵ ˙1✓˙2isC a pure imaginary matrix, which 1 B ˙ ˙ C i✓1✓2 is consistent with its anti-Hermitian property. For instance, when Lagrangian includes 2 ✓@↵✓ ✓↵✓ , DA↵ includes ✓↵✓ ,whichisanti- T T Hermitian as (✓↵✓ ) =(✓ ✓ ) =(✓ ✓↵) = ✓↵✓ = ✓ ✓↵.Wenotethatthetransposedefinedbyreplacingsubscriptsofthe 0 ˙ 1 † ⇤ ↵⇤ i✓1✓1 elements of6 a matrix implies a property (EF)T =( ) E F F T ET . The product of two real fermionic variables | is|| not| real but imaginary because of the Grassmann property (11)andshouldbealways i✓ ✓˙ accompanied by the imaginary unit i in (Grassmann-even real) Lagrangian. Then, the matrix D is a pure imaginaryi matrix,i which B 2 2 C ↵ L = GI (q , q˙ )xI , where x = B C , (66) is consistent with its anti-Hermitian property. For instance, when Lagrangian includes 1 ✓ ✓ ✓˙ ✓˙ , D includes ✓ ✓ ,whichisanti- ˙ 2 ↵ ↵ ↵ ↵ B i✓1✓2 C T T Hermitian as (✓↵✓ ) =(✓ ✓ ) =(✓ ✓↵) = ✓↵✓ = ✓ ✓↵.Wenotethatthetransposedefinedbyreplacingsubscriptsofthe B C † ⇤ ↵⇤ ˙ T E F T T B i✓2✓1 C elements of a matrix implies a property (EF) =( )| || |F E . B C B i✓˙ ✓˙ C B 1 2 C B ˙ ˙ C B✓1✓2✓1✓2C @ A and G (I =1, 2, , 8) are functions depending on qi andq ˙i only. Therefore, we obtain I ···
D12 = iG7 + G8✓1✓2 . (67)
7 Applying the degeneracy condition, we have G7 = 0. The momenta are now
@xI @xI ⇡1 = GI = iG3✓1 iG6✓2 + G8✓1✓2✓˙2 , ⇡2 = GI = iG5✓1 iG4✓2 G8✓1✓2✓˙1 . (68) @✓˙1 @✓˙2
The explicit form of Aij is
@2G A = A(0) + AI x , where AI = I ,A(0) = A1 . (69) ij ij ij I ij @q˙j@q˙i ij ij XI>1
7 When we do not require the degeneracy condition, iG7✓˙2 and iG7✓˙1,doesappearin(68), which makes them solvable for ✓˙2 and ✓˙1, and extra degrees of freedom, corresponding to the fermionic ghost, remain in the fermionic sector. 2
Inverse matrix: whether a matrix can be invertible or not plays an important role in degenerate theories as we will see in Sec. IV. The condition for the existence of an inverse matrix of a Grassmann valued square matrix is obtained as follows. We introduce two square matrices that are functions of the variables qi and ✓↵, which can be in general written as
A(qi, ✓↵)=A (qi)+A (qi)✓↵ + A (qi)✓↵✓ + , (12) 0 ↵ ↵ ··· B(qi, ✓↵)=B (qi)+B (qi)✓↵ + B (qi)✓↵✓ + , (13) 0 ↵ ↵ ··· i where A0,A↵,...,B0,B↵,... are matrices depending on q . The condition that B is the inverse of A is given by AB = I,whereI is the identity matrix, leads to the following equations, 8
A(0)0B0 =0, (14) The inverse is easily obtained since we have assumed Aij has the inverse and xI (I>1) have at least one of ✓1 and ✓2 except for I = 7, which does not contributeA0B↵ + becauseA↵B0 of=0 the, degeneracy condition. (15) 1 8 A0B↵ + (A↵B jk A B↵j )+jmA↵ (0) B0 =0I , lk(0) (16) 2 (0) A = l A AmlxI A (70) The inverse is easily obtained since we have assumed Aij has the inverse and xI (I>1) have! at least one of ✓1 XI>1 . and ✓2 except for I = 7, which does not contribute because of the degeneracy condition.. Each component of the coe cient in the left hand side of Eq. (63) is calculated straightforwardly and we have Ajk = j Ajm(0) ij AI x Alk(0) ij (70) 1 1 1 Dl 11 1 1iA j1 ml= DI22 2iA j2 =0, A =A0 A0 A↵A 0 C I>B1 ! C B 8 ijX jk(0) D12 1iA j2 = G8 + G3,q˙j G4,q˙k + G6,q˙j G5,q˙k A ✓1✓2 , 1 C B (0) Each component of the coe cientThe in inverse the1 left is easily hand obtained1 side of since Eq.1 we ( have63) isassumed calculated1 A 1has straightforwardly the inverse1 and xI 1(I> and1) havewe have at least one of ✓1 + A0 (A↵A0 A A0 ij A h A0 A↵Aij0 ij) A0 A↵ A0 + i (17) and 2✓2 except forDI21= 7, which2iA doesj 1 not= contribute(D12 because1iA of j the2) . degeneracy condition. ··· (71) D Aij = D ACij =0B , C B 11 C1i Bj1 22 C2i Bj2 Since D A 1 has terms with ✓ ✓ jk,weneedtomultiply(j jm(0) I 63)bylk(0) ✓ or by ✓ to have relations1 among the The above equations can beij solved in succession if1 andA2 = only l ifA A0 hasjk(0)Aml anxI inverse.A 1 Then, assuming2 A0 (70) exists, D12 1iA C j2 =B G8 + G3,q˙j G4,q˙k + G6 ,q˙j G 5,q˙k A ✓1!✓2 , we get canonical C variables.B For instance, we multiply it by ✓1,XI> and1 however, we cannot have reasonable ones since they i i Dhave ✓1 ⇡AEachij↵, which component= h ( meansD of the@⇡ coe↵A /ijcient@z (z in) = the. q left,p hand, ✓1, side✓2 ) of cannot Eq. (i63) be is calculated determined straightforwardly uniquely(71) as and we we can have add arbitrary 21 C2i Bj1 12 C1i Bj2 functions proportional1 to ✓1, ij ij 1 B0 = A0 , D11 1iA j1 = D22 2iA j2 =0, (18) Since D A has terms with ✓1✓2,weneedtomultiply( C B 63)by C✓1 orB by ✓2 to have relations among the 1 1 ij @⇡↵ @⇡↵ jk(0) C B B = A A A D12, 1iA j2 = G8 + G3,q˙j G4,q˙k + G6,q˙j G5,q˙k A ✓1✓2 , (19) canonical variables. For instance,↵ we multiply0 ↵ it by0 ✓1 ,C and however,B we cannot+ haveg↵z(q, reasonable p)✓1 . ones since they (72) i i ij h @z ! @zij i have ✓1 ⇡↵, which means @⇡↵/@z (z1= q ,p , ✓1,D✓221) cannot2iA j be1 = determined(D12 1iA uniquelyj2) . as we can add arbitrary (71) 1 1 C 1B 1 C B 1 1 1 B↵ = A (A↵A A A A A A↵A ) A A↵ A 8, (20) functions proportional toTherefore,✓1, no phase0 space1 variable0 0 is properly constrained0 0 by these0 relations0 . A quite similar discussion applies Since 2D A has terms with ✓1 ✓2,weneedtomultiply( 63)by✓1 or by ✓2 to have relations among the C B ˙ 1 for whencanonical we. multiply variables.@⇡ (63 For)by@⇡ instance,✓2. To we avoid multiply such it by a✓1 situation,, and however, the we coe cannot cient have matrixreasonable of ones ✓ sincein (63 they), D A , . ↵ ↵ i i C B must vanishhave. for✓1 ⇡N↵, which= 2 case, means and@⇡+↵/ weg@↵zz then((zq,= pq) have✓,p1 ., ✓1 two, ✓2) primary cannot be constraints, determined uniquely as we can(72) add arbitrary functions proportional@z ! to@z✓1,
ij @⇡ij↵ @⇡↵ 8 k ij Therefore, no phase space variable is properly= ⇡ constrainedA p by+ theseA relationsL j k .L A↵ quitek q similar discussionA L j + appliesL ↵ ✓ =0, (73) which also satisfy BA = I. Therefore,↵ ↵ we↵ concludei j that↵i amatrixq˙ q + gA✓↵˙ z(qhasq, p)✓1 the. inverse↵i ifq˙ ✓ and only✓˙ ✓ if A(72)0 has the for when we multiply (63)by✓ . To avoid such C a situation, theC coe@z !cient@z matrix of ✓˙ inC (63), D A 1 , inverse, i.e., 2 ⇣ ⌘ ⇣ ⌘ must vanish for N = 2 case, andTherefore, we then no have phase two space primary variable is constraints, properly constrained by these relations 8. A quite similar C discussionB applies whose number is su cient to eliminate half degrees of freedom in phase space. 1 for when we multiply (63)by✓ . To avoid such a situation, the coe cient matrix of ✓˙ in (63), D A , 2 C B ij must vanish forijdet(N = 2A case,0) =0 and, we thenwherek have twoA0 primaryij= A ✓ constraints,=0. (21)7 7 ↵ = ⇡↵ ↵iA pj + ↵iA Lq˙j qk L6 ✓˙↵qk q ↵iA Lq˙j ✓| + L✓˙↵✓ ✓ =0, (73) NC 3 case: C ij MAXIMALLY ijC k ij = ⇡ A p + A L j k L ˙↵ k q A L j + L ˙↵ ✓ =0, (73) • as discussed in⇣ Sec. II.↵ A. ↵ C↵i j⌘ C↵i ⇣ q˙ q ✓ q C↵i⌘ q˙ ✓ ✓ ✓ as discussed in Sec. II. A. ij i whose number is su cientLetMultiplied to us eliminate consider by theN half inverse degrees3 case. of A In ofij, this freedomA , case, the⇣ first in the phase line degenerate of space.(57) can⌘ condition be solved⇣ foris no q˙ longeras enough⌘ to eliminate all the extra whose number DEGENERATE is su cient to eliminateij half degrees of freedom CONDITION in phase space. i degreesMultiplied of freedom, by the and inverseB. the Hamiltonian of analysisAij, A becomes, the formalism first quite line involved. of (57) can Thus, be we solved just for comment q˙ as on the general analysis in i ij ij ˙↵ ij k ij ↵ Appendix A and concentrate q˙ on= A the p casej A wherej↵ ✓ all theA L extraq˙j qk q degrees+ A L ofq˙j ✓ freedom↵ ✓ , are eliminated only(62) by primary i ij B ij ˙↵ ij k ij ↵ N 3 case: N 3 case: Removing q ˙ dependence= A pj fromA thej↵ second✓ A lineL qof˙j q ＊k ,q +iA Lq˙j ✓↵ ✓ , (62) Now• we move on to Hamiltonianconstraintsand plugging• from formalismthis into now the on, second with as similarly line both of (57n done)Grassmann gives in bosonicB even case [ variables41]. Here weq ( supposet) and thatN Grassmann all the elements odd in the Let us consider N 3 case. In this case, the degenerate condition is no longer enough to eliminate all the extra Let↵ us consider N 3 case.andcoe plugging Incient this matrix case, this the ofinto the degenerate the left second hand condition line side of in ( (5763 is)) no gives vanish, longer enough to eliminate all the extra ones ✓ (t). In the present paper,degrees we consider of freedom, the and Lagrangian the analysisij becomes˙ containing quite involved.ij up Thus, to first weij just derivatives, comment on the namely,k general analysis in degrees of freedom, and the analysis becomes(D quite↵ involved.↵iA j ) Thus,✓ = ⇡ we↵ just↵iA comment pj + ↵ oniA theLq˙j q generalk L✓˙↵qk analysis q in Appendix A and concentrate C onB the case where allC the extraij degreesC of freedom are eliminated only by primary Appendix A and concentrate onconstraints the case where from now all on, the as extra similarlyij degrees done˙ in D of↵ bosonic freedomij ↵ caseiA [ are41].j⇣ij eliminated Here=0 we, suppose onlyij that by⌘ all primary the elements ink the (74) dependence(D↵ on↵iA ✓⇡˙ j ) ✓ =↵ ⇡i A↵CLq˙j ✓ ↵+iBAL ˙ ↵ p j + ✓ ↵iA Lq˙j qk L ˙↵ k q (63)(22) constraints from now on, as similarlycoe cient done matrix in of bosonic the left C hand case side [41B ]. in ( Here63 ) vanish,C we suppose C that✓ ✓ allC the elements in✓ theq ij coe cient matrix of thewhich left hand yields sideN inprimary (63) vanish, constraints, j ˙↵⇣ ⌘ Now we would like to consider the situation suchD thatA the↵ijcanonicaliA velocities=0Lq˙ ,✓ variables+✓ L✓cannot˙↵✓ ✓ be expressed in terms(74) of other (63) ↵ ↵↵iC j canonical variables, that is, thet coe2 cient matrix of C✓˙ doesB not have the inverse, equivalent to imposing the degeneracy condition, ijij i i ↵ij ˙↵ k ij ˙↵ Now wewhich would yields↵ =If likeN ⇡ D primary ↵S↵ to = consider ↵ constraints, i↵ A i A L p ( j the q j+ , =0q ˙ situation ↵ , i ✓ A ,, ,L✓q˙j )q suchkdt. L that✓˙↵qk the q velocities↵iA L✓q˙j ✓cannot + L✓˙(74)↵ be✓ expressed ✓ =0. in(23) terms(75) of other C C B C ↵ C canonical variables, that is,t the1 coe cient matrix of ✓˙ does not have the inverse, equivalent to imposing the ij (0)⇣ ij (0)k⌘ ⇣ij ⌘ = ⇡ Z detA Dp + =0A, L j wherek L ↵ kD q = D A L˙ j . + L ↵ ✓ =0. (75) (64) which yields N primarydegeneracy constraints, condition,↵ ↵ ↵i j↵ ↵i q˙ q ✓˙ q ↵ ↵↵ i ✓,✓=0q˙ ✓ ✓˙ ✓ The Lagrangian should beA an straightforward even and real calculation function, C shows and that theC they dynamical actually satisfy variables C the| integrableqi and ✓↵ conditionare taken (including to be real the1 case of NWe= consider 2), and the therefore, cases of N they= 1, haveN = the 2, and integrated⇣N 3 separately. form, ⌘ ⇣ ⌘ ij A straightforwardij calculationIntegrability showsA k thatcondition(0)i they↵ij actually satisfy the integrable(0) condition (including the case of throughout this paper. ⇡↵ The↵iA variation pj+ ↵ withiA L respectj k L ˙ to↵ kz qdet=(Dq ↵,i✓A=0)L yields, j +whereL the˙↵ Euler-LagrangeD ✓ =0= D. ˙ equations,. (75) (64) C N =1N =caseC 2), and: therefore,q˙ q they✓ q have the integratedC↵ form,q˙ ✓ ✓ ✓ ↵ ↵ |✓,✓=0 • ↵ = ⇡↵ F↵(q, p, ✓)=0. (76) ⇣ ⌘ ⇣ ↵ = ⇡↵ F↵(q, p, ✓)=0⌘ . 1 (76) We considerLet us start the with cases a single of dN = fermionic 1,@LN = variable, 2, and@L NN = 1. 3 In separately. this case, both D and A are zero due to the Grass- A straightforward calculation shows that they actually satisfyN the primary integrable=0 ,constraints condition are obtained. (including C theB case of (24) In AppendixmannIn Appendix property,C, weC and give, we the give an degeneracy alternative an alternativeA condition proof proofA of of of the the the equivalence kinetic equivalence matrix of (74 is of) automatically and (74 the) and existence the satisfied. existence of the More primary of importantly, the con- primary con- N = 2), and therefore, they have the integrateddt form,@z˙1 @z theN =1 coe casecient: matrix D A always vanishes, and we have a primary constraint = ⇡ f (q, p, ✓), which straints.• straints. ✓C B◆ 1 1 1 will remove the fermionic ghost properly. (0) 1 ToLet summarize, us start the with ↵ degeneracy= a⇡ single↵ condition,F fermionic↵(q, p, ✓ det)=0 variable,D(0).= 0, isN equivalent= 1. In thisto the case, maximally-degenerate both D and (76)A condition, areD zero due to the Grass- To summarize, the degeneracy condition, det D↵ = 0, is equivalent to the maximally-degenerateC B condition, D Nmann1=2case property,: and the degeneracy condition↵ of the kinetic matrix is automatically satisfied. More importantly, 1 A = 0, for N = 1 and N = 2 cases. For N 3, the latter one is a su cient (but not necessary) condition for A =•C 0, forB N = 1 and N = 2 cases.1 For N 3, the latter one1 is a su cient (but not necessary) condition for 1 In Appendix C, we give anthe alternativeWhenthe former. coeN In=cient 2,the proof the following, matrix matrix of the weDD equivalence simplyis noA longer adoptalways the zero, of condition (74 which, vanishes,) andD in the general, andA existence we has= have 0 forthe of any a following primary theN. primary form, constraint con- 1 = ⇡1 f1(q, p, ✓), which Since complex variablestheC can former.B be decomposed In the following, into real we and simply imaginaryC adoptB parts the condition and expressed DC inBA terms1 = of 0 for a set any ofN two. real variables, we can alwaysstraints. identify complex variableswill with remove doubled the real fermionic variables ghost without properly. loss of generality. 0 D C B (0) D = 12 . (65) To summarize, the degeneracy condition,8 det D = 0, is equivalent to the maximally-degenerate condition, D ThisN =2 requirementcase is: referred↵ to as “regularity condition”, where the JacobianD12 matrix0 of the M 0 (independent) constraints with respect to 1 • the canonical variables should have rank M ,andhence,theconstraintsproperlyreducethedimensionofthephasespace,asexplained✓ ◆ A = 0, for N = 1 and N = 2 cases. For N 3, the latter0 one is a su cient (but not necessary) condition for C B 8 This requirementinLetWhen [44 us]. is explicitlyN referred= 2, the to write as matrix “regularity the LagrangianD is condition”, no for longer1 this where case.zero, the which, Jacobian in matrix general, of the hasM the(independent) following form, constraints with respect to the former. In the following, we simply adopt the condition D A = 0 for any N. 0 the canonical variables should have rank M 0,andhence,theconstraintsproperlyreducethedimensionofthephasespace,asexplained C B 0 D 1 in [44]. D = 12 . (65) D 0 i✓1✓2 12 0 1 ✓ i✓◆1✓˙1 8 This requirement is referred to as “regularity condition”, where the Jacobian matrix of the M 0 (independent) constraints˙ with respect to Let us explicitly write the Lagrangiani i for this case. B i✓2✓2 C the canonical variables should have rank M 0,andhence,theconstraintsproperlyreducethedimensionofthephasespace,asexplainedL = GI (q , q˙ )xI , where x = B ˙ C , (66) B i✓1✓2 C in [44]. B C B i✓ ✓˙ C 1 B 2 1 C B i✓˙ ✓˙ C i✓ ✓ B 1 2 C 1 2 B ˙ ˙0C ˙ 1 B✓1✓2✓1✓2C i✓1✓1 @ A i✓ ✓˙ and G (I =1, 2, , 8) are functions dependingi i on qi andq ˙i only. Therefore,B we obtain2 2 C I L = GI (q , q˙ )xI , where x = B ˙ C , (66) ··· B i✓1✓2 C B C D12 = iG7 + G8✓1✓2 . B i✓ ✓˙ C (67) B 2 1 C 7B i✓˙1✓˙2 C Applying the degeneracy condition, we have G7 = 0. The momenta are now B C B✓1✓2✓˙1✓˙2C @x @x B C ⇡ = G I = iG ✓ iG ✓ + G ✓ ✓ ✓˙ , ⇡ = G I = iG@✓ iG ✓A G ✓ ✓ ✓˙ . (68) 1 I ˙ 3 1 6 2 8 1 2 2 2 i I ˙ i 5 1 4 2 8 1 2 1 and G (I =1,@2✓,1 , 8) are functions depending on q and@✓q ˙2 only. Therefore, we obtain I ··· The explicit form of Aij is D12 = iG7 + G8✓1✓2 . (67) 2 (0) @ GI (0) A = A + AI x , where AI = ,A = A17 . (69) Applying the degeneracyij ij condition,ij weI have G7 = 0. Theij momenta@q˙j@q˙i areij nowij XI>1 @xI @xI ⇡1 = GI = iG3✓1 iG6✓2 + G8✓1✓2✓˙2 , ⇡2 = GI = iG5✓1 iG4✓2 G8✓1✓2✓˙1 . (68) @✓˙1 @✓˙2 7 When we do not require the degeneracy condition, iG7✓˙2 and iG7✓˙1,doesappearin(68), which makes them solvable for ✓˙2 and ✓˙1, and extraThe degrees explicit of freedom, form of correspondingAij is to the fermionic ghost, remain in the fermionic sector. @2G A = A(0) + AI x , where AI = I ,A(0) = A1 . (69) ij ij ij I ij @q˙j@q˙i ij ij XI>1
7 When we do not require the degeneracy condition, iG7✓˙2 and iG7✓˙1,doesappearin(68), which makes them solvable for ✓˙2 and ✓˙1, and extra degrees of freedom, corresponding to the fermionic ghost, remain in the fermionic sector. 12 12 where we have used (101) and (103) in the second line, and (106) in the third line. Therefore, we find where we have used (101) and (103) in the second line, and (106) in the third line. Therefore, we find @ ↵ @ ↵ ij @ ↵ Y = ↵, = Y + A j Yi , (108) @✓˙ @{ ↵ } @✓˙ @ B↵ @q˙ ij ˙ @ ↵ q,p,✓ Y = ↵, q, q, ˙ ✓ = Y + Aq,✓,✓ j Y , (108) ˙ { } ˙ B @q˙i @✓ q,p,✓ @✓ q,q,˙ ✓ q,✓,✓˙ where we again used (106) in the right hand side of (105), and where we again used (106) in the right hand side of (105), and (0) (0) C↵ = J↵ . (109) C(0) = J (0) . (109) ↵ ↵ (0) As a result, we explicitly see that the invertibility of C↵ ,(84), coincides with the non-zero determinant of J↵ , (0) the uniquenessAs condition a result, ( we95). explicitly In other see words, that the the condition invertibility that of allC↵ the,(84 Lagrange), coincides multipliers with the are non-zero uniquely determinant fixed is of J↵ , equivalent to thethe condition uniqueness that condition the time (95 evolution). In other of words, the system the condition is uniquely that determined. all the Lagrange multipliers are uniquely fixed is equivalent to the condition that the time evolution of the system is uniquely determined.
V. CONCRETE MODELS V. CONCRETE MODELS In the previous section, we have derived the conditions to successfully eliminate unwanted degrees of freedom in the In the previous section, we have derived the conditions to successfully eliminate unwanted degrees of freedom in the fermionic sector. In this section, we provide some examples of degenerate (boson-)fermion system, having n + N/2 fermionic sector. In this section, we provide some examples of degenerate (boson-)fermion system, having n + N/2 physical degrees of freedom. physical degrees of freedom.
Example 1 :ExampleLet us first 1 : considerLet us first the considersimplest the example, simplest where example, the bosonic where the sector bosonic is absent. sector In is thisabsent. case, In one this case, one can immediatelycan notice immediately that the notice Lagrangian that the should Lagrangian be linear should in the be linear time derivatives in the time of derivatives fermions from of fermions the maximally- from the maximally- degenerate conditiondegenerate (76). condition Then, the (76 most). Then, general the most Lagrangian general in Lagrangian this case is in given this case by is given by
L = if (✓ )✓˙↵ , ↵ (110) ↵ L = if↵(✓ )✓˙ , (110)
where f↵ are arbitrarywhere f↵ Grassmann-oddare arbitrary Grassmann-odd functions of ✓ functions. The momenta of ✓ . The are momenta easily found are easily as found as
⇡↵ = if↵ , ⇡ = if , (111) (111) ↵ ↵ which lead towhich the constraints, lead to the constraints,↵ = ⇡↵ + if↵ .↵ As= ⇡ long↵ + if as↵ the. As matrix, long as the matrix,
@f↵ @f @f↵ @f C↵ = i C i= i , i , (112) (112) @✓ ↵ @✓ ↵ @✓ @✓↵ is invertible, theis invertible, number of the degrees number of freedomof degrees is ofN/ freedom2. is N/2.
Example 2 :ExampleThe second 2 : exampleThe second is a Lagrangian example is a for Lagrangiann = 1 and forNn== 2, 1 and N = 2, 1 1 2 2 ˙ L = q˙ + iq˙L(✓=+ ✓q˙ )+✓˙ iq,˙(✓1 + ✓2)✓1 , (113) (113) 2 1 2 2 1 which satisfieswhich the condition satisfies the (76). condition The momenta (76). The are momenta given by are given by ˙ p =˙q + i(✓ +p ✓=˙)q✓˙+,i(✓1⇡+=✓2)✓i1q˙(,✓ +⇡1✓=) , iq˙(⇡✓1 =0+ ✓2), , ⇡2 =0, (114) (114) 1 2 1 1 1 2 2 where the last two ones lead to the primary constraints, 1 = ⇡1 + ip(✓1 + ✓2) and 2 = ⇡2. Then, the constraint where the last two ones lead to the primary constraints, 1 = ⇡1 + ip(✓1 + ✓2) and 2 = ⇡2. Then, the constraint matrix C↵ is invertible (for a non-zero value of p)since matrix C↵ is invertible (for a non-zero value of p)since AN EXAMPLE2 OF 2det C↵ = p . (115) det C↵ = p . (115) Thus, the total number of degrees of freedom is 2 = 1 + 2 1/2. Thus, the total number of degrees of freedomHEALTHY is 2 = 1 + 2 1/2. THEORIES⇥ ⇥ Example 3 : An example for n = 1 and arbitrary N is given by Example 3 : An exampleExample: for n = 1 and 1 scalar arbitrary + N fermionsN is given by 1 1 L = q˙2 + i f (q, ✓ )+f (q, ✓ )˙q ✓ ✓˙↵ + g(q, ✓ )✓ ✓ ✓˙↵✓˙ . (116) 1 2 1 ˙2↵ 1 ↵ ˙↵ ˙ ↵ L = q˙ + i f1(q,2✓ )+f2(q, ✓ )˙q ✓↵✓ + g(q, ✓ )✓↵✓ 2✓ ✓ . 13 (116) 2 2 13 The maximally-degenerate condition, 2 The maximally-degenerate condition,Maximally degenerate condition 13 implies g = f2 , which we suppose from now on.2 The conjugate momenta are 1 2 implies g = f , which we supposeL from˙↵ ˙ + nowL ˙↵ on.L TheL ˙ conjugate = g ( momentaf ) ✓ ✓ are=0, (117) 2 ✓ 1✓ ✓ q˙ q˙q˙ q˙✓ 2 2 ↵ g= f2 or -f2 2 L✓˙↵✓˙ + L✓˙↵q˙Lq ˙q˙ Lq˙✓˙ = g (f2) ✓↵✓ =0, (117) implies g = f2 , which we suppose from now on. The˙↵ conjugate momenta are p =˙q + if2✓↵✓ , p =˙q + if ✓ ✓˙↵ , (118) (118) 2 ↵ ↵ p =˙q + if2✓↵✓˙ , (118) Momenta ˙ ⇡ = i(f + f q˙)✓ + g✓ ✓ ✓˙ = i(f + f p)✓ , (119) ⇡↵ = i(f1 + f2q˙)✓↵ + g✓↵✓ ✓ ↵= i(f11 + f22p)✓↵↵ , ↵ 1 2 ↵ (119) ⇡↵ = i(f1 + f2q˙)✓↵ + g✓↵✓ ✓˙ = i(f1 + f2p)✓↵ , (119) where the last line is again regarded as the primary constraints, ↵ = ⇡↵ + i(f1 + f2p)✓↵. As long as (f1 + f2p) ✓=0 = where the last line is again regarded as(0) the(0) primaryConstraints constraints, ↵ = ⇡↵ + i(f1 + f2p)✓↵. As long as (f1 + f2p) ✓=0 = | f + f p = 0, the constraint matrix C↵ is invertible since | (0) (0) where the last line1 is again2 regarded6 as the primary constraints, ↵ = ⇡↵ + i(f1 + f2p)✓↵. As long as (f1 + f2p) ✓=0 = f1 + f2 p = 0, the(0) constraint(0) matrix C↵ is invertible since | 6 f1 + f2 p = 0, the constraint matrix C↵ is invertible since (0) (0) (0) N 6 Quadratic terms of the timedet derivativeC = 2ofi( ffermions1 + f2 pare) included!. (120) (0) (0) (0) N ↵ det C = 2i(f (0)+ f p) (0). (0) N (120) ↵ det1C = 2 2i(f + f p ) . (120) Then, the system has N second↵ class constraints,1 2 and the total number of degrees of freedom is 1 + N/2. Then, the system has N second class constraints, and the total number of degrees of freedom is 1 + N/2. Then, the systemExample has N second 4 : A class similar constraints, but practically and the di total↵erent number model toof degreesthe previous of freedom one is is 1 + N/2. Example 4 : A similarExample but practically 4 : A similar di but↵erent practically model di to↵erent the previous model to theone previous1 is one is i L = (˙q + i✏ ✓↵✓˙ )2 + ✓ ✓˙↵ . (121) 2 ↵ 2 ↵ 1 1 i ↵ ˙ 2 i ˙↵ L↵=˙ (˙2q + i✏↵ ✓˙↵✓ ) + ✓↵✓ . (121) We shouldL = note(˙q that+ i✏ it↵ is✓ not✓ 2) an+ essentially✓↵✓ . new model2 since there exists an invertible(121) transformation as q q + ↵ 2 ↵ ↵ 2 2 ↵ ! (i/2)✏↵ ✓ ✓ and ✓ ✓ between this Lagrangian and L =˙q /2+(i/2) ✓↵✓˙ . However, it would be worthwhile We should note thatto examine it is not this an model essentially! because new we model have found since a there field exists theoretical an invertible extension transformation of this model as as exhibitedq q + in Appendix E. We should note that it is not↵ an essentially↵ ↵ new model since there exists an2 invertible transformation˙↵ as q q + ! (i/2)✏↵ ✓ ✓ andThe✓ conjugate✓ between momenta this Lagrangianare and L =˙q /2+(i/2) ✓↵✓ . However, it would be worthwhile ↵ ↵ ↵ ! 2 ↵ ! (i/2)✏↵ ✓ ✓ and ✓ to examine✓ between this model this because Lagrangian we have and foundL a=˙ fieldq / theoretical2+(i/2) ✓ extension↵✓˙ . However, of this model it would as exhibited be worthwhile in Appendix E. ! ↵ to examine this modelThe because conjugate we momenta have found are a field theoretical extension of thisp =˙q model+ i✏↵ ✓ as✓˙ exhibited, in Appendix E. (122)
The conjugate momenta are ↵ p =˙q + i✏↵ ✓ ✓˙ , (122) ↵ i i p =˙q + i✏ ✓ ⇡✓˙↵ =, i✏↵ ✓ (˙q + i✏ ✓ ✓˙ ) ✓↵ = i✏↵ ✓ p ✓↵ . (122) (123) ↵ 2 2 i i ⇡ = i✏ ✓ (˙q + i✏ ✓ ✓˙ ) ✓ = i✏ ✓ p ✓ . (123) Therefore, N primary↵ constraints↵ are found as2 ↵↵ = ⇡↵ ↵ i✏↵ ✓2 p↵+(i/2)✓↵. The Poisson brackets, i i ⇡ = i✏ ✓ (˙q + i✏ ✓ ✓˙ ) ✓ = i✏ ✓ p ✓ . (123) Therefore, N primary↵ ↵ constraints are found as ↵= ⇡ ↵ i✏ ✓ p +(↵,i/ ↵ 2)✓=. Thei ↵ Poisson, brackets, (124) 2↵ ↵ ↵ { 2 } ↵ imply the invertibility since , = i , (124) Therefore, N primary constraints are found as ↵ = ⇡↵ i✏↵ ✓{ p↵ +( }i/2) ✓↵.↵ The Poisson brackets, N imply the invertibility since det C↵ =( i) . (125) ↵, = i ↵ , (124) { } N ˙↵ As a result, the number of degreesdet C of↵ freedom=( i) is 1. + N/2. If the canonical kinetic term, (i/2)✓(125)↵✓ , is absent, this imply the invertibility since system will generate secondary constraints and/or have first class constraints since ↵, vanishes. The use of ↵ ˙ ˙↵ { } As a result, the number(i/2)✏↵ of✓ ✓ degreesinstead of of freedom (i/2)✓↵ is✓ 1also + N/ gives2. If the the vanishing canonical Poisson kinetic brackets term, and (i/2) does✓ ✓˙↵ not, is work absent, as well. this In those cases, N ↵ system will generatewe will secondary have a less constraintsdet numberC↵ =( of and/or degreei) have. of freedom first class than constraints 1 + N/2, which since shows ↵, the explicitvanishes. di(125)↵erence The use from of Example 3. In ↵ a field theoretical↵ extension given in Appendix E, the standard Weyl kinetic{ term} plays the same role with (i/2)✓ ✓˙↵. (i/2)✏↵ ✓ ✓˙ instead of (i/2)✓↵✓˙ also gives the vanishing Poisson brackets and does not work as well. In those cases, ↵ ↵ As a result, the numberwe will of have degrees a less of number freedom of degree is 1 + of freedomN/2. If than the 1 canonical + N/2, which kinetic shows term, the explicit (i/2)✓ di↵↵✓˙erence, is absent, from Example this 3. In ˙↵ system will generatea field secondary theoretical constraints extension and/or given in Appendix have firstE class, the standard constraints Weyl since kinetic term↵, plays vanishes. the same role The with use (i/ of2)✓↵✓ . ↵ ↵ VI. SUMMARY{ } (i/2)✏↵ ✓ ✓˙ instead of (i/2)✓↵✓˙ also gives the vanishing Poisson brackets and does not work as well. In those cases, we will have a less number of degree ofAs freedom mentioned than in [44 1 +],N/ even2, when which the shows Lagrangian the explicit contains di only↵erence up to from first derivatives Example 3.of fermions, In non-degenerate VI. SUMMARY ↵ a field theoretical extension given infermionic Appendix systemE, the always standard su↵ers from Weyl the kinetic presence term of negative plays norm the same states, role which with come (i/ as2) a✓ consequence↵✓˙ . of the existence of extra degrees of freedom. Although the situation in fermionic case is more involved because of the Grassmann As mentioned inproperty [44], even of fermionic when the variables, Lagrangian this contains can be contrasted only up towith first non-degenerate derivatives of bosonic fermions, system non-degenerate containing second or higher fermionic system alwaysderivatives su↵ers in from the Lagrangian. the presence In of suchnegative bosonic norm system, states, the which Hamiltonian come as a consequence should include of the terms existence linear in momentum, of extra degrees ofmaking freedom. the Hamiltonian AlthoughVI. SUMMARY the unbounded situation from in fermionic below. This case is is what more is involvedcalled Ostrogradsky’s because of the ghost Grassmann instability. So far, there property of fermionicseems variables, to be no this obvious can be criteria contrasted to determine with non-degenerate the existence bosonic of the ghosts system in containing fermionic second system or at higher classical level, which derivatives in theare, Lagrangian. in turn, transparently In such bosonic observed system, as the negative Hamiltonian norm states should once include the system terms is linear quantized. in momentum, (The relation between As mentioned in [making44], even the when Hamiltonian the Lagrangian unbounded contains from below. only This up is to what first is called derivatives Ostrogradsky’s of fermions, ghost non-degenerate instability. So far, there fermionic system always su↵ers fromthe the Ostrogradsky’s presence of negative ghosts and norm negative states, norm which states come is more as a obvious consequence in bosonic of the system existence as shown in [48].) To avoid seems to be no obviousthese negative criteria norm to determine states, the the fermionic existence system of the must ghosts be in degenerate fermionic and system contain at classical a su cient level, number which of constraints to of extra degrees ofare, freedom. in turn, Although transparentlyeliminate the half observedsituation degrees as in of negative fermionicfreedom norm in phase case states is space, more once whose the involved system situation because is is quantized. similar of for the (The bosonic Grassmann relation Lagrangian between including second property of fermionicthe variables, Ostrogradsky’s thisderivatives can ghosts be contrasted and as investigated negative with norm innon-degenerate [ states41]. is more obviousbosonic in system bosonic containing system as second shown in or [48 higher].) To avoid derivatives in the Lagrangian.these negative In norm such states, bosonic the fermionic system, systemthe Hamiltonian must be degenerate should and include contain terms a su linearcient number in momentum, of constraints to making the Hamiltonianeliminate unbounded half degrees from of below.freedom This in phase is what space, is whose called situation Ostrogradsky’s is similar ghost for bosonic instability. Lagrangian So far, including there second seems to be no obviousderivatives criteria as to investigated determine in the [41]. existence of the ghosts in fermionic system at classical level, which are, in turn, transparently observed as negative norm states once the system is quantized. (The relation between the Ostrogradsky’s ghosts and negative norm states is more obvious in bosonic system as shown in [48].) To avoid these negative norm states, the fermionic system must be degenerate and contain a su cient number of constraints to eliminate half degrees of freedom in phase space, whose situation is similar for bosonic Lagrangian including second derivatives as investigated in [41]. SUMMARY
We have …
discussed that, when we construct new interactions between bosons and fermions, we need to avoid what we call fermionic ghosts, coming from the extra degrees of freedom in fermionic sector. Analogy with Ostrogradsky’s ghost may be pointed out. (See also J.Phys. A35 (2002) 6169-6182)
found ghost free conditions, which eliminate such the ghosts, in general formulation and non-trivial examples. Remarkably, some of them include quadratic terms of the time derivative of fermionic variables. Thank you for your attention