On the Hydrodynamic Canonical Formalism of the Gross-Pitaevskii Field

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Inserting Schrödinger equation. nrdc h arnino h aybd ytm[14– system many-body the of Lagrangian the introduce cteiglnt and length scattering ytetn h aybd aeucina rdc of states: product system, particle a as single the wavefunction identical of many-body mean- dynamics the a true treating by as the to retrieved approximation be field may equation Gross-Pitaevskii ic ecfrhorcnenle nywt h meanfield, the with only lies concern our henceforth Since L pre- be may density, Lagrangian as meanfield sented, the (5), and = h aitna fadlt lu of cloud dilute a of Hamiltonian The Ohberg ¨ i 2 ~ H I H RS-IAVKIFIELD GROSS-PITAEVSKII THE II. R p ˆ ˆ s ftecnest omacanonical a form condensate the of ase i d ψ H = 3 ˆ naino h rs-iavkifield Gross-Pitaevskii the of entation l sasnua arninsystem Lagrangian singular a as eld ∗ = L r MF ψ X L ˙ | MF s h ade-akwmethod Faddeev-Jackiw The ds. ( Ψ φ i − − = ψ ( i r ψ r ~ = Z ) = 1 2 ψ ∇ | p ˆ , 2 ˙ m Z ∗ i 2 r i Y r =1 N 2 .T os,i scnein to convenient is it so, do To 1. = i − + , , d − · · · MF d 3 V 2 g ~ V 3 r m 2 2 r ~ ψ m , sa xenlptnil The potential. external an is ( 4 = i 2 ∇ ∗ m Ψ r nacrac ihEs (4) Eqs. with accordance In . r ψ i ∇ N 2 ∗ ) ( i = ) + ψ ~ r π = ) i ∂ ~ + · ~ V t 2 si field vskii ∇ ∂ ∗ i Y a/m − X i =1 N + t 6= ilstemany-body the yields √ ψ − H j ˆ H ∗ φ g Nφ 2 ˆ gδ MF H , − ( MF ˆ | ψ r a ( i g 2 ( | r ) N 2 r Ψ cso the on acts , i sthe is , | ,as ), ψ, ψ − , oeatoms Bose ψ. | 4 r j − ) , V s -wave | ψ (4) (3) (1) (2) (6) (5) | 2 , 2 where we have denoted partial differentiation with re- Dynamical systems with this property are called singu- spect to time by a dot and performed the transforma- lar Lagrangian systems or constrained Hamiltonian sys- i~ ∗ tion 2 ∂t (ψ ψ), such that IR. Inserting tems [21, 22]. Two equivalent [23] methods have been de- the LagrangianL→L− density (6) into the Euler-LagrangeL ∈ field vised to eliminate redundant variables and construct a re- equation for ψ∗, duced phase space for such systems: the Dirac-Bergmann method [22, 24–32] and the Faddeev-Jackiw method δL ∂ δL =0, (7) [28, 33–36]. In the particular case of the Schrödinger field, δψ∗ − ∂t ˙∗ δψ an alternative route is made available by performing a yields the Gross-Pitaevskii equation suitable canonical transformation [20, 37], where one be- gins by decomposing the field into real and imaginary 2 ~ 2 2 parts and then supplements the resulting Lagrangian by i~∂tψ = ∇ + V + g ψ ψ, (8) −2m | | a total time derivative to obtain a single pair of real conjugate variables. Several other field transformations yield- where δL/δψ = ∂ /∂ψ ∇ (∂ /∂ (∇ψ)) and δL/δψ˙ = ing Schrödinger’s equation from a canonical field equa- L − · L ∂ /∂ψ˙ are functional derivatives [19, 20]. Carrying out tion, can also be found in the literature [16, 38]. How- L the same procedure for ψ, yields the complex conjugate ever, these depend either on one or two complex pairs of of the Gross-Pitaevskii equation. conjugate variables, meaning redundant variables have not entirely been eliminated. For a review of these for- malisms, and an application of the Dirac-Bergmann and III. THE GROSS-PITAEVSKII FIELD AS A the Faddeev-Jackiw methods to the Schrödinger field, see SINGULAR LAGRANGIAN SYSTEM [39]. In this paper, we take a different approach and derive Notice how a complex field variable, ψ, automatically a canonical formalism in terms of the single pair of real requires that also depend on ψ∗ in order for the action variables (ρ,θ), namely the density and the phase of the L to be real [20]. Yet, the fact that the Gross-Pitaevskii matter-field ψ. These constitute the natural pair of vari- equation is first order in time, signifies that ψ, ψ∗, ψ˙ and ables connecting the field and fluid descriptions of a con- ψ˙ ∗ are not independent and that an excess of dynamical densate. Accordingly, we shall refer to this formalism as variables are contained in the Lagrangian [16, 20]. This the “hydrodynamic canonical formalism”. is invariably the situation when the Lagrangian density is linear in the time derivatives of the fields. To see this, let ˙ φα, ∇φα, φα be a multi-component Lagrangian den- IV. HYDRODYNAMIC FORMALISM FOR THE L sity which is linear in the time derivatives of the fields, GROSS-PITAEVSKII FIELD where φα φ1, , φn. In this situation, the total La- ≡ ··· grangian of the system, may be written as Following up on previous discussions, let us represent

3 the Gross-Pitaevskii ﬁeld in the polar, or Madelung form L φα, φ˙α = d r i (φα, ∇φα) φ˙i U [φα] , (9) A − i i θ ∗ −i θ h i Z X ψ = √ρe ~ , ψ = √ρe ~ , (13) where U is an interaction functional of the ﬁeld components: and treat (ρ,θ) as the independent variables subject to

3 the process of variation. From a dynamical perspective, U [φα]= d r (φα, ∇φα) . (10) U a clear role is played by the new field variables: the first Z defines the distribution or amplitude of the matter-field, When L is of the form (9), the canonical momenta, while the second dictates the flow of this distribution. πi = i (φα, ∇φα), can not be treated as independent This may be seen by substituting Eq. (13) into the cur- A dynamical variables. Indeed, upon constructing the to- rent density tal Hamiltonian ~ n i ∇ ∗ ∗∇ 3 ˙ ˙ J = (ψ ψ ψ ψ) , (14) H [φα, πα]= d r πiφi L φα, φα , (11) 2m − =1 − Z i h i X which yields the more perspicuous form the field velocities φ˙i no longer appear on the right hand side of Eq. (11) as they would typically. Hence, the ρ J = ∇θ, (15) Hamiltonian reduces to the interaction functional, H = m U, and it is not possible to invert φ˙i as a function of φα and πα, since and allows for the identification of the velocity field 2 ∂ ∂ i ∇ L = A =0. (12) θ ˙ ˙ ˙ v = . (16) ∂φj ∂φi ∂φj m 3

A. Lagrangian formalism according to ∂ ρθ Under substitution (13), the Lagrangian density (6), + (24) L → L ∂t 2 becomes 2 2 2 2 1 (∇θ) g ~ 2 (∇θ) g ~ 2 = θρ˙ ρθ˙ ρ + ρ + V (∇ρ) . = ρ θ˙ + + ρ + V (∇ρ) . (17) 2 − − 2m 2 − 8mρ L − 2m 2 − 8mρ ! The Euler-Lagrange ﬁeld equations for θ and ρ, yield re- This will allow for the possibility of readily identifying spectively, the wave equations a canonical transformation (54), which separates out the physical pair of conjugate variables from the redundant

∂tρ + ∇ J =0, (18) pair of conjugate variables, the latter representing the · constraints of the theory. The first order nature of the 1 2 ∂tθ + mv + gρ + V + Q =0, (19) Lagrangian density (24), means that the canonical mo- 2 menta where v = v is the modulus of the superfluid flow from ∂ θ ∂ ρ Eq. (16), J| is| given by Eq. (15) and πρ = L = , πθ = L = , (25) ∂ρ˙ 2 ∂θ˙ − 2 2 ~2 √ρ cannot be treated as independent dynamical variables. In Q = ∇ , (20) −2m √ρ other words, the phase space variables are restricted by the following equations: is the quantum potential [40]. The coupled wave equa- ρ θ tions (18) and (19), express, respectively, the conserva- C1 = πθ + =0, C2 = πρ =0. (26) tion of mass and momentum. They are entirely equiv- 2 − 2 alent to the Gross-Pitaevskii Eq. (8), where transfor- In the Dirac treatment [24, 25] of singular Lagrangian sys- mation (13) defines the mapping between both sets of tems, the relations from Eq. (26) define a constraint hy- equations. persurface Γc in the full phase space (ρ,θ,πρ, πθ). Since the constraints are primary, only two canonical variables play a physical role in the dynamical description of the B. Canonical formalism system. Before constructing the reduced phase space, it is instruc- Let us turn our attention to the construction of a tive to examine the form of the dynamical equations on canonical formalism for the field. In particular, we will Γc, as embedded in the full phase space of the system. show that the reduced phase space of the system com- Although the duality of the Legendre transform is de- prises the single pair of conjugate variables (ρ,θ), dynam- stroyed by the singular nature of the system, let us de- ically governed by the following canonical field equations: fine a canonical Hamiltonian density, , according to the usual prescription H δH ρ˙ (r)= , (21) ˙ δθ (r) = πρρ˙ + πθθ , (27) H − L δH θ˙ (r)= , (22) which on Γc, is given by r −δρ ( ) 2 2 (∇θ) g ~ 2 ρ + ρ + V + (∇ρ) , (28) where H is the total canonical Hamiltonian of the field, in- H≈ 2m 2 8mρ troduced further in the text. In turn, the Poisson bracket of two dynamical variables f and g on the reduced phase where the symbol ’ ’ denotes weak equality [25]. The pri- ≈ space, takes the form mary Hamiltonian [22], which incorporates the primary constraints, may be defined as 3 δf (x) δg (y) δf (x) δg (y) f (x) ,g (y) = d r . 3 { } δρ (r) δθ (r) − δθ (r) δρ (r) Hp = d r + θC˙ 1 +ρC ˙ 2 , (29) Z (23) H Z In the following section, we retrieve the above expression where the Lagrange multipliers are identified [26] with for the Poisson bracket as the reduced Dirac bracket on the field velocities θ˙ andρ ˙, which are unknown functions the full phase. on phase space. Note also that p . On the full phase space, the time evolution ofH an≈ arbitrary H dynamical variable f is generated by the primary Hamiltonian 1. The Dirac-Bergmann algorithm rather than the canonical Hamiltonian [25–28], through the Poisson bracket As a starting point to applying the Dirac-Bergmann algorithm, let us transform the Lagrangian density (17), f˙ f,Hp , (30) ≈{ } 4

where f,g f (x) ,g (y) . Note that we will occa- Furthermore, in contrast to Eq. (30), time-evolution is sionally{ use}≡{ the shorthand} form f,g = f,g + now generated by the canonical Hamiltonian, through { } { }ρ,πρ f,g . The unknown functionsρ ˙ and θ˙, may be ob- the Dirac bracket { }θ,πθ tained on Γc, from the consistency requirement that the ˙ f = f,H D . (39) constraint equations must be preserved in time: C˙ i 0. { } ≈ To see this, let us replace f in Eq. (30) by Ci and expand Recalling Eq. (26) for the constraints, let us explic- out the Poisson bracket, which gives ′ itly construct Qij (r, r ) for our system. In view of the anti-symmetric property of the Poisson bracket, the di- ′ ˙ 3 ′ δCi (r) δHp δCi (r) δHp agonal elements are Q11 = C1 (r) , C1 (r ) = Q22 = Ci (r) d r ′ ′ ′ ′ ′ { } ≈ δρ (r ) δπρ (r ) − δπρ (r ) δρ (r ) C2 (r) , C2 (r ) = 0. For the off-diagonal elements, we Z { } δC (r) δH δC (r) δH find + i p i p . (31) δθ (r′) δπ (r′) − δπ (r′) δθ (r′) ′ ′ θ θ Q12 = C1 (r) , C2 (r ) + C1 (r) , C2 (r ) { }ρ,πρ { }θ,πθ ′ For the constraint C1 = πθ +ρ/2, the first and last of the = δ (r r )= Q21. (40) four terms under the integral survive, yielding − − Thus, the constraint Poisson bracket matrix, takes the 1 δHp δHp form C˙ 1 (r) , (32) ≈ 2 δπ (r) − δθ (r) ρ 0 1 Q (r, r′)= δ (r r′) , (41) 1 0 − while for C2 = πρ θ/2, the other two terms survive, − yielding − whose inverse is given by ˙ δHp 1 δHp C2 (r) . (33) −1 ′ 0 1 ′ ≈−δρ (r) − 2 δπθ (r) Q (r, r )= − δ (r r ) , (42) 1 0 − Using Eq. (29) for H , with C given by Eq. (26), we p i in accordance with Eq. (38). For the present problem, then find evaluation of the Dirac bracket (36) involves the two off- ρ˙ δH ρ˙ diagonal terms R12 and R21. Since both constraints (26) C˙ 1 + 0, (34) ≈ 2 − δθ 2 ≈ depend on two phase space variables, only two out of four δH θ˙ θ˙ terms are retained in any Poisson bracket comprised in C˙ 2 0. (35) R12 and R21. Substituting Eqs. (26) and (42) into Eq. ≈− δρ − 2 − 2 ≈ (37) and making use of standard delta function relations, Hence the consistency equations for the constraints are gives equivalent to the canonical system of equations (21) and 3 1 δf (x) δf (x) δg (y) 1 δg (y) (22). However, these have emerged as weak equalities R12 = d r + + , −2 δπρ (r) δθ (r) δρ (r) 2 δπθ (r) on Γc, where substitution of the canonical Hamiltonian Z density (28) into Eqs. (34) and (35), yields, respectively, (43) Eqs. (18) and (19), also in the form of weak equalities. while pursuing a similar procedure for R21, leads to

3 δf (x) 1 δf (x) 1 δg (y) δg (y) Let us now proceed with the phase space reduction. This R21 = d r + . δρ (r) 2 δπ (r) 2 δπ (r) − δθ (r) may be achieved by appropriate implementation of Dirac Z θ ρ brackets instead of Poisson brackets. The Dirac bracket (44) of two phase space variables, is given by [22, 24–30] Hence, from Eqs. (36), (43) and (44), we ﬁnd that the Dirac bracket takes the form

f (x) ,g (y) = f (x) ,g (y) Rij , (36) { }D { }− 1 i,j f,g = f,g + f,g + f,g X { }D { }ρ,θ 2 { }ρ,πρ { }θ,πθ h 1 i where we have defined + f,g , (45) 4 { }πρ,πθ 3 3 ′ −1 ′ ′ Rij = d rd r f (x) , Ci (r) Q (r, r ) Cj (r ) ,g (y) , where we have used the antisymmetry of brackets under { } ij { } ZZ exchange of field variables, e.g. f,g = f,g . (37) { }ρ,θ −{ }θ,ρ ′ ′ From here, the Dirac brackets of the canonical field vari- and Qij (r, r ) = Ci (r) , Cj (r ) is a matrix with elements given by the{ Poisson brackets} of the constraints, ables may be read off, as which satisfies ρ (r) ,ρ (r′) = θ (r) ,θ (r′) =0, (46) { }D { }D 3 ′′ ′′ −1 ′′ ′ ′ d r Qij (r, r ) Q (r , r )= δikδ (r r ) . (38) jk − ′ ′ j πρ (r) , πρ (r ) = πθ (r) , πθ (r ) = 0 (47) X Z { }D { }D 5

′ ′ ρ (r) , πθ (r ) = θ (r) , πρ (r ) =0, (48) variables with respect to the old variables, should satisfy { }D { }D ′ ′ Qi (r) ,Qj (r ) = Pi (r) , Pj (r ) =0, ′ ′ 1 ′ { ′ } { ′ } ρ (r) , πρ (r ) D = θ (r) , πθ (r ) D = δ (r r ) , (49) Qi (r) , Pj (r ) = δij δ (r r ) . (53) { } { } 2 − { } − Given these observations, it is not difficult to see that the ′ 1 ′ πρ (r) , πθ (r ) = δ (r r ) , (50) transformation { }D 4 − ρ θ Q1 = ρ/2 πθ Q2 = πθ + ρ/2 − , ′ ′ πρ πθ → P1 = θ/2+ πρ P2 = πρ θ/2 ρ (r) ,θ (r ) D = δ (r r ) . (51) − { } − (54) The reduction of phase space to the physical degrees of transforms the second pair of conjugate variables into freedom of the system may be achieved by a suitable the constraints and is canonical. Let us examine the canonical transformation. In particular, we would like to form of the Dirac bracket (45) written in terms of the find a transformation new canonical variables. For this purpose, it is useful to express the functional derivatives with respect to the old ρ θ Q1 Q2 , (52) variables, in terms of the new ones. Using simple chain πρ πθ → P1 P2 rules, we find which transforms one of the set of conjugate variables, δ 1 δ δ δ 1 δ δ (θ, πθ) say, into the pair of constraints (26), such that = + , = , δρ 2 δQ1 δQ2 δθ 2 δP1 − δP2 Q2 = πθ + ρ/2 and P2 = πρ θ/2. The existence of − δ δ δ δ δ δ such a transformation is guaranteed by a theorem due to = + , = + . (55) Maskawa and Nakajima [31]. Under the transformation, δπρ δP1 δP2 δπθ −δQ1 δQ2 the Dirac bracket on the full phase space should reduce to the Poisson bracket on the reduced phase space [32], From the sign of each of these terms and the ordering of

so that f,g D = f,g Q1,P1 . In order for the transfor- the ﬁelds in the brackets appearing in the Dirac bracket mation to{ be} canonical,{ } the Poisson brackets of the new (45), each bracket contribution may be read oﬀ, as

1 f,g = f,g f,g + f,g f,g , (56) { }ρ,θ 4 { }Q1,P1 −{ }Q1,P2 { }Q2,P1 −{ }Q2,P2 1 1 h i f,g = f,g + f,g + f,g + f,g , (57) 2 { }ρ,πρ 4 { }Q1,P1 { }Q1,P2 { }Q2,P1 { }Q2,P2 1 1 h i f,g = f,g f,g f,g + f,g , (58) 2 { }θ,πθ 4 { }Q1,P1 −{ }Q1,P2 −{ }Q2,P1 { }Q2,P2 1 1 h i f,g = f,g + f,g f,g f,g . (59) 4 { }πρ,πθ 4 { }Q1,P1 { }Q1,P2 −{ }Q2,P1 −{ }Q2,P2 h i

Therefore, the Dirac bracket (45), reduces to ﬁeld equations

δH ρ˙ (r)= ρ (r) ,H = , (61) { }D δθ (r)

f,g D = f,g Q1,P1 , (60) δH { } { } θ˙ (r)= θ (r) ,H = . (62) { }D −δρ (r)

The canonical Hamiltonian density (28) becomes a strong namely, the Poisson bracket on the reduced phase space. equality on the reduced phase space, with Implementing the constraints (26) directly in transformation (54), gives Q1 = ρ and P1 = θ, so that the Dirac 2 2 (∇θ) g ~ 2 bracket reduces to the Poisson bracket from Eq. (23), = ρ + ρ + V + (∇ρ) . (63) where the reduced phase space of the system comprises H " 2m 2 # 8mρ the single pair of conjugate variables ρ and θ. Hence from Eq. (39), it follows that the time-evolution of the The wave equations generated by the canonical ﬁeld Eqs. conjugate pair of variables, is governed by the canonical (61) and (62) are, respectively, Eqs. (18) and (19). 6

2. The Faddeev-Jackiw method Thus, given the canonical Hamiltonian of a particular first order Lagrangian with linear functions i, the asso- A When the Lagrangian is at most first order in the time ciated canonical field equations may be obtained simply derivatives of the fields, the Faddeev-Jackiw method pro- by evaluating the inverse of the antisymmetric part of vides a more direct approach to the Dirac-Bergmann al- the constant matrix ω characterising the system. For the gorithm. In fact, the method was designed specifically particular case of the Gross-Pitaevskii field described by for such Lagrangians. Here we follow closely Jackiw’s the hydrodynamical Lagrangian density (17), let us de- paper “Quantization without tears” [34], which we tai- note the field components by φ1 = ρ and φ2 = θ. Then, lor to the specific system of interest. The essence of the in accordance with Eq. (67), it is clear that ω should Faddeev-Jackiw approach, follows from the observation solve the system of equations that the canonical momenta may be viewed as additional 1 positional variables subject to their own Euler-Lagrange 1 = (ρω11 + θω21)=0 (70) equations. In other words, the canonical equations of A 2 1 motion are equivalent to two Euler-Lagrange equations 2 = (ρω12 + θω22)= ρ, (71) [41]: A 2 − ∂L d ∂L ∂H so that = p˙i =0, (64) ∂qi − dt ∂q˙i − ∂qi − 0 2 ω = . (72) ∂L d ∂L ∂H 0− 0 =q ˙i =0. (65) ∂pi − dt ∂p˙i − ∂pi The antisymmetric part of this matrix, is Therefore, given a Hamiltonian description governed by H (q,p), it is always possible to construct a first order 0 1 ωA = , (73) Lagrangian 1− 0 L (q,p)= piq˙i H (q,p) , (66) − and should be identified with ω appearing in Eq. (69). i X Notice that dropping the symmetric part of ω is equiva- whose configuration space is identical to the Hamiltonian lent to performing transformation (24) on the Lagrangian phase space [34], where the Euler-Lagrange equations for density. From the inverse of Eq. (73), we find that the the Lagrangian L (q,p) coincide with the canonical equa- canonical equations (69) are equivalent to those of Eqs. tions associated with H (q,p). Hence, if the Lagrangian (21) and (22). For completeness, we note that Poisson presents itself in a first order form, one can readily iden- brackets are defined so as to reproduce the canonical tify the conjugate pairs of variables from the linear form equations through Poisson commutation with the Hamil- in the velocities. Let us examine how this works in prac- tonian, so that [34] tice for the present problem. Consider the situation where the functions i from Eq. 3 δf 1 δg A f,g = d r ω− , (74) (9) are linear in the fields, such that { } δφ ij δφ i,j Z i j 1 X i = φj ωji, (67) A 2 which again yields expression (23) as the Poisson bracket. j X where ω is a matrix of constant coefficients. Notice that the Lagrangian density (17) of the Gross-Pitaevskii field V. CONCLUDING REMARKS appears in this form. In such instances, the Lagrangian density may be presented as We have derived a hydrodynamic canonical formalism 1 for the Gross-Pitaevskii field and eliminated redundant = φj ωjiφ˙i . (68) L 2 − H variables. The physically relevant conjugate variables are i,j X the density and the phase of the condensate, which form As a further observation, we note that the symmetric a conjugate pair. These results were obtained using both part of ω is equivalent to a total time derivative in and, the Dirac-Bergmann and Faddeev-Jackiw methods. The L ˙ therefore, may be discarded [34]. Indeed, i,j φj ωij φi = Faddeev-Jackiw method is a more direct approach, which lays out the problem in a much simpler form. For in- ∂t i,j φj ωij φi/2 when ωij = ωji. Hence, only the antisymmetric part of ω should be retained.P Taking this stance, in addition to being more computationally de- intoP account, the Euler-Lagrange equations for a field de- manding, the Dirac-Bergmann method relied on the ad- scribed by a Lagrangian density of the form (68), are dition of a suitably chosen total time derivative to the equivalent to the following system of equations: Lagrangian, in order to easily identify, at a later point, a canonical transformation which separates out the physi- ˙ −1 δH cal conjugate pair of variables from the pair of constraints. φj = ωjk . (69) δφk k In the Faddeev-Jackiw method, the same transformation X 7 arises by retaining the anti-symmetric part of ω, which is appeared in the Faddeev-Jackiw method, highlighting assumed in the procedure. Furthermore, no constraints the non-physical character of the constraints which appeared in the Dirac-Bergmann method.

[1] Mike H Anderson, Jason R Ensher, Michael R Matthews, [20] Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Carl E Wieman, and Eric A Cornell. Observation of Bose- Grynberg, and Patricia Thickstun. Atom-photon inter- Einstein condensation in a dilute atomic vapor. science, actions: basic processes and applications. Wiley Online 269(5221):198–201, 1995. Library, 1992. [2] Kendall B Davis, M-O Mewes, Michael R Andrews, [21] Marc Henneaux and Claudio Teitelboim. Quantization NJ Van Druten, DS Durfee, DM Kurn, and Wolfgang of gauge systems. Princeton university press, 1992. Ketterle. Bose-Einstein condensation in a gas of sodium [22] Kurt Sundermeyer. Constrained dynamics with applica- atoms. Physical Review Letters, 75(22):3969, 1995. tions to Yang-Mills theory, general relativity, classical [3] Cl C Bradley, CA Sackett, JJ Tollett, and Randall G spin, dual string model. Springer-Verlag, Berlin, 1982. Hulet. Evidence of Bose-Einstein condensation in an [23] J Barcelos-Neto and C Wotzasek. Symplectic quantiza- atomic gas with attractive interactions. Physical Review tion of constrained systems. Modern Physics Letters A, Letters, 75(9):1687, 1995. 7(19):1737–1747, 1992. [4] Lev Pitaevskii and Sandro Stringari. Bose-Einstein con- [24] Paul AM Dirac. Generalized hamiltonian dynamics. Pro- densation and superfluidity, volume 164. Oxford Univer- ceedings of the Royal Society of London A: Mathemat- sity Press, 2016. ical, Physical and Engineering Sciences, 246(1246):326– [5] E Madelung. Quantum theory in hydrodynamic form. Z. 332, 1958. Phys., 40:322–326, 1926. [25] Paul Adrien Maurice Dirac. Lectures on quantum me- [6] Fritz London. Superfluids. Wiley, 1954. chanics, volume 2. Courier Corporation, 2001. [7] Christopher J Pethick and Henrik Smith. Bose-Einstein [26] Ashok Das. Lectures on quantum field theory. World condensation in dilute gases. Cambridge university press, Scientific, 2008. 2002. [27] Dmitri Gitman and Igor V Tyutin. Quantization of fields [8] VE Zakharov. Hamiltonian formalism for waves in non- with constraints. Springer Science & Business Media, linear media with dispersion. Nauchnaia Shkola po Ne- 2012. lineinym Kolebaniiam i Volnam v Raspredelennykh Sis- [28] Heinz J Rothe and Klaus Dieter Rothe. Classical and temakh, 2 nd, Gorki, USSR, Mar. 1973.) Radiofizika, quantum dynamics of constrained Hamiltonian systems, 17(4):431–453, 1974. volume 81. World Scientific, 2010. [9] V Zakharov and EA Kuznetsov. Hamiltonian formalism [29] AndréBurnel. Canonical quantization for constrained for nonlinear waves. Physics-Uspekhi, 40(11):1087–1116, systems. In Noncovariant Gauges in Canonical Formal- 1997. ism, pages 1–23. Springer, 2009. [10] Maxim Olshanii, Hélène Perrin, and Vincent Lorent. Ex- [30] Parthasarathi Mitra. Symmetries and symmetry breaking ample of a quantum anomaly in the physics of ultracold in field theory. CRC Press, 2014. gases. Phys. Rev. Lett., 105:095302, Aug 2010. [31] Toshihide Maskawa and Hideo Nakajima. Singular La- [11] VL Pokrovskii and M Khalatnikov. Hamiltonian formal- grangian and the Dirac-Faddeev method: Existence the- ism in classical and two-fluid. JETP Lett, 23(11), 1976. orem of constraints in ’standard form’. Progress of The- [12] IM Khalatnikov and VV Lebedev. Canonical equations oretical Physics, 56(4):1295–1309, 1976. of hydrodynamics of quantum liquids. Journal of Low [32] Steven Weinberg. The quantum theory of fields. Vol. 1: Temperature Physics, 32(5-6):789–801, 1978. Foundations. Cambridge University Press, 1995. [13] IM Khalatnikov. Hydrodynamics of classical and quan- [33] L Faddeev and R Jackiw. Hamiltonian reduction of un- tum liquid with free surface. Journal of Low Temperature constrained and constrained systems. Physical Review Physics, 82(1-2):93–101, 1991. Letters, 60(17):1692, 1988. [14] Yakovic Ilich Frenkel. Wave mechanics: advanced general [34] R Jackiw. Quantization without tears, in constraint the- theory. ory and quantization methods, 1994. [15] Paul Adrien Maurice Dirac. The principles of quantum [35] Peter D Drummond and Mark Hillery. The quantum mechanics. Number 27. Oxford university press, 1981. theory of nonlinear optics. Cambridge University Press, [16] Leonard I Schiff. Quantum Mechanics 3rd. New York: 2014. McGraw-Hill, 1968. [36] Harald JW Mãller Kirsten. Introduction to Quantum Me- [17] CG Gray, G Karl, and VA Novikov. Progress in classical chanics: Schrödinger Equation and Path Integral. World and quantum variational principles. Reports on Progress Scientific Publishing Company, 2006. in Physics, 67(2):159, 2004. [37] AK Kerman and SE Koonin. Hamiltonian formulation of [18] E Deumens, A Diz, R Longo, and Y Ohrn.¨ Time- time-dependent variational principles for the many-body dependent theoretical treatments of the dynamics of elec- system. Annals of Physics, 100(1-2):332–358, 1976. trons and nuclei in molecular systems. Reviews of Modern [38] Ernest-M Henley and Walter Thirring. Elementary quan- Physics, 66(3):917, 1994. tum field theory. McGraw-Hill Book Co., Inc., PWN- [19] AndréMercier. Analytical and canonical formalism in Polish Scientific Publishers, 1959. physics. Courier Corporation, 2004. [39] László A´ Gergely. On Hamiltonian formulations of the Schrödinger system. Annals of Physics, 298(2):394–402, 8

2002. [41] Cornelius Lanczos. The variational principles of mechan- [40] David Bohm. A suggested interpretation of the quantum ics. Courier Corporation, 2012. theory in terms of” hidden” variables. i. Physical Review, 85(2):166, 1952.