4028:, and therefore anything that weakly vanishes must be strongly equal to a linear combination of the constraints. One can demonstrate that the Poisson bracket of two first-class quantities must also be first class. The first-class constraints are intimately connected with the unphysical degrees of freedom mentioned earlier. Namely, the number of independent first-class constraints is equal to the number of unphysical degrees of freedom, and furthermore, the primary first-class constraints generate gauge transformations. Dirac further postulated that all secondary first-class constraints are generators of gauge transformations, which turns out to be false; however, typically one operates under the assumption that all first-class constraints generate gauge transformations when using this treatment.
3092:. Upon finding the secondary constraint one should add it to the extended Hamiltonian and check the new consistency conditions, which may result in still more constraints. Iterate this process until there are no more constraints. The distinction between primary and secondary constraints is largely an artificial one (i.e. a constraint for the same system can be primary or secondary depending on the Lagrangian), so this article does not distinguish between them from here on. Assuming the consistency condition has been iterated until all of the constraints have been found, then
2889:(functions of the velocity) exist; this causes no problems since the contribution weakly vanishes. Now, there are some consistency conditions which must be satisfied in order for this formalism to make sense. If the constraints are going to be satisfied, then their equations of motion must weakly vanish, that is, we require
6266:
4281:
are constraints that must vanish on physical states, whereas the right-hand side cannot vanish. This example illustrates the need for some generalization of the
Poisson bracket which respects the system's constraints, and which leads to a consistent quantization procedure. This new bracket should be
6462:
If one always uses the Dirac bracket instead of the
Poisson bracket, then there is no issue about the order of applying constraints and evaluating expressions, since the Dirac bracket of anything weakly zero is strongly equal to zero. This means that one can just use the naive Hamiltonian with Dirac
3931:
Above is everything needed to find the equations of motion in Dirac's modified
Hamiltonian procedure. Having the equations of motion, however, is not the endpoint for theoretical considerations. If one wants to canonically quantize a general system, then one needs the Dirac brackets. Before defining
3922:
Later, the extended
Hamiltonian is introduced. For gauge-invariant (physically measurable quantities) quantities, all of the Hamiltonians should give the same time evolution, since they are all weakly equivalent. It is only for non gauge-invariant quantities that the distinction becomes important.
1573:
The new procedure works as follows, start with a
Lagrangian and define the canonical momenta in the usual way. Some of those definitions may not be invertible and instead give a constraint in phase space (as above). Constraints derived in this way or imposed from the beginning of the problem are
1898:
2039:
5424:
5252:
5651:
1115:, which is one of the conditions under which the standard Hamiltonian procedure breaks down. While this example has been motivated as an approximation, the Lagrangian under consideration is legitimate and leads to consistent equations of motion in the Lagrangian formalism.
4631:
3103:
will index all of them. Note this article uses secondary constraint to mean any constraint that was not initially in the problem or derived from the definition of canonical momenta; some authors distinguish between secondary constraints, tertiary constraints, et cetera.
2517:
2641:
6457:
5052:
3029:
1768:, consider how one gets the equations of motion from the naive Hamiltonian in the standard procedure. One expands the variation of the Hamiltonian out in two ways and sets them equal (using a somewhat abbreviated notation with suppressed indices and sums):
621:
3129:
are not completely determined, then that means there are unphysical (gauge) degrees of freedom in the system. Once all of the constraints (primary and secondary) are added to the naive
Hamiltonian and the solutions to the consistency conditions for the
1514:
to the
Lagrangian to account for them. The extra terms vanish when the constraints are satisfied, thereby forcing the path of stationary action to be on the constraint surface. In this case, going to the Hamiltonian formalism introduces a constraint on
1493:
dimensions, that is sometimes expressing the coordinates as momenta and sometimes as coordinates. However, this is neither a general nor rigorous solution. This gets to the heart of the matter: that the definition of the canonical momenta implies a
2873:
3475:(which is the same as the number of constraints) minus the number of consistency conditions of the fourth type (in previous subsection). This is the number of unphysical degrees of freedom in the system. Labeling the linear independent solutions
280:
6114:
4873:
3291:
The above equation must possess at least one solution, since otherwise the initial
Lagrangian is inconsistent; however, in systems with gauge degrees of freedom, the solution will not be unique. The most general solution is of the form
6103:
6017:
3286:
809:
715:
1903:
where the second equality holds after simplifying with the Euler-Lagrange equations of motion and the definition of canonical momentum. From this equality, one deduces the equations of motion in the
Hamiltonian formalism from
1299:
3767:
1207:
2371:
2283:
1774:
5868:
5792:
1106:
1910:
1458:
5719:
4419:
1029:
2155:
1304:
which are unusual in that they are not invertible to the velocities; instead, they are constrained to be functions of the coordinates: the four phase-space variables are linearly dependent, so the variable basis is
5258:
7663:
5086:
4667:
It is straightforward to check that the above definition of the Dirac bracket satisfies all of the desired properties, and especially the last one, of vanishing for an argument which is a second-class constraint.
443:
direction. Here, the hats indicate unit vectors. Later in the article, however, they are used to distinguish quantum mechanical operators from their classical analogs. The usage should be clear from the context.
3459:
956:
7557:
430:
6630:
58:; specifically, when constraints are at hand, so that the number of apparent variables exceeds that of dynamical ones. More abstractly, the two-form implied from the Dirac bracket is the restriction of the
7880:
5478:
7291:
7400:
4062:. The extended Hamiltonian gives the most general possible time evolution for any gauge-dependent quantities, and may actually generalize the equations of motion from those of the Lagrangian formalism.
3576:
1712:
7773:
6721:
4438:
2396:
2523:
2044:
where the weak equality symbol is no longer displayed explicitly, since by definition the equations of motion only hold weakly. In the present context, one cannot simply set the coefficients of
3917:
3840:
7447:
s through the two constraints ab initio, which would obey plain
Poisson brackets. The Dirac brackets add simplicity and elegance, at the cost of excessive (constrained) phase-space variables.
4163:
3464:
The most general solution will be a linear combination of linearly independent solutions to the above homogeneous equation. The number of linearly independent solutions equals the number of
4008:
6530:
7035:
1728:
are not constants but functions of the coordinates and momenta. Since this new Hamiltonian is the most general function of coordinates and momenta weakly equal to the naive Hamiltonian,
7950:
6292:
4884:
2895:
457:
6810:
7192:
6927:
4685:. Since the Dirac bracket respects the constraints, one need not be careful about evaluating all brackets before using any weak equations, as is the case with the Poisson bracket.
4323:
3349:
2728:
2693:
161:
6261:{\displaystyle M^{-1}={\frac {c}{qB}}\left({\begin{matrix}0&-1\\1&0\end{matrix}}\right)\quad \Rightarrow \quad M_{ab}^{-1}=-{\frac {c}{qB_{0}}}\varepsilon _{ab},}
4257:
4746:
5451:. Therefore, there are no secondary constraints and the arbitrary coefficients are completely determined, indicating that there are no unphysical degrees of freedom.
4056:
3633:
3606:
4752:
6028:
5900:
3174:
723:
4282:
bilinear, antisymmetric, satisfy the Jacobi identity as does the Poisson bracket, reduce to the Poisson bracket for unconstrained systems, and, additionally,
2056:
separately to zero, since the variations are somewhat restricted by the constraints. In particular, the variations must be tangent to the constraint surface.
632:
1893:{\displaystyle \delta H={\frac {\partial H}{\partial q}}\delta q+{\frac {\partial H}{\partial p}}\delta p\approx {\dot {q}}\delta p-{\dot {p}}\delta q~,}
1213:
3652:
1124:
2289:
2204:
5798:
2034:{\displaystyle \left({\frac {\partial H}{\partial q}}+{\dot {p}}\right)\delta q+\left({\frac {\partial H}{\partial p}}-{\dot {q}}\right)\delta p=0~,}
6466:
To quantize the system, the Dirac brackets between all of the phase space variables are needed. The nonvanishing Dirac brackets for this system are
5725:
4688:
Note that while the Poisson bracket of bosonic (Grassmann even) variables with itself must vanish, the Poisson bracket of fermions represented as a
1616:, in the usual way via a Legendre transformation, exactly as in the above example. Note that the Hamiltonian can always be written as a function of
1035:
1322:
5657:
5419:{\displaystyle \{\phi _{2},H\}_{PB}+\sum _{j}u_{j}\{\phi _{2},\phi _{j}\}_{PB}=-{\frac {\partial V}{\partial y}}-u_{1}{\frac {qB}{c}}\approx 0.}
4331:
967:
5247:{\displaystyle \{\phi _{1},H\}_{PB}+\sum _{j}u_{j}\{\phi _{1},\phi _{j}\}_{PB}=-{\frac {\partial V}{\partial x}}+u_{2}{\frac {qB}{c}}\approx 0}
2065:
7568:
4174:
times their classical Poisson bracket. Assuming there are no ordering issues that give rise to new quantum corrections, this implies that
3382:
2710:
The Legendre transform between the Lagrangian formalism and the Hamiltonian formalism has been saved at the cost of adding new variables.
852:
7482:
354:
6536:
5646:{\displaystyle {\dot {x}}=\{x,H\}_{PB}+u_{1}\{x,\phi _{1}\}_{PB}+u_{2}\{x,\phi _{2}\}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}}
8256:
4168:
Now, suppose one wishes to employ canonical quantization, then the phase-space coordinates become operators whose commutators become
8194:
8114:
8073:
7779:
7198:
7297:
3500:
4626:{\displaystyle \{f,g\}_{DB}=\{f,g\}_{PB}-\sum _{a,b}\{f,{\tilde {\phi }}_{a}\}_{PB}M_{ab}^{-1}\{{\tilde {\phi }}_{b},g\}_{PB}~,}
2512:{\displaystyle {\dot {p}}_{j}=-{\frac {\partial H}{\partial q_{j}}}-\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial q_{j}}}}
1642:
2636:{\displaystyle {\dot {q}}_{j}={\frac {\partial H}{\partial p_{j}}}+\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial p_{j}}}}
8266:
8093:
7688:
6641:
2707:
are functions of coordinates and velocities that can be determined, in principle, from the second equation of motion above.
3855:
8261:
4069:. Second class constraints are constraints that have a nonvanishing Poisson bracket with at least one other constraint.
1470:
The Hamiltonian procedure has broken down. One might try to fix the problem by eliminating two of the components of the
1636:
Dirac argues that we should generalize the Hamiltonian (somewhat analogously to the method of Lagrange multipliers) to
93:
When the Lagrangian is at most linear in the velocity of at least one coordinate; in which case, the definition of the
4100:
3955:
3949:
of coordinates and momenta first class if its Poisson bracket with all of the constraints weakly vanishes, that is,
6472:
6452:{\displaystyle \{f,g\}_{DB}=\{f,g\}_{PB}+{\frac {c\varepsilon _{ab}}{qB}}\{f,\phi _{a}\}_{PB}\{\phi _{b},g\}_{PB}.}
5047:{\displaystyle H^{*}=V(x,y)+u_{1}\left(p_{x}+{\tfrac {qB}{2c}}y\right)+u_{2}\left(p_{y}-{\tfrac {qB}{2c}}x\right).}
3778:
3024:{\displaystyle {\dot {\phi _{j}}}\approx \{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.}
3494:
to the number of unphysical degrees of freedom, the general solution to the consistency conditions is of the form
616:{\displaystyle L={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})+{\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,}
8068:. Belfer Graduate School of Science Monographs Series. Vol. 2. Belfer Graduate School of Science, New York.
7991:
6938:
7891:
6463:
brackets, instead, to thus get the correct equations of motion, which one can easily confirm on the above ones.
7986:
101:. This is the most frequent reason to resort to Dirac brackets. For instance, the Lagrangian (density) for any
39:
6736:
7132:
81:. Details of Dirac's modified Hamiltonian formalism are also summarized to put the Dirac bracket in context.
7071:
6816:
1313:
1118:
Following the Hamiltonian procedure, however, the canonical momenta associated with the coordinates are now
7981:
7961:
6727:
4672:
4292:
4066:
4058:
as the first-class primary constraints are added to arrive at the total Hamiltonian, then one obtains the
2868:{\displaystyle {\dot {f}}\approx \{f,H^{*}\}_{PB}\approx \{f,H\}_{PB}+\sum _{k}u_{k}\{f,\phi _{k}\}_{PB},}
78:
47:
7966:
7075:
3298:
275:{\displaystyle L={\tfrac {1}{2}}m{\vec {v}}^{2}+{\frac {q}{c}}{\vec {A}}\cdot {\vec {v}}-V({\vec {r}}),}
74:
51:
43:
2647:
8134:; Trugenberger, C. (1991). "Self-dual Chern-Simons solitons and two-dimensional nonlinear equations".
3077:
The first case indicates that the starting Lagrangian gives inconsistent equations of motion, such as
8224:
8143:
7996:
3088:
3086:
The third case gives new constraints in phase space. A constraint derived in this manner is called a
1511:
1507:
448:
94:
70:
55:
3635:
corresponds to a gauge transformation, and should leave the physical state of the system unchanged.
1556:
are equal independently of the constraints being satisfied, they are called strongly equal, written
3048:
An equation that places new constraints on our coordinates and momenta, but is independent of the
125:
59:
8044:
6283:
4265:
On one hand, canonical quantization gives the above commutation relation, but on the other hand
4180:
144:
plane with a strong constant, homogeneous perpendicular magnetic field, so then pointing in the
89:
The standard development of Hamiltonian mechanics is inadequate in several specific situations:
7115:. From a plain kinetic Lagrangian, it is evident that their momenta are perpendicular to them,
8190:
8159:
8110:
8089:
8069:
4710:
4868:{\displaystyle \phi _{1}=p_{x}+{\tfrac {qB}{2c}}y,\qquad \phi _{2}=p_{y}-{\tfrac {qB}{2c}}x.}
8232:
8151:
8034:
8001:
6098:{\displaystyle M={\frac {qB}{c}}\left({\begin{matrix}0&1\\-1&0\end{matrix}}\right),}
4689:
2195:
1306:
301:
20:
8083:
6012:{\displaystyle \{\phi _{1},\phi _{2}\}_{PB}=-\{\phi _{2},\phi _{1}\}_{PB}={\frac {qB}{c}},}
4034:
3611:
3584:
8079:
7971:
4704:
Returning to the above example, the naive Hamiltonian and the two primary constraints are
4031:
When the first-class secondary constraints are added into the Hamiltonian with arbitrary
3045:
An equation that is identically true, possibly after using one of our primary constraints.
31:
3281:{\displaystyle \{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.}
804:{\displaystyle m{\ddot {y}}=-{\frac {\partial V}{\partial y}}-{\frac {qB}{c}}{\dot {x}}.}
8228:
8215:
Corrigan, E.; Zachos, C. K. (1979). "Non-local charges for the supersymmetric σ-model".
8147:
710:{\displaystyle m{\ddot {x}}=-{\frac {\partial V}{\partial x}}+{\frac {qB}{c}}{\dot {y}}}
4675:
on a constrained Hamiltonian system, the commutator of the operators is supplanted by
340:
is an arbitrary external scalar potential; one could easily take it to be quadratic in
4065:
For the purposes of introducing the Dirac bracket, of more immediate interest are the
8250:
8236:
8048:
7976:
2718:
The equations of motion become more compact when using the Poisson bracket, since if
16:
Quantization method for constrained Hamiltonian systems with second-class constraints
8127:
1294:{\displaystyle p_{y}={\frac {\partial L}{\partial {\dot {y}}}}={\frac {qB}{2c}}x~,}
846:. One may then drop the kinetic term to produce a simple approximate Lagrangian,
435:
as our vector potential; this corresponds to a uniform and constant magnetic field
109:
3762:{\displaystyle H_{T}=H+\sum _{k}U_{k}\phi _{k}+\sum _{a,k}v_{a}V_{k}^{a}\phi _{k}}
1202:{\displaystyle p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=-{\frac {qB}{2c}}y}
8063:
2366:{\displaystyle B_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial p_{n}}},}
1628:
s only, even if the velocities cannot be inverted into functions of the momenta.
8131:
5873:
which are self-consistent and coincide with the Lagrangian equations of motion.
2278:{\displaystyle A_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial q_{n}}}}
1467:, which means that equations of motion (Hamilton's equations) are inconsistent.
63:
5863:{\displaystyle {\dot {p}}_{y}=-{\frac {1}{2}}{\frac {\partial V}{\partial y}},}
1568:
no weak equations may be used before evaluating derivatives or Poisson brackets
5787:{\displaystyle {\dot {p}}_{x}=-{\frac {1}{2}}{\frac {\partial V}{\partial x}}}
4284:
the bracket of any second-class constraint with any other quantity must vanish
115:
When there are any other constraints that one wishes to impose in phase space.
35:
8155:
3034:
There are four different types of conditions that can result from the above:
1101:{\displaystyle {\dot {x}}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}~.}
7885:
whence, instantly, virtually by inspection, oscillation for both variables,
1453:{\displaystyle H(x,y,p_{x},p_{y})={\dot {x}}p_{x}+{\dot {y}}p_{y}-L=V(x,y).}
8163:
8039:
8022:
155:
The Lagrangian for this system with an appropriate choice of parameters is
7126:. Thus the corresponding Dirac Brackets are likewise simple to work out,
5714:{\displaystyle {\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}}
4414:{\displaystyle M_{ab}=\{{\tilde {\phi }}_{a},{\tilde {\phi }}_{b}\}_{PB}.}
1024:{\displaystyle {\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}}
4262:
where the hats emphasize the fact that the constraints are on operators.
4017:. Note that the only quantities that weakly vanish are the constraints
2150:{\displaystyle \sum _{n}A_{n}\delta q_{n}+\sum _{n}B_{n}\delta p_{n}=0,}
3608:
are completely arbitrary functions of time. A different choice of the
102:
7658:{\displaystyle {\ddot {z}}=-z{\frac {{\dot {z}}^{2}}{1-z^{2}}}=-z2E~,}
4692:
with itself need not vanish. This means that in the fermionic case it
1540:
when the constraints are satisfied, but not throughout the phase space
1498:(between momenta and coordinates) that was never taken into account.
8105:
See pages 48-58 of Ch. 2 in Henneaux, Marc and Teitelboim, Claudio,
4696:
possible for there to be an odd number of second class constraints.
3454:{\displaystyle \sum _{k}V_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.}
951:{\displaystyle L={\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,}
7552:{\displaystyle L={\frac {1}{2}}{\frac {{\dot {z}}^{2}}{1-z^{2}}}~,}
1734:
is the broadest generalization of the Hamiltonian possible so that
4424:
In this case, the Dirac bracket of two functions on phase space,
1566:. It is important to note that, in order to get the right answer,
425:{\displaystyle {\vec {A}}={\frac {B}{2}}(x{\hat {y}}-y{\hat {x}})}
6625:{\displaystyle \{x,p_{x}\}_{DB}=\{y,p_{y}\}_{DB}={\tfrac {1}{2}}}
7668:
an oscillation; whereas the equivalent constrained system with
112:(or other unphysical) degrees of freedom which need to be fixed.
3168:
must solve a set of inhomogeneous linear equations of the form
7875:{\displaystyle {\dot {p}}^{i}=\{p^{i},H\}_{DB}=-x^{i}~p^{2}~,}
7074:. (Since the two coordinates do not commute, there will be an
7286:{\displaystyle \{x_{i},p_{j}\}_{DB}=\delta _{ij}-x_{i}x_{j},}
1522:
Before proceeding, it is useful to understand the notions of
7395:{\displaystyle \{p_{i},p_{j}\}_{DB}=x_{j}p_{i}-x_{i}p_{j}~.}
4289:
At this point, the second class constraints will be labeled
3571:{\displaystyle u_{k}\approx U_{k}+\sum _{a}v_{a}V_{k}^{a},}
3376:
is the most general solution to the homogeneous equation
1707:{\displaystyle H^{*}=H+\sum _{j}c_{j}\phi _{j}\approx H,}
8176:
See page 8 in Henneaux and Teitelboim in the references.
7768:{\displaystyle {\dot {x}}^{i}=\{x^{i},H\}_{DB}=p^{i}~,}
7439:
unconstrained variables, had one eliminated one of the
6716:{\displaystyle \{p_{x},p_{y}\}_{DB}=-{\frac {qB}{4c}}.}
1519:
in Hamiltonian mechanics, but the solution is similar.
6611:
6154:
6058:
5012:
4951:
4838:
4783:
3845:
The time evolution of a function on the phase space,
172:
7894:
7782:
7691:
7571:
7485:
7300:
7201:
7135:
6941:
6819:
6739:
6644:
6539:
6475:
6295:
6117:
6031:
5903:
5801:
5728:
5660:
5481:
5261:
5089:
5057:
The next step is to apply the consistency conditions
4887:
4755:
4713:
4441:
4334:
4295:
4183:
4103:
4037:
3958:
3858:
3781:
3655:
3614:
3587:
3503:
3385:
3301:
3177:
2898:
2731:
2722:
is some function of the coordinates and momenta then
2650:
2526:
2399:
2292:
2207:
2068:
1913:
1777:
1645:
1325:
1216:
1127:
1038:
970:
855:
726:
635:
460:
357:
164:
7476:
from the circle constraint yields the unconstrained
7040:This example has a nonvanishing commutator between
5472:, then one can see that the equations of motion are
3083:. The second case does not contribute anything new.
50:. It is an important part of Dirac's development of
4878:Therefore, the extended Hamiltonian can be written
3912:{\displaystyle {\dot {f}}\approx \{f,H_{T}\}_{PB}.}
835:Now, in the limit of a very large magnetic field,
69:This article assumes familiarity with the standard
7944:
7874:
7767:
7657:
7551:
7394:
7285:
7186:
7029:
6921:
6804:
6715:
6624:
6524:
6451:
6260:
6097:
6011:
5862:
5786:
5713:
5645:
5418:
5246:
5046:
4867:
4740:
4625:
4413:
4317:
4251:
4157:
4072:For instance, consider second-class constraints
4050:
4002:
3911:
3834:
3761:
3627:
3600:
3570:
3453:
3343:
3280:
3023:
2867:
2687:
2635:
2511:
2390:Using this result, the equations of motion become
2365:
2277:
2149:
2033:
1892:
1706:
1452:
1293:
1201:
1100:
1023:
950:
803:
709:
615:
424:
274:
2878:if one assumes that the Poisson bracket with the
5433:secondary constraints, but conditions that fix
4158:{\displaystyle \{\phi _{1},\phi _{2}\}_{PB}=c~.}
85:Inadequacy of the standard Hamiltonian procedure
8189:, Volume 1. Cambridge University Press, 1995.
4003:{\displaystyle \{f,\phi _{j}\}_{PB}\approx 0,}
3643:At this point, it is natural to introduce the
3038:An equation that is inherently false, such as
6525:{\displaystyle \{x,y\}_{DB}=-{\frac {c}{qB}}}
6286:. Thus, the Dirac brackets are defined to be
3835:{\displaystyle H'=H+\sum _{k}U_{k}\phi _{k}.}
8:
7825:
7805:
7734:
7714:
7328:
7301:
7229:
7202:
7163:
7136:
7094:Similarly, for free motion on a hypersphere
6672:
6645:
6595:
6575:
6560:
6540:
6489:
6476:
6434:
6414:
6402:
6382:
6337:
6324:
6309:
6296:
5976:
5949:
5931:
5904:
5599:
5580:
5555:
5535:
5510:
5497:
5344:
5317:
5282:
5262:
5172:
5145:
5110:
5090:
4605:
4576:
4543:
4514:
4483:
4470:
4455:
4442:
4396:
4351:
4131:
4104:
4090:whose Poisson bracket is simply a constant,
3979:
3959:
3894:
3874:
3433:
3406:
3260:
3233:
3198:
3178:
3003:
2976:
2941:
2921:
2850:
2830:
2795:
2782:
2767:
2747:
7030:{\displaystyle =-i{\frac {\hbar qB}{4c}}~.}
1506:In Lagrangian mechanics, if the system has
7945:{\displaystyle {\ddot {x}}^{i}=-x^{i}2E~.}
7450:For example, for free motion on a circle,
814:For a harmonic potential, the gradient of
120:Example of a Lagrangian linear in velocity
8205:See Henneaux and Teitelboim, pages 18-19.
8038:
7924:
7908:
7897:
7896:
7893:
7860:
7847:
7828:
7812:
7796:
7785:
7784:
7781:
7753:
7737:
7721:
7705:
7694:
7693:
7690:
7625:
7608:
7597:
7596:
7593:
7573:
7572:
7570:
7534:
7517:
7506:
7505:
7502:
7492:
7484:
7380:
7370:
7357:
7347:
7331:
7321:
7308:
7299:
7274:
7264:
7248:
7232:
7222:
7209:
7200:
7166:
7156:
7143:
7134:
7070:, which means this structure specifies a
6998:
6980:
6969:
6968:
6958:
6947:
6946:
6940:
6909:
6894:
6883:
6882:
6867:
6866:
6851:
6840:
6839:
6824:
6823:
6818:
6782:
6759:
6758:
6744:
6743:
6738:
6726:Therefore, the correct implementation of
6690:
6675:
6665:
6652:
6643:
6610:
6598:
6588:
6563:
6553:
6538:
6507:
6492:
6474:
6437:
6421:
6405:
6395:
6362:
6352:
6340:
6312:
6294:
6246:
6233:
6220:
6205:
6197:
6153:
6134:
6122:
6116:
6057:
6038:
6030:
5991:
5979:
5969:
5956:
5934:
5924:
5911:
5902:
5837:
5827:
5815:
5804:
5803:
5800:
5764:
5754:
5742:
5731:
5730:
5727:
5691:
5676:
5662:
5661:
5659:
5623:
5608:
5593:
5574:
5558:
5548:
5529:
5513:
5483:
5482:
5480:
5395:
5389:
5362:
5347:
5337:
5324:
5311:
5301:
5285:
5269:
5260:
5223:
5217:
5190:
5175:
5165:
5152:
5139:
5129:
5113:
5097:
5088:
5011:
5002:
4987:
4950:
4941:
4926:
4892:
4886:
4837:
4828:
4815:
4782:
4773:
4760:
4754:
4712:
4608:
4592:
4581:
4580:
4567:
4559:
4546:
4536:
4525:
4524:
4502:
4486:
4458:
4440:
4399:
4389:
4378:
4377:
4367:
4356:
4355:
4339:
4333:
4309:
4298:
4297:
4294:
4222:
4211:
4210:
4200:
4189:
4188:
4182:
4134:
4124:
4111:
4102:
4042:
4036:
3982:
3972:
3957:
3897:
3887:
3860:
3859:
3857:
3823:
3813:
3803:
3780:
3753:
3743:
3738:
3728:
3712:
3699:
3689:
3679:
3660:
3654:
3619:
3613:
3592:
3586:
3559:
3554:
3544:
3534:
3521:
3508:
3502:
3436:
3426:
3413:
3400:
3390:
3384:
3332:
3319:
3306:
3300:
3263:
3253:
3240:
3227:
3217:
3201:
3185:
3176:
3006:
2996:
2983:
2970:
2960:
2944:
2928:
2906:
2900:
2899:
2897:
2853:
2843:
2824:
2814:
2798:
2770:
2760:
2733:
2732:
2730:
2655:
2649:
2624:
2609:
2599:
2593:
2583:
2567:
2549:
2540:
2529:
2528:
2525:
2500:
2485:
2475:
2469:
2459:
2443:
2425:
2413:
2402:
2401:
2398:
2351:
2336:
2326:
2320:
2310:
2297:
2291:
2266:
2251:
2241:
2235:
2225:
2212:
2206:
2132:
2119:
2109:
2096:
2083:
2073:
2067:
2059:One can demonstrate that the solution to
1997:
1996:
1973:
1943:
1942:
1919:
1912:
1867:
1866:
1846:
1845:
1816:
1787:
1776:
1689:
1679:
1669:
1650:
1644:
1414:
1399:
1398:
1389:
1374:
1373:
1361:
1348:
1324:
1262:
1245:
1244:
1230:
1221:
1215:
1176:
1156:
1155:
1141:
1132:
1126:
1111:Note that this approximate Lagrangian is
1072:
1057:
1040:
1039:
1037:
1001:
986:
972:
971:
969:
907:
906:
889:
888:
862:
854:
787:
786:
771:
748:
731:
730:
725:
696:
695:
680:
657:
640:
639:
634:
572:
571:
554:
553:
527:
515:
504:
503:
493:
482:
481:
467:
459:
408:
407:
390:
389:
373:
359:
358:
356:
255:
254:
234:
233:
219:
218:
208:
199:
188:
187:
171:
163:
6805:{\displaystyle =-i{\frac {\hbar c}{qB}}}
8013:
7187:{\displaystyle \{x_{i},x_{j}\}_{DB}=0,}
7001:
6911:
6785:
4237:
3118:. If, at the end of this process, the
3062:An equation that serves to specify the
2194:(assuming the constraints satisfy some
1463:Note that this "naive" Hamiltonian has
626:which leads to the equations of motion
7090:Further Illustration for a hypersphere
6922:{\displaystyle ==i{\frac {\hbar }{2}}}
6730:dictates the commutation relations,
4657:'s inverse matrix. Dirac proved that
3139:are plugged in, the result is called
3107:Finally, the last case helps fix the
1538:, are weakly equal if they are equal
961:with first-order equations of motion
348:, without loss of generality. We use
323:is the speed of light in vacuum; and
7:
8109:. Princeton University Press, 1992.
6932:with the cross terms vanishing, and
5876:A simple calculation confirms that
5454:If one plugs in with the values of
4700:Illustration on the example provided
4318:{\displaystyle {\tilde {\phi }}_{a}}
77:formalisms, and their connection to
46:, and to thus allow them to undergo
5894:are second class constraints since
3940:constraints need to be introduced.
1489:, down to a reduced phase space of
8023:"Generalized Hamiltonian dynamics"
7413:constrained phase-space variables
6635:while the cross-terms vanish, and
5848:
5840:
5775:
5767:
5702:
5694:
5634:
5626:
5373:
5365:
5201:
5193:
3344:{\displaystyle u_{k}=U_{k}+V_{k},}
2617:
2602:
2560:
2552:
2493:
2478:
2436:
2428:
2344:
2329:
2259:
2244:
1984:
1976:
1930:
1922:
1827:
1819:
1798:
1790:
1241:
1233:
1152:
1144:
1083:
1075:
1012:
1004:
759:
751:
668:
660:
14:
2688:{\displaystyle \phi _{j}(q,p)=0,}
1502:Generalized Hamiltonian procedure
820:amounts to just the coordinates,
54:to elegantly handle more general
1530:. Two functions on phase space,
38:to treat classical systems with
8027:Canadian Journal of Mathematics
6192:
6188:
4810:
4325:. Define a matrix with entries
23:, also known as Dirac notation.
6986:
6974:
6952:
6942:
6900:
6888:
6872:
6863:
6857:
6845:
6829:
6820:
6770:
6764:
6749:
6740:
6189:
4916:
4904:
4735:
4723:
4586:
4530:
4383:
4361:
4303:
4228:
4216:
4194:
4184:
3365:is a particular solution and
2673:
2661:
2182:restricted by the constraints
1474:-dimensional phase space, say
1444:
1432:
1367:
1329:
1316:then produces the Hamiltonian
939:
927:
918:
882:
604:
592:
583:
547:
521:
477:
419:
413:
395:
383:
364:
266:
260:
251:
239:
224:
193:
1:
8107:Quantization of Gauge Systems
8065:Lectures on quantum mechanics
7104:coordinates are constrained,
62:to the constraint surface in
8237:10.1016/0370-2693(79)90465-9
8187:The Quantum Theory of Fields
6108:which is easily inverted to
6022:hence the matrix looks like
5080:, which in this case become
1632:Generalizing the Hamiltonian
1465:no dependence on the momenta
4252:{\displaystyle =i\hbar ~c,}
1757:To further illuminate the
1578:. The constraints, labeled
30:is a generalization of the
8283:
8062:Dirac, Paul A. M. (1964).
1510:, then one generally adds
128:is a particle with charge
18:
8257:Mathematical quantization
7562:with equations of motion
4662:will always be invertible
2387:are arbitrary functions.
1496:constraint on phase space
148:-direction with strength
8156:10.1103/PhysRevD.43.1332
8021:Dirac, P. A. M. (1950).
7987:Second class constraints
4741:{\displaystyle H=V(x,y)}
4067:second class constraints
1113:linear in the velocities
304:for the magnetic field,
40:second class constraints
19:Not to be confused with
7072:noncommutative geometry
3147:Determination of the
1314:Legendre transformation
8040:10.4153/CJM-1950-012-1
7982:First class constraint
7962:Canonical quantization
7946:
7876:
7769:
7659:
7553:
7430:simpler Dirac brackets
7396:
7287:
7188:
7031:
6923:
6806:
6728:canonical quantization
6717:
6626:
6526:
6453:
6262:
6099:
6013:
5864:
5788:
5715:
5647:
5420:
5248:
5048:
4869:
4742:
4681:times their classical
4673:canonical quantization
4627:
4415:
4319:
4253:
4159:
4052:
4004:
3913:
3836:
3763:
3629:
3602:
3572:
3455:
3345:
3282:
3025:
2869:
2714:Consistency conditions
2689:
2637:
2513:
2367:
2279:
2151:
2035:
1894:
1708:
1589:, must weakly vanish,
1454:
1295:
1203:
1102:
1025:
952:
805:
711:
617:
426:
276:
79:canonical quantization
48:canonical quantization
8267:Hamiltonian mechanics
7967:Hamiltonian mechanics
7947:
7877:
7770:
7660:
7554:
7397:
7288:
7189:
7076:uncertainty principle
7032:
6924:
6807:
6718:
6627:
6527:
6454:
6263:
6100:
6014:
5865:
5789:
5716:
5648:
5421:
5249:
5049:
4870:
4743:
4628:
4416:
4320:
4254:
4160:
4053:
4051:{\displaystyle v_{a}}
4005:
3914:
3837:
3764:
3639:The total Hamiltonian
3630:
3628:{\displaystyle v_{a}}
3603:
3601:{\displaystyle v_{a}}
3573:
3456:
3346:
3283:
3141:the total Hamiltonian
3026:
2870:
2690:
2638:
2514:
2368:
2280:
2196:regularity conditions
2152:
2036:
1895:
1709:
1508:holonomic constraints
1455:
1296:
1204:
1103:
1026:
953:
806:
712:
618:
427:
277:
52:Hamiltonian mechanics
44:Hamiltonian mechanics
7997:Symplectic structure
7892:
7780:
7689:
7569:
7483:
7298:
7199:
7133:
6939:
6817:
6737:
6642:
6537:
6473:
6293:
6115:
6029:
5901:
5799:
5726:
5658:
5479:
5259:
5087:
4885:
4753:
4711:
4439:
4332:
4293:
4181:
4101:
4060:extended Hamiltonian
4035:
3956:
3856:
3779:
3772:and what is denoted
3653:
3612:
3585:
3501:
3383:
3299:
3175:
3089:secondary constraint
2896:
2729:
2648:
2524:
2397:
2290:
2205:
2066:
1911:
1775:
1643:
1608:Next, one finds the
1512:Lagrange multipliers
1323:
1214:
1125:
1036:
968:
853:
724:
633:
458:
355:
162:
8262:Symplectic geometry
8229:1979PhLB...88..273C
8148:1991PhRvD..43.1332D
6213:
4690:Grassmann variables
4575:
3943:We call a function
3748:
3564:
2160:for the variations
1576:primary constraints
126:classical mechanics
8185:Weinberg, Steven,
7942:
7872:
7765:
7655:
7549:
7392:
7283:
7184:
7027:
6919:
6802:
6713:
6622:
6620:
6522:
6449:
6284:Levi-Civita symbol
6258:
6193:
6182:
6095:
6086:
6009:
5860:
5784:
5711:
5643:
5416:
5306:
5244:
5134:
5044:
5031:
4970:
4865:
4857:
4802:
4738:
4623:
4555:
4513:
4411:
4315:
4249:
4155:
4048:
4000:
3909:
3832:
3808:
3759:
3734:
3723:
3684:
3625:
3598:
3568:
3550:
3539:
3451:
3395:
3341:
3278:
3222:
3021:
2965:
2865:
2819:
2685:
2633:
2588:
2509:
2464:
2363:
2315:
2275:
2230:
2147:
2114:
2078:
2031:
1890:
1704:
1674:
1450:
1291:
1199:
1098:
1021:
948:
801:
707:
613:
422:
272:
181:
95:canonical momentum
8217:Physics Letters B
8136:Physical Review D
7938:
7905:
7868:
7855:
7793:
7761:
7702:
7651:
7632:
7605:
7581:
7545:
7541:
7514:
7500:
7443:s and one of the
7388:
7023:
7019:
6977:
6955:
6917:
6891:
6875:
6848:
6832:
6800:
6767:
6752:
6708:
6619:
6520:
6380:
6240:
6147:
6051:
6004:
5855:
5835:
5812:
5782:
5762:
5739:
5709:
5689:
5670:
5641:
5621:
5491:
5408:
5380:
5297:
5236:
5208:
5125:
5030:
4969:
4856:
4801:
4619:
4589:
4533:
4498:
4386:
4364:
4306:
4242:
4219:
4197:
4151:
3927:The Dirac bracket
3868:
3849:, is governed by
3799:
3708:
3675:
3645:total Hamiltonian
3530:
3386:
3213:
2956:
2915:
2810:
2741:
2631:
2579:
2574:
2537:
2507:
2455:
2450:
2410:
2358:
2306:
2273:
2221:
2105:
2069:
2027:
2005:
1991:
1951:
1937:
1886:
1875:
1854:
1834:
1805:
1665:
1610:naive Hamiltonian
1407:
1382:
1287:
1280:
1257:
1253:
1194:
1168:
1164:
1094:
1090:
1070:
1048:
1019:
999:
980:
944:
915:
897:
880:
795:
784:
766:
739:
704:
693:
675:
648:
609:
580:
562:
545:
512:
490:
475:
416:
398:
381:
367:
263:
242:
227:
216:
196:
180:
8274:
8241:
8240:
8212:
8206:
8203:
8197:
8183:
8177:
8174:
8168:
8167:
8142:(4): 1332–1345.
8123:
8117:
8103:
8097:
8087:
8059:
8053:
8052:
8042:
8018:
8002:Overcompleteness
7951:
7949:
7948:
7943:
7936:
7929:
7928:
7913:
7912:
7907:
7906:
7898:
7881:
7879:
7878:
7873:
7866:
7865:
7864:
7853:
7852:
7851:
7836:
7835:
7817:
7816:
7801:
7800:
7795:
7794:
7786:
7774:
7772:
7771:
7766:
7759:
7758:
7757:
7745:
7744:
7726:
7725:
7710:
7709:
7704:
7703:
7695:
7681:
7664:
7662:
7661:
7656:
7649:
7633:
7631:
7630:
7629:
7613:
7612:
7607:
7606:
7598:
7594:
7583:
7582:
7574:
7558:
7556:
7555:
7550:
7543:
7542:
7540:
7539:
7538:
7522:
7521:
7516:
7515:
7507:
7503:
7501:
7493:
7475:
7467:and eliminating
7466:
7456:
7446:
7442:
7438:
7427:
7412:
7401:
7399:
7398:
7393:
7386:
7385:
7384:
7375:
7374:
7362:
7361:
7352:
7351:
7339:
7338:
7326:
7325:
7313:
7312:
7292:
7290:
7289:
7284:
7279:
7278:
7269:
7268:
7256:
7255:
7240:
7239:
7227:
7226:
7214:
7213:
7193:
7191:
7190:
7185:
7174:
7173:
7161:
7160:
7148:
7147:
7125:
7114:
7103:
7099:
7085:
7081:
7069:
7068:
7067:
7066:
7061:
7054:
7053:
7052:
7051:
7046:
7036:
7034:
7033:
7028:
7021:
7020:
7018:
7010:
6999:
6985:
6984:
6979:
6978:
6970:
6963:
6962:
6957:
6956:
6948:
6928:
6926:
6925:
6920:
6918:
6910:
6899:
6898:
6893:
6892:
6884:
6877:
6876:
6868:
6856:
6855:
6850:
6849:
6841:
6834:
6833:
6825:
6811:
6809:
6808:
6803:
6801:
6799:
6791:
6783:
6769:
6768:
6760:
6754:
6753:
6745:
6722:
6720:
6719:
6714:
6709:
6707:
6699:
6691:
6683:
6682:
6670:
6669:
6657:
6656:
6631:
6629:
6628:
6623:
6621:
6612:
6606:
6605:
6593:
6592:
6571:
6570:
6558:
6557:
6531:
6529:
6528:
6523:
6521:
6519:
6508:
6500:
6499:
6458:
6456:
6455:
6450:
6445:
6444:
6426:
6425:
6413:
6412:
6400:
6399:
6381:
6379:
6371:
6370:
6369:
6353:
6348:
6347:
6320:
6319:
6281:
6267:
6265:
6264:
6259:
6254:
6253:
6241:
6239:
6238:
6237:
6221:
6212:
6204:
6187:
6183:
6148:
6146:
6135:
6130:
6129:
6104:
6102:
6101:
6096:
6091:
6087:
6052:
6047:
6039:
6018:
6016:
6015:
6010:
6005:
6000:
5992:
5987:
5986:
5974:
5973:
5961:
5960:
5942:
5941:
5929:
5928:
5916:
5915:
5893:
5884:
5869:
5867:
5866:
5861:
5856:
5854:
5846:
5838:
5836:
5828:
5820:
5819:
5814:
5813:
5805:
5793:
5791:
5790:
5785:
5783:
5781:
5773:
5765:
5763:
5755:
5747:
5746:
5741:
5740:
5732:
5720:
5718:
5717:
5712:
5710:
5708:
5700:
5692:
5690:
5688:
5677:
5672:
5671:
5663:
5652:
5650:
5649:
5644:
5642:
5640:
5632:
5624:
5622:
5620:
5609:
5598:
5597:
5579:
5578:
5566:
5565:
5553:
5552:
5534:
5533:
5521:
5520:
5493:
5492:
5484:
5471:
5462:
5450:
5441:
5425:
5423:
5422:
5417:
5409:
5404:
5396:
5394:
5393:
5381:
5379:
5371:
5363:
5355:
5354:
5342:
5341:
5329:
5328:
5316:
5315:
5305:
5293:
5292:
5274:
5273:
5253:
5251:
5250:
5245:
5237:
5232:
5224:
5222:
5221:
5209:
5207:
5199:
5191:
5183:
5182:
5170:
5169:
5157:
5156:
5144:
5143:
5133:
5121:
5120:
5102:
5101:
5079:
5053:
5051:
5050:
5045:
5040:
5036:
5032:
5029:
5021:
5013:
5007:
5006:
4992:
4991:
4979:
4975:
4971:
4968:
4960:
4952:
4946:
4945:
4931:
4930:
4897:
4896:
4874:
4872:
4871:
4866:
4858:
4855:
4847:
4839:
4833:
4832:
4820:
4819:
4803:
4800:
4792:
4784:
4778:
4777:
4765:
4764:
4747:
4745:
4744:
4739:
4680:
4660:
4656:
4652:
4646:
4632:
4630:
4629:
4624:
4617:
4616:
4615:
4597:
4596:
4591:
4590:
4582:
4574:
4566:
4554:
4553:
4541:
4540:
4535:
4534:
4526:
4512:
4494:
4493:
4466:
4465:
4432:, is defined as
4431:
4427:
4420:
4418:
4417:
4412:
4407:
4406:
4394:
4393:
4388:
4387:
4379:
4372:
4371:
4366:
4365:
4357:
4347:
4346:
4324:
4322:
4321:
4316:
4314:
4313:
4308:
4307:
4299:
4280:
4268:
4258:
4256:
4255:
4250:
4240:
4227:
4226:
4221:
4220:
4212:
4205:
4204:
4199:
4198:
4190:
4173:
4164:
4162:
4161:
4156:
4149:
4142:
4141:
4129:
4128:
4116:
4115:
4093:
4089:
4080:
4057:
4055:
4054:
4049:
4047:
4046:
4027:
4016:
4009:
4007:
4006:
4001:
3990:
3989:
3977:
3976:
3948:
3932:Dirac brackets,
3918:
3916:
3915:
3910:
3905:
3904:
3892:
3891:
3870:
3869:
3861:
3848:
3841:
3839:
3838:
3833:
3828:
3827:
3818:
3817:
3807:
3789:
3768:
3766:
3765:
3760:
3758:
3757:
3747:
3742:
3733:
3732:
3722:
3704:
3703:
3694:
3693:
3683:
3665:
3664:
3634:
3632:
3631:
3626:
3624:
3623:
3607:
3605:
3604:
3599:
3597:
3596:
3577:
3575:
3574:
3569:
3563:
3558:
3549:
3548:
3538:
3526:
3525:
3513:
3512:
3493:
3489:
3486:where the index
3485:
3474:
3460:
3458:
3457:
3452:
3444:
3443:
3431:
3430:
3418:
3417:
3405:
3404:
3394:
3375:
3364:
3350:
3348:
3347:
3342:
3337:
3336:
3324:
3323:
3311:
3310:
3287:
3285:
3284:
3279:
3271:
3270:
3258:
3257:
3245:
3244:
3232:
3231:
3221:
3209:
3208:
3190:
3189:
3157:
3138:
3128:
3117:
3102:
3082:
3072:
3058:
3041:
3030:
3028:
3027:
3022:
3014:
3013:
3001:
3000:
2988:
2987:
2975:
2974:
2964:
2952:
2951:
2933:
2932:
2917:
2916:
2911:
2910:
2901:
2888:
2874:
2872:
2871:
2866:
2861:
2860:
2848:
2847:
2829:
2828:
2818:
2806:
2805:
2778:
2777:
2765:
2764:
2743:
2742:
2734:
2721:
2706:
2694:
2692:
2691:
2686:
2660:
2659:
2642:
2640:
2639:
2634:
2632:
2630:
2629:
2628:
2615:
2614:
2613:
2600:
2598:
2597:
2587:
2575:
2573:
2572:
2571:
2558:
2550:
2545:
2544:
2539:
2538:
2530:
2518:
2516:
2515:
2510:
2508:
2506:
2505:
2504:
2491:
2490:
2489:
2476:
2474:
2473:
2463:
2451:
2449:
2448:
2447:
2434:
2426:
2418:
2417:
2412:
2411:
2403:
2386:
2372:
2370:
2369:
2364:
2359:
2357:
2356:
2355:
2342:
2341:
2340:
2327:
2325:
2324:
2314:
2302:
2301:
2284:
2282:
2281:
2276:
2274:
2272:
2271:
2270:
2257:
2256:
2255:
2242:
2240:
2239:
2229:
2217:
2216:
2193:
2181:
2170:
2156:
2154:
2153:
2148:
2137:
2136:
2124:
2123:
2113:
2101:
2100:
2088:
2087:
2077:
2055:
2049:
2040:
2038:
2037:
2032:
2025:
2012:
2008:
2007:
2006:
1998:
1992:
1990:
1982:
1974:
1958:
1954:
1953:
1952:
1944:
1938:
1936:
1928:
1920:
1899:
1897:
1896:
1891:
1884:
1877:
1876:
1868:
1856:
1855:
1847:
1835:
1833:
1825:
1817:
1806:
1804:
1796:
1788:
1767:
1753:
1743:
1733:
1727:
1713:
1711:
1710:
1705:
1694:
1693:
1684:
1683:
1673:
1655:
1654:
1627:
1621:
1615:
1604:
1588:
1565:
1555:
1551:
1547:
1537:
1533:
1492:
1488:
1477:
1473:
1459:
1457:
1456:
1451:
1419:
1418:
1409:
1408:
1400:
1394:
1393:
1384:
1383:
1375:
1366:
1365:
1353:
1352:
1300:
1298:
1297:
1292:
1285:
1281:
1279:
1271:
1263:
1258:
1256:
1255:
1254:
1246:
1239:
1231:
1226:
1225:
1208:
1206:
1205:
1200:
1195:
1193:
1185:
1177:
1169:
1167:
1166:
1165:
1157:
1150:
1142:
1137:
1136:
1107:
1105:
1104:
1099:
1092:
1091:
1089:
1081:
1073:
1071:
1069:
1058:
1050:
1049:
1041:
1030:
1028:
1027:
1022:
1020:
1018:
1010:
1002:
1000:
998:
987:
982:
981:
973:
957:
955:
954:
949:
942:
917:
916:
908:
899:
898:
890:
881:
879:
871:
863:
845:
831:
819:
810:
808:
807:
802:
797:
796:
788:
785:
780:
772:
767:
765:
757:
749:
741:
740:
732:
716:
714:
713:
708:
706:
705:
697:
694:
689:
681:
676:
674:
666:
658:
650:
649:
641:
622:
620:
619:
614:
607:
582:
581:
573:
564:
563:
555:
546:
544:
536:
528:
520:
519:
514:
513:
505:
498:
497:
492:
491:
483:
476:
468:
451:amounts to just
447:Explicitly, the
431:
429:
428:
423:
418:
417:
409:
400:
399:
391:
382:
374:
369:
368:
360:
347:
343:
339:
337:
336:
335:
330:
322:
318:
317:
316:
315:
310:
302:vector potential
299:
298:
297:
296:
291:
281:
279:
278:
273:
265:
264:
256:
244:
243:
235:
229:
228:
220:
217:
209:
204:
203:
198:
197:
189:
182:
173:
151:
147:
143:
139:
136:confined to the
135:
131:
105:is of this form.
21:bra-ket notation
8282:
8281:
8277:
8276:
8275:
8273:
8272:
8271:
8247:
8246:
8245:
8244:
8214:
8213:
8209:
8204:
8200:
8184:
8180:
8175:
8171:
8125:
8124:
8120:
8104:
8100:
8076:
8061:
8060:
8056:
8020:
8019:
8015:
8010:
7972:Poisson bracket
7958:
7920:
7895:
7890:
7889:
7856:
7843:
7824:
7808:
7783:
7778:
7777:
7749:
7733:
7717:
7692:
7687:
7686:
7669:
7621:
7614:
7595:
7567:
7566:
7530:
7523:
7504:
7481:
7480:
7474:
7468:
7464:
7458:
7451:
7444:
7440:
7433:
7424:
7420:
7414:
7406:
7376:
7366:
7353:
7343:
7327:
7317:
7304:
7296:
7295:
7270:
7260:
7244:
7228:
7218:
7205:
7197:
7196:
7162:
7152:
7139:
7131:
7130:
7121:
7116:
7110:
7105:
7101:
7095:
7092:
7083:
7079:
7062:
7059:
7058:
7057:
7056:
7047:
7044:
7043:
7042:
7041:
7011:
7000:
6967:
6945:
6937:
6936:
6881:
6838:
6815:
6814:
6792:
6784:
6735:
6734:
6700:
6692:
6671:
6661:
6648:
6640:
6639:
6594:
6584:
6559:
6549:
6535:
6534:
6512:
6488:
6471:
6470:
6433:
6417:
6401:
6391:
6372:
6358:
6354:
6336:
6308:
6291:
6290:
6280:
6272:
6242:
6229:
6225:
6181:
6180:
6175:
6169:
6168:
6160:
6149:
6139:
6118:
6113:
6112:
6085:
6084:
6079:
6070:
6069:
6064:
6053:
6040:
6027:
6026:
5993:
5975:
5965:
5952:
5930:
5920:
5907:
5899:
5898:
5892:
5886:
5883:
5877:
5847:
5839:
5802:
5797:
5796:
5774:
5766:
5729:
5724:
5723:
5701:
5693:
5681:
5656:
5655:
5633:
5625:
5613:
5589:
5570:
5554:
5544:
5525:
5509:
5477:
5476:
5470:
5464:
5461:
5455:
5449:
5443:
5440:
5434:
5397:
5385:
5372:
5364:
5343:
5333:
5320:
5307:
5281:
5265:
5257:
5256:
5225:
5213:
5200:
5192:
5171:
5161:
5148:
5135:
5109:
5093:
5085:
5084:
5077:
5067:
5058:
5022:
5014:
4998:
4997:
4993:
4983:
4961:
4953:
4937:
4936:
4932:
4922:
4888:
4883:
4882:
4848:
4840:
4824:
4811:
4793:
4785:
4769:
4756:
4751:
4750:
4709:
4708:
4702:
4676:
4658:
4654:
4648:
4645:
4637:
4634:
4604:
4579:
4542:
4523:
4482:
4454:
4437:
4436:
4429:
4425:
4395:
4376:
4354:
4335:
4330:
4329:
4296:
4291:
4290:
4279:
4273:
4271:
4266:
4209:
4187:
4179:
4178:
4169:
4130:
4120:
4107:
4099:
4098:
4091:
4088:
4082:
4079:
4073:
4038:
4033:
4032:
4026:
4018:
4014:
3978:
3968:
3954:
3953:
3944:
3929:
3893:
3883:
3854:
3853:
3846:
3819:
3809:
3782:
3777:
3776:
3749:
3724:
3695:
3685:
3656:
3651:
3650:
3641:
3615:
3610:
3609:
3588:
3583:
3582:
3540:
3517:
3504:
3499:
3498:
3491:
3487:
3484:
3476:
3473:
3465:
3432:
3422:
3409:
3396:
3381:
3380:
3374:
3366:
3363:
3355:
3328:
3315:
3302:
3297:
3296:
3259:
3249:
3236:
3223:
3197:
3181:
3173:
3172:
3167:
3159:
3156:
3148:
3136:
3131:
3127:
3119:
3116:
3108:
3101:
3093:
3078:
3071:
3063:
3057:
3049:
3039:
3002:
2992:
2979:
2966:
2940:
2924:
2902:
2894:
2893:
2887:
2879:
2849:
2839:
2820:
2794:
2766:
2756:
2727:
2726:
2719:
2716:
2704:
2699:
2651:
2646:
2645:
2620:
2616:
2605:
2601:
2589:
2563:
2559:
2551:
2527:
2522:
2521:
2496:
2492:
2481:
2477:
2465:
2439:
2435:
2427:
2400:
2395:
2394:
2385:
2377:
2347:
2343:
2332:
2328:
2316:
2293:
2288:
2287:
2262:
2258:
2247:
2243:
2231:
2208:
2203:
2202:
2198:) is generally
2191:
2183:
2180:
2172:
2169:
2161:
2128:
2115:
2092:
2079:
2064:
2063:
2051:
2045:
1983:
1975:
1972:
1968:
1929:
1921:
1918:
1914:
1909:
1908:
1826:
1818:
1797:
1789:
1773:
1772:
1766:
1758:
1750:
1745:
1735:
1729:
1726:
1718:
1685:
1675:
1646:
1641:
1640:
1634:
1623:
1617:
1613:
1598:
1590:
1587:
1579:
1557:
1553:
1549:
1543:
1535:
1531:
1528:strong equality
1504:
1490:
1487:
1479:
1475:
1471:
1410:
1385:
1357:
1344:
1321:
1320:
1272:
1264:
1240:
1232:
1217:
1212:
1211:
1186:
1178:
1151:
1143:
1128:
1123:
1122:
1082:
1074:
1062:
1034:
1033:
1011:
1003:
991:
966:
965:
872:
864:
851:
850:
836:
821:
815:
773:
758:
750:
722:
721:
682:
667:
659:
631:
630:
537:
529:
502:
480:
456:
455:
353:
352:
345:
341:
331:
328:
327:
326:
324:
320:
311:
308:
307:
306:
305:
292:
289:
288:
287:
286:
186:
160:
159:
149:
145:
141:
137:
133:
129:
122:
108:When there are
87:
60:symplectic form
32:Poisson bracket
24:
17:
12:
11:
5:
8280:
8278:
8270:
8269:
8264:
8259:
8249:
8248:
8243:
8242:
8207:
8198:
8178:
8169:
8118:
8098:
8074:
8054:
8012:
8011:
8009:
8006:
8005:
8004:
7999:
7994:
7989:
7984:
7979:
7974:
7969:
7964:
7957:
7954:
7953:
7952:
7941:
7935:
7932:
7927:
7923:
7919:
7916:
7911:
7904:
7901:
7883:
7882:
7871:
7863:
7859:
7850:
7846:
7842:
7839:
7834:
7831:
7827:
7823:
7820:
7815:
7811:
7807:
7804:
7799:
7792:
7789:
7775:
7764:
7756:
7752:
7748:
7743:
7740:
7736:
7732:
7729:
7724:
7720:
7716:
7713:
7708:
7701:
7698:
7666:
7665:
7654:
7648:
7645:
7642:
7639:
7636:
7628:
7624:
7620:
7617:
7611:
7604:
7601:
7592:
7589:
7586:
7580:
7577:
7560:
7559:
7548:
7537:
7533:
7529:
7526:
7520:
7513:
7510:
7499:
7496:
7491:
7488:
7472:
7462:
7422:
7418:
7403:
7402:
7391:
7383:
7379:
7373:
7369:
7365:
7360:
7356:
7350:
7346:
7342:
7337:
7334:
7330:
7324:
7320:
7316:
7311:
7307:
7303:
7293:
7282:
7277:
7273:
7267:
7263:
7259:
7254:
7251:
7247:
7243:
7238:
7235:
7231:
7225:
7221:
7217:
7212:
7208:
7204:
7194:
7183:
7180:
7177:
7172:
7169:
7165:
7159:
7155:
7151:
7146:
7142:
7138:
7119:
7108:
7091:
7088:
7038:
7037:
7026:
7017:
7014:
7009:
7006:
7003:
6997:
6994:
6991:
6988:
6983:
6976:
6973:
6966:
6961:
6954:
6951:
6944:
6930:
6929:
6916:
6913:
6908:
6905:
6902:
6897:
6890:
6887:
6880:
6874:
6871:
6865:
6862:
6859:
6854:
6847:
6844:
6837:
6831:
6828:
6822:
6812:
6798:
6795:
6790:
6787:
6781:
6778:
6775:
6772:
6766:
6763:
6757:
6751:
6748:
6742:
6724:
6723:
6712:
6706:
6703:
6698:
6695:
6689:
6686:
6681:
6678:
6674:
6668:
6664:
6660:
6655:
6651:
6647:
6633:
6632:
6618:
6615:
6609:
6604:
6601:
6597:
6591:
6587:
6583:
6580:
6577:
6574:
6569:
6566:
6562:
6556:
6552:
6548:
6545:
6542:
6532:
6518:
6515:
6511:
6506:
6503:
6498:
6495:
6491:
6487:
6484:
6481:
6478:
6460:
6459:
6448:
6443:
6440:
6436:
6432:
6429:
6424:
6420:
6416:
6411:
6408:
6404:
6398:
6394:
6390:
6387:
6384:
6378:
6375:
6368:
6365:
6361:
6357:
6351:
6346:
6343:
6339:
6335:
6332:
6329:
6326:
6323:
6318:
6315:
6311:
6307:
6304:
6301:
6298:
6276:
6269:
6268:
6257:
6252:
6249:
6245:
6236:
6232:
6228:
6224:
6219:
6216:
6211:
6208:
6203:
6200:
6196:
6191:
6186:
6179:
6176:
6174:
6171:
6170:
6167:
6164:
6161:
6159:
6156:
6155:
6152:
6145:
6142:
6138:
6133:
6128:
6125:
6121:
6106:
6105:
6094:
6090:
6083:
6080:
6078:
6075:
6072:
6071:
6068:
6065:
6063:
6060:
6059:
6056:
6050:
6046:
6043:
6037:
6034:
6020:
6019:
6008:
6003:
5999:
5996:
5990:
5985:
5982:
5978:
5972:
5968:
5964:
5959:
5955:
5951:
5948:
5945:
5940:
5937:
5933:
5927:
5923:
5919:
5914:
5910:
5906:
5890:
5881:
5871:
5870:
5859:
5853:
5850:
5845:
5842:
5834:
5831:
5826:
5823:
5818:
5811:
5808:
5794:
5780:
5777:
5772:
5769:
5761:
5758:
5753:
5750:
5745:
5738:
5735:
5721:
5707:
5704:
5699:
5696:
5687:
5684:
5680:
5675:
5669:
5666:
5653:
5639:
5636:
5631:
5628:
5619:
5616:
5612:
5607:
5604:
5601:
5596:
5592:
5588:
5585:
5582:
5577:
5573:
5569:
5564:
5561:
5557:
5551:
5547:
5543:
5540:
5537:
5532:
5528:
5524:
5519:
5516:
5512:
5508:
5505:
5502:
5499:
5496:
5490:
5487:
5468:
5459:
5447:
5438:
5427:
5426:
5415:
5412:
5407:
5403:
5400:
5392:
5388:
5384:
5378:
5375:
5370:
5367:
5361:
5358:
5353:
5350:
5346:
5340:
5336:
5332:
5327:
5323:
5319:
5314:
5310:
5304:
5300:
5296:
5291:
5288:
5284:
5280:
5277:
5272:
5268:
5264:
5254:
5243:
5240:
5235:
5231:
5228:
5220:
5216:
5212:
5206:
5203:
5198:
5195:
5189:
5186:
5181:
5178:
5174:
5168:
5164:
5160:
5155:
5151:
5147:
5142:
5138:
5132:
5128:
5124:
5119:
5116:
5112:
5108:
5105:
5100:
5096:
5092:
5073:
5063:
5055:
5054:
5043:
5039:
5035:
5028:
5025:
5020:
5017:
5010:
5005:
5001:
4996:
4990:
4986:
4982:
4978:
4974:
4967:
4964:
4959:
4956:
4949:
4944:
4940:
4935:
4929:
4925:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4900:
4895:
4891:
4876:
4875:
4864:
4861:
4854:
4851:
4846:
4843:
4836:
4831:
4827:
4823:
4818:
4814:
4809:
4806:
4799:
4796:
4791:
4788:
4781:
4776:
4772:
4768:
4763:
4759:
4748:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4701:
4698:
4671:When applying
4641:
4622:
4614:
4611:
4607:
4603:
4600:
4595:
4588:
4585:
4578:
4573:
4570:
4565:
4562:
4558:
4552:
4549:
4545:
4539:
4532:
4529:
4522:
4519:
4516:
4511:
4508:
4505:
4501:
4497:
4492:
4489:
4485:
4481:
4478:
4475:
4472:
4469:
4464:
4461:
4457:
4453:
4450:
4447:
4444:
4434:
4422:
4421:
4410:
4405:
4402:
4398:
4392:
4385:
4382:
4375:
4370:
4363:
4360:
4353:
4350:
4345:
4342:
4338:
4312:
4305:
4302:
4277:
4269:
4260:
4259:
4248:
4245:
4239:
4236:
4233:
4230:
4225:
4218:
4215:
4208:
4203:
4196:
4193:
4186:
4166:
4165:
4154:
4148:
4145:
4140:
4137:
4133:
4127:
4123:
4119:
4114:
4110:
4106:
4086:
4077:
4045:
4041:
4022:
4011:
4010:
3999:
3996:
3993:
3988:
3985:
3981:
3975:
3971:
3967:
3964:
3961:
3928:
3925:
3920:
3919:
3908:
3903:
3900:
3896:
3890:
3886:
3882:
3879:
3876:
3873:
3867:
3864:
3843:
3842:
3831:
3826:
3822:
3816:
3812:
3806:
3802:
3798:
3795:
3792:
3788:
3785:
3770:
3769:
3756:
3752:
3746:
3741:
3737:
3731:
3727:
3721:
3718:
3715:
3711:
3707:
3702:
3698:
3692:
3688:
3682:
3678:
3674:
3671:
3668:
3663:
3659:
3640:
3637:
3622:
3618:
3595:
3591:
3579:
3578:
3567:
3562:
3557:
3553:
3547:
3543:
3537:
3533:
3529:
3524:
3520:
3516:
3511:
3507:
3480:
3469:
3462:
3461:
3450:
3447:
3442:
3439:
3435:
3429:
3425:
3421:
3416:
3412:
3408:
3403:
3399:
3393:
3389:
3370:
3359:
3352:
3351:
3340:
3335:
3331:
3327:
3322:
3318:
3314:
3309:
3305:
3289:
3288:
3277:
3274:
3269:
3266:
3262:
3256:
3252:
3248:
3243:
3239:
3235:
3230:
3226:
3220:
3216:
3212:
3207:
3204:
3200:
3196:
3193:
3188:
3184:
3180:
3165:
3158:
3152:
3145:
3134:
3123:
3112:
3097:
3075:
3074:
3067:
3060:
3053:
3046:
3043:
3032:
3031:
3020:
3017:
3012:
3009:
3005:
2999:
2995:
2991:
2986:
2982:
2978:
2973:
2969:
2963:
2959:
2955:
2950:
2947:
2943:
2939:
2936:
2931:
2927:
2923:
2920:
2914:
2909:
2905:
2883:
2876:
2875:
2864:
2859:
2856:
2852:
2846:
2842:
2838:
2835:
2832:
2827:
2823:
2817:
2813:
2809:
2804:
2801:
2797:
2793:
2790:
2787:
2784:
2781:
2776:
2773:
2769:
2763:
2759:
2755:
2752:
2749:
2746:
2740:
2737:
2715:
2712:
2702:
2696:
2695:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2658:
2654:
2643:
2627:
2623:
2619:
2612:
2608:
2604:
2596:
2592:
2586:
2582:
2578:
2570:
2566:
2562:
2557:
2554:
2548:
2543:
2536:
2533:
2519:
2503:
2499:
2495:
2488:
2484:
2480:
2472:
2468:
2462:
2458:
2454:
2446:
2442:
2438:
2433:
2430:
2424:
2421:
2416:
2409:
2406:
2381:
2374:
2373:
2362:
2354:
2350:
2346:
2339:
2335:
2331:
2323:
2319:
2313:
2309:
2305:
2300:
2296:
2285:
2269:
2265:
2261:
2254:
2250:
2246:
2238:
2234:
2228:
2224:
2220:
2215:
2211:
2187:
2176:
2165:
2158:
2157:
2146:
2143:
2140:
2135:
2131:
2127:
2122:
2118:
2112:
2108:
2104:
2099:
2095:
2091:
2086:
2082:
2076:
2072:
2042:
2041:
2030:
2024:
2021:
2018:
2015:
2011:
2004:
2001:
1995:
1989:
1986:
1981:
1978:
1971:
1967:
1964:
1961:
1957:
1950:
1947:
1941:
1935:
1932:
1927:
1924:
1917:
1901:
1900:
1889:
1883:
1880:
1874:
1871:
1865:
1862:
1859:
1853:
1850:
1844:
1841:
1838:
1832:
1829:
1824:
1821:
1815:
1812:
1809:
1803:
1800:
1795:
1792:
1786:
1783:
1780:
1762:
1748:
1722:
1715:
1714:
1703:
1700:
1697:
1692:
1688:
1682:
1678:
1672:
1668:
1664:
1661:
1658:
1653:
1649:
1633:
1630:
1594:
1583:
1503:
1500:
1483:
1461:
1460:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1417:
1413:
1406:
1403:
1397:
1392:
1388:
1381:
1378:
1372:
1369:
1364:
1360:
1356:
1351:
1347:
1343:
1340:
1337:
1334:
1331:
1328:
1302:
1301:
1290:
1284:
1278:
1275:
1270:
1267:
1261:
1252:
1249:
1243:
1238:
1235:
1229:
1224:
1220:
1209:
1198:
1192:
1189:
1184:
1181:
1175:
1172:
1163:
1160:
1154:
1149:
1146:
1140:
1135:
1131:
1109:
1108:
1097:
1088:
1085:
1080:
1077:
1068:
1065:
1061:
1056:
1053:
1047:
1044:
1031:
1017:
1014:
1009:
1006:
997:
994:
990:
985:
979:
976:
959:
958:
947:
941:
938:
935:
932:
929:
926:
923:
920:
914:
911:
905:
902:
896:
893:
887:
884:
878:
875:
870:
867:
861:
858:
812:
811:
800:
794:
791:
783:
779:
776:
770:
764:
761:
756:
753:
747:
744:
738:
735:
729:
718:
717:
703:
700:
692:
688:
685:
679:
673:
670:
665:
662:
656:
653:
647:
644:
638:
624:
623:
612:
606:
603:
600:
597:
594:
591:
588:
585:
579:
576:
570:
567:
561:
558:
552:
549:
543:
540:
535:
532:
526:
523:
518:
511:
508:
501:
496:
489:
486:
479:
474:
471:
466:
463:
433:
432:
421:
415:
412:
406:
403:
397:
394:
388:
385:
380:
377:
372:
366:
363:
283:
282:
271:
268:
262:
259:
253:
250:
247:
241:
238:
232:
226:
223:
215:
212:
207:
202:
195:
192:
185:
179:
176:
170:
167:
124:An example in
121:
118:
117:
116:
113:
106:
86:
83:
15:
13:
10:
9:
6:
4:
3:
2:
8279:
8268:
8265:
8263:
8260:
8258:
8255:
8254:
8252:
8238:
8234:
8230:
8226:
8222:
8218:
8211:
8208:
8202:
8199:
8196:
8195:0-521-55001-7
8192:
8188:
8182:
8179:
8173:
8170:
8165:
8161:
8157:
8153:
8149:
8145:
8141:
8137:
8133:
8129:
8122:
8119:
8116:
8115:0-691-08775-X
8112:
8108:
8102:
8099:
8095:
8091:
8085:
8081:
8077:
8075:9780486417134
8071:
8067:
8066:
8058:
8055:
8050:
8046:
8041:
8036:
8032:
8028:
8024:
8017:
8014:
8007:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7983:
7980:
7978:
7977:Moyal bracket
7975:
7973:
7970:
7968:
7965:
7963:
7960:
7959:
7955:
7939:
7933:
7930:
7925:
7921:
7917:
7914:
7909:
7902:
7899:
7888:
7887:
7886:
7869:
7861:
7857:
7848:
7844:
7840:
7837:
7832:
7829:
7821:
7818:
7813:
7809:
7802:
7797:
7790:
7787:
7776:
7762:
7754:
7750:
7746:
7741:
7738:
7730:
7727:
7722:
7718:
7711:
7706:
7699:
7696:
7685:
7684:
7683:
7680:
7676:
7672:
7652:
7646:
7643:
7640:
7637:
7634:
7626:
7622:
7618:
7615:
7609:
7602:
7599:
7590:
7587:
7584:
7578:
7575:
7565:
7564:
7563:
7546:
7535:
7531:
7527:
7524:
7518:
7511:
7508:
7497:
7494:
7489:
7486:
7479:
7478:
7477:
7471:
7461:
7454:
7448:
7437:
7431:
7425:
7410:
7389:
7381:
7377:
7371:
7367:
7363:
7358:
7354:
7348:
7344:
7340:
7335:
7332:
7322:
7318:
7314:
7309:
7305:
7294:
7280:
7275:
7271:
7265:
7261:
7257:
7252:
7249:
7245:
7241:
7236:
7233:
7223:
7219:
7215:
7210:
7206:
7195:
7181:
7178:
7175:
7170:
7167:
7157:
7153:
7149:
7144:
7140:
7129:
7128:
7127:
7123:
7112:
7098:
7089:
7087:
7077:
7073:
7065:
7050:
7024:
7015:
7012:
7007:
7004:
6995:
6992:
6989:
6981:
6971:
6964:
6959:
6949:
6935:
6934:
6933:
6914:
6906:
6903:
6895:
6885:
6878:
6869:
6860:
6852:
6842:
6835:
6826:
6813:
6796:
6793:
6788:
6779:
6776:
6773:
6761:
6755:
6746:
6733:
6732:
6731:
6729:
6710:
6704:
6701:
6696:
6693:
6687:
6684:
6679:
6676:
6666:
6662:
6658:
6653:
6649:
6638:
6637:
6636:
6616:
6613:
6607:
6602:
6599:
6589:
6585:
6581:
6578:
6572:
6567:
6564:
6554:
6550:
6546:
6543:
6533:
6516:
6513:
6509:
6504:
6501:
6496:
6493:
6485:
6482:
6479:
6469:
6468:
6467:
6464:
6446:
6441:
6438:
6430:
6427:
6422:
6418:
6409:
6406:
6396:
6392:
6388:
6385:
6376:
6373:
6366:
6363:
6359:
6355:
6349:
6344:
6341:
6333:
6330:
6327:
6321:
6316:
6313:
6305:
6302:
6299:
6289:
6288:
6287:
6285:
6279:
6275:
6255:
6250:
6247:
6243:
6234:
6230:
6226:
6222:
6217:
6214:
6209:
6206:
6201:
6198:
6194:
6184:
6177:
6172:
6165:
6162:
6157:
6150:
6143:
6140:
6136:
6131:
6126:
6123:
6119:
6111:
6110:
6109:
6092:
6088:
6081:
6076:
6073:
6066:
6061:
6054:
6048:
6044:
6041:
6035:
6032:
6025:
6024:
6023:
6006:
6001:
5997:
5994:
5988:
5983:
5980:
5970:
5966:
5962:
5957:
5953:
5946:
5943:
5938:
5935:
5925:
5921:
5917:
5912:
5908:
5897:
5896:
5895:
5889:
5880:
5874:
5857:
5851:
5843:
5832:
5829:
5824:
5821:
5816:
5809:
5806:
5795:
5778:
5770:
5759:
5756:
5751:
5748:
5743:
5736:
5733:
5722:
5705:
5697:
5685:
5682:
5678:
5673:
5667:
5664:
5654:
5637:
5629:
5617:
5614:
5610:
5605:
5602:
5594:
5590:
5586:
5583:
5575:
5571:
5567:
5562:
5559:
5549:
5545:
5541:
5538:
5530:
5526:
5522:
5517:
5514:
5506:
5503:
5500:
5494:
5488:
5485:
5475:
5474:
5473:
5467:
5458:
5452:
5446:
5437:
5432:
5413:
5410:
5405:
5401:
5398:
5390:
5386:
5382:
5376:
5368:
5359:
5356:
5351:
5348:
5338:
5334:
5330:
5325:
5321:
5312:
5308:
5302:
5298:
5294:
5289:
5286:
5278:
5275:
5270:
5266:
5255:
5241:
5238:
5233:
5229:
5226:
5218:
5214:
5210:
5204:
5196:
5187:
5184:
5179:
5176:
5166:
5162:
5158:
5153:
5149:
5140:
5136:
5130:
5126:
5122:
5117:
5114:
5106:
5103:
5098:
5094:
5083:
5082:
5081:
5076:
5071:
5066:
5062:
5041:
5037:
5033:
5026:
5023:
5018:
5015:
5008:
5003:
4999:
4994:
4988:
4984:
4980:
4976:
4972:
4965:
4962:
4957:
4954:
4947:
4942:
4938:
4933:
4927:
4923:
4919:
4913:
4910:
4907:
4901:
4898:
4893:
4889:
4881:
4880:
4879:
4862:
4859:
4852:
4849:
4844:
4841:
4834:
4829:
4825:
4821:
4816:
4812:
4807:
4804:
4797:
4794:
4789:
4786:
4779:
4774:
4770:
4766:
4761:
4757:
4749:
4732:
4729:
4726:
4720:
4717:
4714:
4707:
4706:
4705:
4699:
4697:
4695:
4691:
4686:
4684:
4683:Dirac bracket
4679:
4674:
4669:
4665:
4663:
4651:
4644:
4640:
4633:
4620:
4612:
4609:
4601:
4598:
4593:
4583:
4571:
4568:
4563:
4560:
4556:
4550:
4547:
4537:
4527:
4520:
4517:
4509:
4506:
4503:
4499:
4495:
4490:
4487:
4479:
4476:
4473:
4467:
4462:
4459:
4451:
4448:
4445:
4433:
4408:
4403:
4400:
4390:
4380:
4373:
4368:
4358:
4348:
4343:
4340:
4336:
4328:
4327:
4326:
4310:
4300:
4287:
4285:
4276:
4263:
4246:
4243:
4234:
4231:
4223:
4213:
4206:
4201:
4191:
4177:
4176:
4175:
4172:
4152:
4146:
4143:
4138:
4135:
4125:
4121:
4117:
4112:
4108:
4097:
4096:
4095:
4085:
4076:
4070:
4068:
4063:
4061:
4043:
4039:
4029:
4025:
4021:
3997:
3994:
3991:
3986:
3983:
3973:
3969:
3965:
3962:
3952:
3951:
3950:
3947:
3941:
3939:
3935:
3926:
3924:
3906:
3901:
3898:
3888:
3884:
3880:
3877:
3871:
3865:
3862:
3852:
3851:
3850:
3829:
3824:
3820:
3814:
3810:
3804:
3800:
3796:
3793:
3790:
3786:
3783:
3775:
3774:
3773:
3754:
3750:
3744:
3739:
3735:
3729:
3725:
3719:
3716:
3713:
3709:
3705:
3700:
3696:
3690:
3686:
3680:
3676:
3672:
3669:
3666:
3661:
3657:
3649:
3648:
3647:
3646:
3638:
3636:
3620:
3616:
3593:
3589:
3565:
3560:
3555:
3551:
3545:
3541:
3535:
3531:
3527:
3522:
3518:
3514:
3509:
3505:
3497:
3496:
3495:
3483:
3479:
3472:
3468:
3448:
3445:
3440:
3437:
3427:
3423:
3419:
3414:
3410:
3401:
3397:
3391:
3387:
3379:
3378:
3377:
3373:
3369:
3362:
3358:
3338:
3333:
3329:
3325:
3320:
3316:
3312:
3307:
3303:
3295:
3294:
3293:
3275:
3272:
3267:
3264:
3254:
3250:
3246:
3241:
3237:
3228:
3224:
3218:
3214:
3210:
3205:
3202:
3194:
3191:
3186:
3182:
3171:
3170:
3169:
3164:
3155:
3151:
3146:
3144:
3142:
3137:
3126:
3122:
3115:
3111:
3105:
3100:
3096:
3091:
3090:
3084:
3081:
3070:
3066:
3061:
3056:
3052:
3047:
3044:
3037:
3036:
3035:
3018:
3015:
3010:
3007:
2997:
2993:
2989:
2984:
2980:
2971:
2967:
2961:
2957:
2953:
2948:
2945:
2937:
2934:
2929:
2925:
2918:
2912:
2907:
2903:
2892:
2891:
2890:
2886:
2882:
2862:
2857:
2854:
2844:
2840:
2836:
2833:
2825:
2821:
2815:
2811:
2807:
2802:
2799:
2791:
2788:
2785:
2779:
2774:
2771:
2761:
2757:
2753:
2750:
2744:
2738:
2735:
2725:
2724:
2723:
2713:
2711:
2708:
2705:
2682:
2679:
2676:
2670:
2667:
2664:
2656:
2652:
2644:
2625:
2621:
2610:
2606:
2594:
2590:
2584:
2580:
2576:
2568:
2564:
2555:
2546:
2541:
2534:
2531:
2520:
2501:
2497:
2486:
2482:
2470:
2466:
2460:
2456:
2452:
2444:
2440:
2431:
2422:
2419:
2414:
2407:
2404:
2393:
2392:
2391:
2388:
2384:
2380:
2360:
2352:
2348:
2337:
2333:
2321:
2317:
2311:
2307:
2303:
2298:
2294:
2286:
2267:
2263:
2252:
2248:
2236:
2232:
2226:
2222:
2218:
2213:
2209:
2201:
2200:
2199:
2197:
2190:
2186:
2179:
2175:
2168:
2164:
2144:
2141:
2138:
2133:
2129:
2125:
2120:
2116:
2110:
2106:
2102:
2097:
2093:
2089:
2084:
2080:
2074:
2070:
2062:
2061:
2060:
2057:
2054:
2048:
2028:
2022:
2019:
2016:
2013:
2009:
2002:
1999:
1993:
1987:
1979:
1969:
1965:
1962:
1959:
1955:
1948:
1945:
1939:
1933:
1925:
1915:
1907:
1906:
1905:
1887:
1881:
1878:
1872:
1869:
1863:
1860:
1857:
1851:
1848:
1842:
1839:
1836:
1830:
1822:
1813:
1810:
1807:
1801:
1793:
1784:
1781:
1778:
1771:
1770:
1769:
1765:
1761:
1755:
1751:
1742:
1738:
1732:
1725:
1721:
1701:
1698:
1695:
1690:
1686:
1680:
1676:
1670:
1666:
1662:
1659:
1656:
1651:
1647:
1639:
1638:
1637:
1631:
1629:
1626:
1620:
1611:
1606:
1602:
1597:
1593:
1586:
1582:
1577:
1571:
1569:
1564:
1560:
1546:
1541:
1529:
1525:
1524:weak equality
1520:
1518:
1513:
1509:
1501:
1499:
1497:
1486:
1482:
1468:
1466:
1447:
1441:
1438:
1435:
1429:
1426:
1423:
1420:
1415:
1411:
1404:
1401:
1395:
1390:
1386:
1379:
1376:
1370:
1362:
1358:
1354:
1349:
1345:
1341:
1338:
1335:
1332:
1326:
1319:
1318:
1317:
1315:
1310:
1308:
1288:
1282:
1276:
1273:
1268:
1265:
1259:
1250:
1247:
1236:
1227:
1222:
1218:
1210:
1196:
1190:
1187:
1182:
1179:
1173:
1170:
1161:
1158:
1147:
1138:
1133:
1129:
1121:
1120:
1119:
1116:
1114:
1095:
1086:
1078:
1066:
1063:
1059:
1054:
1051:
1045:
1042:
1032:
1015:
1007:
995:
992:
988:
983:
977:
974:
964:
963:
962:
945:
936:
933:
930:
924:
921:
912:
909:
903:
900:
894:
891:
885:
876:
873:
868:
865:
859:
856:
849:
848:
847:
843:
839:
833:
829:
825:
818:
798:
792:
789:
781:
777:
774:
768:
762:
754:
745:
742:
736:
733:
727:
720:
719:
701:
698:
690:
686:
683:
677:
671:
663:
654:
651:
645:
642:
636:
629:
628:
627:
610:
601:
598:
595:
589:
586:
577:
574:
568:
565:
559:
556:
550:
541:
538:
533:
530:
524:
516:
509:
506:
499:
494:
487:
484:
472:
469:
464:
461:
454:
453:
452:
450:
445:
442:
438:
410:
404:
401:
392:
386:
378:
375:
370:
361:
351:
350:
349:
334:
314:
303:
295:
269:
257:
248:
245:
236:
230:
221:
213:
210:
205:
200:
190:
183:
177:
174:
168:
165:
158:
157:
156:
153:
127:
119:
114:
111:
107:
104:
100:
96:
92:
91:
90:
84:
82:
80:
76:
72:
67:
65:
61:
57:
53:
49:
45:
41:
37:
34:developed by
33:
29:
28:Dirac bracket
22:
8223:(3–4): 273.
8220:
8216:
8210:
8201:
8186:
8181:
8172:
8139:
8135:
8121:
8106:
8101:
8064:
8057:
8030:
8026:
8016:
7884:
7678:
7674:
7670:
7667:
7561:
7469:
7459:
7452:
7449:
7435:
7429:
7416:
7408:
7404:
7117:
7106:
7096:
7093:
7086:positions.)
7063:
7048:
7039:
6931:
6725:
6634:
6465:
6461:
6277:
6273:
6270:
6107:
6021:
5887:
5878:
5875:
5872:
5465:
5456:
5453:
5444:
5435:
5430:
5428:
5074:
5069:
5064:
5060:
5056:
4877:
4703:
4693:
4687:
4682:
4677:
4670:
4666:
4661:
4649:
4647:denotes the
4642:
4638:
4635:
4435:
4423:
4288:
4283:
4274:
4264:
4261:
4170:
4167:
4083:
4074:
4071:
4064:
4059:
4030:
4023:
4019:
4012:
3945:
3942:
3938:second-class
3937:
3933:
3930:
3921:
3844:
3771:
3644:
3642:
3580:
3481:
3477:
3470:
3466:
3463:
3371:
3367:
3360:
3356:
3353:
3290:
3162:
3160:
3153:
3149:
3140:
3132:
3124:
3120:
3113:
3109:
3106:
3098:
3094:
3087:
3085:
3079:
3076:
3068:
3064:
3054:
3050:
3033:
2884:
2880:
2877:
2717:
2709:
2700:
2697:
2389:
2382:
2378:
2375:
2188:
2184:
2177:
2173:
2166:
2162:
2159:
2058:
2052:
2046:
2043:
1902:
1763:
1759:
1756:
1746:
1740:
1736:
1730:
1723:
1719:
1716:
1635:
1624:
1618:
1609:
1607:
1600:
1595:
1591:
1584:
1580:
1575:
1572:
1567:
1562:
1558:
1544:
1539:
1527:
1523:
1521:
1516:
1505:
1495:
1484:
1480:
1469:
1464:
1462:
1311:
1307:overcomplete
1303:
1117:
1112:
1110:
960:
841:
837:
834:
827:
823:
816:
813:
625:
446:
440:
436:
434:
332:
312:
293:
284:
154:
123:
98:
88:
68:
27:
25:
8126:Dunne, G.;
8033:: 129–014.
3934:first-class
3581:where the
1517:phase space
97:leads to a
75:Hamiltonian
64:phase space
56:Lagrangians
8251:Categories
8128:Jackiw, R.
8094:0486417131
8088:; Dover,
8008:References
7992:Lagrangian
7432:than the
7428:obey much
5429:These are
3490:runs from
2698:where the
2376:where the
1717:where the
1542:, denoted
449:Lagrangian
99:constraint
71:Lagrangian
36:Paul Dirac
8132:Pi, S. Y.
8049:119748805
7918:−
7903:¨
7841:−
7791:˙
7700:˙
7638:−
7619:−
7603:˙
7588:−
7579:¨
7528:−
7512:˙
7364:−
7258:−
7246:δ
7002:ℏ
6993:−
6975:^
6953:^
6912:ℏ
6889:^
6873:^
6846:^
6830:^
6786:ℏ
6777:−
6765:^
6750:^
6688:−
6505:−
6419:ϕ
6393:ϕ
6360:ε
6244:ε
6218:−
6207:−
6190:⇒
6163:−
6124:−
6074:−
5967:ϕ
5954:ϕ
5947:−
5922:ϕ
5909:ϕ
5849:∂
5841:∂
5825:−
5810:˙
5776:∂
5768:∂
5752:−
5737:˙
5703:∂
5695:∂
5668:˙
5635:∂
5627:∂
5606:−
5591:ϕ
5546:ϕ
5489:˙
5411:≈
5383:−
5374:∂
5366:∂
5360:−
5335:ϕ
5322:ϕ
5299:∑
5267:ϕ
5239:≈
5202:∂
5194:∂
5188:−
5163:ϕ
5150:ϕ
5127:∑
5095:ϕ
5009:−
4894:∗
4835:−
4813:ϕ
4758:ϕ
4653:entry of
4587:~
4584:ϕ
4569:−
4531:~
4528:ϕ
4500:∑
4496:−
4384:~
4381:ϕ
4362:~
4359:ϕ
4304:~
4301:ϕ
4238:ℏ
4217:^
4214:ϕ
4195:^
4192:ϕ
4122:ϕ
4109:ϕ
3992:≈
3970:ϕ
3872:≈
3866:˙
3821:ϕ
3801:∑
3751:ϕ
3710:∑
3697:ϕ
3677:∑
3532:∑
3515:≈
3446:≈
3424:ϕ
3411:ϕ
3388:∑
3273:≈
3251:ϕ
3238:ϕ
3215:∑
3183:ϕ
3016:≈
2994:ϕ
2981:ϕ
2958:∑
2926:ϕ
2919:≈
2913:˙
2904:ϕ
2841:ϕ
2812:∑
2780:≈
2762:∗
2745:≈
2739:˙
2653:ϕ
2618:∂
2607:ϕ
2603:∂
2581:∑
2561:∂
2553:∂
2535:˙
2494:∂
2483:ϕ
2479:∂
2457:∑
2453:−
2437:∂
2429:∂
2423:−
2408:˙
2345:∂
2334:ϕ
2330:∂
2308:∑
2260:∂
2249:ϕ
2245:∂
2223:∑
2126:δ
2107:∑
2090:δ
2071:∑
2014:δ
2003:˙
1994:−
1985:∂
1977:∂
1960:δ
1949:˙
1931:∂
1923:∂
1879:δ
1873:˙
1864:−
1858:δ
1852:˙
1843:≈
1837:δ
1828:∂
1820:∂
1808:δ
1799:∂
1791:∂
1779:δ
1696:≈
1687:ϕ
1667:∑
1652:∗
1421:−
1405:˙
1380:˙
1251:˙
1242:∂
1234:∂
1174:−
1162:˙
1153:∂
1145:∂
1084:∂
1076:∂
1055:−
1046:˙
1013:∂
1005:∂
978:˙
922:−
913:˙
901:−
895:˙
793:˙
769:−
760:∂
752:∂
746:−
737:¨
702:˙
669:∂
661:∂
655:−
646:¨
587:−
578:˙
566:−
560:˙
510:˙
488:˙
414:^
402:−
396:^
365:→
261:→
246:−
240:→
231:⋅
225:→
194:→
132:and mass
8164:10013503
7956:See also
7078:for the
4013:for all
3787:′
8225:Bibcode
8144:Bibcode
8084:2220894
7682:yields
7457:, for
7060:∧
7045:∧
6282:is the
3946:f(q, p)
1744:when
1574:called
439:in the
300:is the
103:fermion
8193:
8162:
8113:
8092:
8082:
8072:
8047:
7937:
7867:
7854:
7760:
7650:
7544:
7387:
7100:, the
7022:
6271:where
4636:where
4618:
4241:
4150:
3354:where
2050:and
2026:
1885:
1622:s and
1548:. If
1286:
1093:
943:
608:
285:where
8045:S2CID
7677:/2 =
7405:The (
7102:n + 1
7055:and
5885:and
5463:and
5442:and
4428:and
4081:and
3161:The
3080:L = q
2171:and
1603:) ≈ 0
1552:and
1545:f ≈ g
1534:and
110:gauge
8191:ISBN
8160:PMID
8111:ISBN
8090:ISBN
8070:ISBN
7411:+ 1)
7082:and
4272:and
3936:and
1739:* ≈
1526:and
1478:and
344:and
73:and
26:The
8233:doi
8152:doi
8035:doi
7465:≡ z
7455:= 1
7421:, p
7124:= 0
7113:= 1
5431:not
5078:≈ 0
3040:1=0
2192:≈ 0
1752:≈ 0
1612:,
1601:p,q
844:≫ 1
42:in
8253::
8231:.
8221:88
8219:.
8158:.
8150:.
8140:43
8138:.
8130:;
8080:MR
8078:.
8043:.
8029:.
8025:.
7673:=
6278:ab
5414:0.
5075:PB
5068:,
4694:is
4678:iħ
4664:.
4650:ab
4643:ab
4286:.
4171:iħ
4094:,
3449:0.
3276:0.
3143:.
3019:0.
2174:δp
2163:δq
2053:δp
2047:δq
1754:.
1747:δϕ
1741:δH
1737:δH
1605:.
1570:.
1561:=
1312:A
1309:.
842:mc
838:qB
832:.
822:−(
325:V(
319:;
152:.
140:-
66:.
8239:.
8235::
8227::
8166:.
8154::
8146::
8096:.
8086:.
8051:.
8037::
8031:2
7940:.
7934:E
7931:2
7926:i
7922:x
7915:=
7910:i
7900:x
7870:,
7862:2
7858:p
7849:i
7845:x
7838:=
7833:B
7830:D
7826:}
7822:H
7819:,
7814:i
7810:p
7806:{
7803:=
7798:i
7788:p
7763:,
7755:i
7751:p
7747:=
7742:B
7739:D
7735:}
7731:H
7728:,
7723:i
7719:x
7715:{
7712:=
7707:i
7697:x
7679:E
7675:p
7671:H
7653:,
7647:E
7644:2
7641:z
7635:=
7627:2
7623:z
7616:1
7610:2
7600:z
7591:z
7585:=
7576:z
7547:,
7536:2
7532:z
7525:1
7519:2
7509:z
7498:2
7495:1
7490:=
7487:L
7473:2
7470:x
7463:1
7460:x
7453:n
7445:p
7441:x
7436:n
7434:2
7426:)
7423:i
7419:i
7417:x
7415:(
7409:n
7407:2
7390:.
7382:j
7378:p
7372:i
7368:x
7359:i
7355:p
7349:j
7345:x
7341:=
7336:B
7333:D
7329:}
7323:j
7319:p
7315:,
7310:i
7306:p
7302:{
7281:,
7276:j
7272:x
7266:i
7262:x
7253:j
7250:i
7242:=
7237:B
7234:D
7230:}
7224:j
7220:p
7216:,
7211:i
7207:x
7203:{
7182:,
7179:0
7176:=
7171:B
7168:D
7164:}
7158:j
7154:x
7150:,
7145:i
7141:x
7137:{
7122:p
7120:i
7118:x
7111:x
7109:i
7107:x
7097:S
7084:y
7080:x
7064:y
7049:x
7025:.
7016:c
7013:4
7008:B
7005:q
6996:i
6990:=
6987:]
6982:y
6972:p
6965:,
6960:x
6950:p
6943:[
6915:2
6907:i
6904:=
6901:]
6896:y
6886:p
6879:,
6870:y
6864:[
6861:=
6858:]
6853:x
6843:p
6836:,
6827:x
6821:[
6797:B
6794:q
6789:c
6780:i
6774:=
6771:]
6762:y
6756:,
6747:x
6741:[
6711:.
6705:c
6702:4
6697:B
6694:q
6685:=
6680:B
6677:D
6673:}
6667:y
6663:p
6659:,
6654:x
6650:p
6646:{
6617:2
6614:1
6608:=
6603:B
6600:D
6596:}
6590:y
6586:p
6582:,
6579:y
6576:{
6573:=
6568:B
6565:D
6561:}
6555:x
6551:p
6547:,
6544:x
6541:{
6517:B
6514:q
6510:c
6502:=
6497:B
6494:D
6490:}
6486:y
6483:,
6480:x
6477:{
6447:.
6442:B
6439:P
6435:}
6431:g
6428:,
6423:b
6415:{
6410:B
6407:P
6403:}
6397:a
6389:,
6386:f
6383:{
6377:B
6374:q
6367:b
6364:a
6356:c
6350:+
6345:B
6342:P
6338:}
6334:g
6331:,
6328:f
6325:{
6322:=
6317:B
6314:D
6310:}
6306:g
6303:,
6300:f
6297:{
6274:ε
6256:,
6251:b
6248:a
6235:0
6231:B
6227:q
6223:c
6215:=
6210:1
6202:b
6199:a
6195:M
6185:)
6178:0
6173:1
6166:1
6158:0
6151:(
6144:B
6141:q
6137:c
6132:=
6127:1
6120:M
6093:,
6089:)
6082:0
6077:1
6067:1
6062:0
6055:(
6049:c
6045:B
6042:q
6036:=
6033:M
6007:,
6002:c
5998:B
5995:q
5989:=
5984:B
5981:P
5977:}
5971:1
5963:,
5958:2
5950:{
5944:=
5939:B
5936:P
5932:}
5926:2
5918:,
5913:1
5905:{
5891:2
5888:ϕ
5882:1
5879:ϕ
5858:,
5852:y
5844:V
5833:2
5830:1
5822:=
5817:y
5807:p
5779:x
5771:V
5760:2
5757:1
5749:=
5744:x
5734:p
5706:x
5698:V
5686:B
5683:q
5679:c
5674:=
5665:y
5638:y
5630:V
5618:B
5615:q
5611:c
5603:=
5600:}
5595:2
5587:,
5584:x
5581:{
5576:2
5572:u
5568:+
5563:B
5560:P
5556:}
5550:1
5542:,
5539:x
5536:{
5531:1
5527:u
5523:+
5518:B
5515:P
5511:}
5507:H
5504:,
5501:x
5498:{
5495:=
5486:x
5469:2
5466:u
5460:1
5457:u
5448:2
5445:u
5439:1
5436:u
5406:c
5402:B
5399:q
5391:1
5387:u
5377:y
5369:V
5357:=
5352:B
5349:P
5345:}
5339:j
5331:,
5326:2
5318:{
5313:j
5309:u
5303:j
5295:+
5290:B
5287:P
5283:}
5279:H
5276:,
5271:2
5263:{
5242:0
5234:c
5230:B
5227:q
5219:2
5215:u
5211:+
5205:x
5197:V
5185:=
5180:B
5177:P
5173:}
5167:j
5159:,
5154:1
5146:{
5141:j
5137:u
5131:j
5123:+
5118:B
5115:P
5111:}
5107:H
5104:,
5099:1
5091:{
5072:}
5070:H
5065:j
5061:Φ
5059:{
5042:.
5038:)
5034:x
5027:c
5024:2
5019:B
5016:q
5004:y
5000:p
4995:(
4989:2
4985:u
4981:+
4977:)
4973:y
4966:c
4963:2
4958:B
4955:q
4948:+
4943:x
4939:p
4934:(
4928:1
4924:u
4920:+
4917:)
4914:y
4911:,
4908:x
4905:(
4902:V
4899:=
4890:H
4863:.
4860:x
4853:c
4850:2
4845:B
4842:q
4830:y
4826:p
4822:=
4817:2
4808:,
4805:y
4798:c
4795:2
4790:B
4787:q
4780:+
4775:x
4771:p
4767:=
4762:1
4736:)
4733:y
4730:,
4727:x
4724:(
4721:V
4718:=
4715:H
4659:M
4655:M
4639:M
4621:,
4613:B
4610:P
4606:}
4602:g
4599:,
4594:b
4577:{
4572:1
4564:b
4561:a
4557:M
4551:B
4548:P
4544:}
4538:a
4521:,
4518:f
4515:{
4510:b
4507:,
4504:a
4491:B
4488:P
4484:}
4480:g
4477:,
4474:f
4471:{
4468:=
4463:B
4460:D
4456:}
4452:g
4449:,
4446:f
4443:{
4430:g
4426:f
4409:.
4404:B
4401:P
4397:}
4391:b
4374:,
4369:a
4352:{
4349:=
4344:b
4341:a
4337:M
4311:a
4278:2
4275:ϕ
4270:1
4267:ϕ
4247:,
4244:c
4235:i
4232:=
4229:]
4224:2
4207:,
4202:1
4185:[
4153:.
4147:c
4144:=
4139:B
4136:P
4132:}
4126:2
4118:,
4113:1
4105:{
4092:c
4087:2
4084:ϕ
4078:1
4075:ϕ
4044:a
4040:v
4024:j
4020:ϕ
4015:j
3998:,
3995:0
3987:B
3984:P
3980:}
3974:j
3966:,
3963:f
3960:{
3907:.
3902:B
3899:P
3895:}
3889:T
3885:H
3881:,
3878:f
3875:{
3863:f
3847:f
3830:.
3825:k
3815:k
3811:U
3805:k
3797:+
3794:H
3791:=
3784:H
3755:k
3745:a
3740:k
3736:V
3730:a
3726:v
3720:k
3717:,
3714:a
3706:+
3701:k
3691:k
3687:U
3681:k
3673:+
3670:H
3667:=
3662:T
3658:H
3621:a
3617:v
3594:a
3590:v
3566:,
3561:a
3556:k
3552:V
3546:a
3542:v
3536:a
3528:+
3523:k
3519:U
3510:k
3506:u
3492:1
3488:a
3482:k
3478:V
3471:k
3467:u
3441:B
3438:P
3434:}
3428:k
3420:,
3415:j
3407:{
3402:k
3398:V
3392:k
3372:k
3368:V
3361:k
3357:U
3339:,
3334:k
3330:V
3326:+
3321:k
3317:U
3313:=
3308:k
3304:u
3268:B
3265:P
3261:}
3255:k
3247:,
3242:j
3234:{
3229:k
3225:u
3219:k
3211:+
3206:B
3203:P
3199:}
3195:H
3192:,
3187:j
3179:{
3166:k
3163:u
3154:k
3150:u
3135:k
3133:u
3125:k
3121:u
3114:k
3110:u
3099:j
3095:ϕ
3073:.
3069:k
3065:u
3059:.
3055:k
3051:u
3042:.
3011:B
3008:P
3004:}
2998:k
2990:,
2985:j
2977:{
2972:k
2968:u
2962:k
2954:+
2949:B
2946:P
2942:}
2938:H
2935:,
2930:j
2922:{
2908:j
2885:k
2881:u
2863:,
2858:B
2855:P
2851:}
2845:k
2837:,
2834:f
2831:{
2826:k
2822:u
2816:k
2808:+
2803:B
2800:P
2796:}
2792:H
2789:,
2786:f
2783:{
2775:B
2772:P
2768:}
2758:H
2754:,
2751:f
2748:{
2736:f
2720:f
2703:k
2701:u
2683:,
2680:0
2677:=
2674:)
2671:p
2668:,
2665:q
2662:(
2657:j
2626:j
2622:p
2611:k
2595:k
2591:u
2585:k
2577:+
2569:j
2565:p
2556:H
2547:=
2542:j
2532:q
2502:j
2498:q
2487:k
2471:k
2467:u
2461:k
2445:j
2441:q
2432:H
2420:=
2415:j
2405:p
2383:m
2379:u
2361:,
2353:n
2349:p
2338:m
2322:m
2318:u
2312:m
2304:=
2299:n
2295:B
2268:n
2264:q
2253:m
2237:m
2233:u
2227:m
2219:=
2214:n
2210:A
2189:j
2185:Φ
2178:n
2167:n
2145:,
2142:0
2139:=
2134:n
2130:p
2121:n
2117:B
2111:n
2103:+
2098:n
2094:q
2085:n
2081:A
2075:n
2029:,
2023:0
2020:=
2017:p
2010:)
2000:q
1988:p
1980:H
1970:(
1966:+
1963:q
1956:)
1946:p
1940:+
1934:q
1926:H
1916:(
1888:,
1882:q
1870:p
1861:p
1849:q
1840:p
1831:p
1823:H
1814:+
1811:q
1802:q
1794:H
1785:=
1782:H
1764:j
1760:c
1749:j
1731:H
1724:j
1720:c
1702:,
1699:H
1691:j
1681:j
1677:c
1671:j
1663:+
1660:H
1657:=
1648:H
1625:p
1619:q
1614:H
1599:(
1596:j
1592:φ
1585:j
1581:φ
1563:g
1559:f
1554:g
1550:f
1536:g
1532:f
1491:2
1485:y
1481:p
1476:y
1472:4
1448:.
1445:)
1442:y
1439:,
1436:x
1433:(
1430:V
1427:=
1424:L
1416:y
1412:p
1402:y
1396:+
1391:x
1387:p
1377:x
1371:=
1368:)
1363:y
1359:p
1355:,
1350:x
1346:p
1342:,
1339:y
1336:,
1333:x
1330:(
1327:H
1289:,
1283:x
1277:c
1274:2
1269:B
1266:q
1260:=
1248:y
1237:L
1228:=
1223:y
1219:p
1197:y
1191:c
1188:2
1183:B
1180:q
1171:=
1159:x
1148:L
1139:=
1134:x
1130:p
1096:.
1087:y
1079:V
1067:B
1064:q
1060:c
1052:=
1043:x
1016:x
1008:V
996:B
993:q
989:c
984:=
975:y
946:,
940:)
937:y
934:,
931:x
928:(
925:V
919:)
910:x
904:y
892:y
886:x
883:(
877:c
874:2
869:B
866:q
860:=
857:L
840:/
830:)
828:y
826:,
824:x
817:V
799:.
790:x
782:c
778:B
775:q
763:y
755:V
743:=
734:y
728:m
699:y
691:c
687:B
684:q
678:+
672:x
664:V
652:=
643:x
637:m
611:,
605:)
602:y
599:,
596:x
593:(
590:V
584:)
575:x
569:y
557:y
551:x
548:(
542:c
539:2
534:B
531:q
525:+
522:)
517:2
507:y
500:+
495:2
485:x
478:(
473:2
470:m
465:=
462:L
441:z
437:B
420:)
411:x
405:y
393:y
387:x
384:(
379:2
376:B
371:=
362:A
346:y
342:x
338:)
333:r
329:→
321:c
313:B
309:→
294:A
290:→
270:,
267:)
258:r
252:(
249:V
237:v
222:A
214:c
211:q
206:+
201:2
191:v
184:m
178:2
175:1
169:=
166:L
150:B
146:z
142:y
138:x
134:m
130:q
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