Gamma-ray burst progenitors

The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac^[1] to treat classical systems with second class constraints in Hamiltonian mechanics, and to thus allow them to undergo canonical quantization. It is an important part of Dirac's development of Hamiltonian mechanics to elegantly handle more general Lagrangians, when constraints and thus more apparent than dynamical variables are at hand.^[2] More abstractly, the two-form implied from the Dirac bracket is the restriction of the symplectic form to the constraint surface in phase space.^[3]

This article assumes familiarity with the standard Lagrangian and Hamiltonian formalisms, and their connection to canonical quantization. Details of Dirac's modified Hamiltonian formalism are also summarized to put the Dirac bracket in context.

Inadequacy of the standard Hamiltonian procedure

The standard development of Hamiltonian mechanics is inadequate in several specific situations:

When the Lagrangian is at most linear in the velocity of at least one coordinate; in which case, the definition of the canonical momentum leads to a constraint. This is the most frequent reason to resort to Dirac brackets. For instance, the Lagrangian (density) for any fermion is of this form.
When there are gauge (or other unphysical) degrees of freedom which need to be fixed.
When there are any other constraints that one wishes to impose in phase space.

Example of a Lagrangian linear in velocity

An example in classical mechanics is a particle with charge q and mass m confined to the x - y plane with a strong constant, homogeneous perpendicular magnetic field, so then pointing in the z-direction with strength B.^[4]

The Lagrangian for this system with an appropriate choice of parameters is

L={\tfrac {1}{2}}m{\vec {v}}^{2}+{\frac {q}{c}}{\vec {A}}\cdot {\vec {v}}-V({\vec {r}}),

where ${\vec {A}}$ is the vector potential for the magnetic field, ${\vec {B}}$ ; c is the speed of light in vacuum; and $V({\vec {r}})$ is an arbitrary external scalar potential; one could easily take it to be quadratic in x and y, without loss of generality. We use

{\vec {A}}={\frac {B}{2}}(x{\hat {y}}-y{\hat {x}})

as our vector potential. 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.

Explicitly, the Lagrangian amounts to just

L={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})+{\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,

which leads to the equations of motion

m{\ddot {x}}=-{\frac {\partial V}{\partial x}}+{\frac {qB}{c}}{\dot {y}}

m{\ddot {y}}=-{\frac {\partial V}{\partial y}}-{\frac {qB}{c}}{\dot {x}}.

For a harmonic potential, the gradient of V amounts to just the coordinates, −(x,y).

Now, in the limit of a very large magnetic field, qB/mc ≫ 1. One may then drop the kinetic term to produce a simple approximate Lagrangian,

L={\frac {qB}{2c}}(x{\dot {y}}-y{\dot {x}})-V(x,y)~,

with first-order equations of motion

{\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}

{\dot {x}}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}~.

Note that this approximate Lagrangian is linear in the velocities, 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.

Following the Hamiltonian procedure, however, the canonical momenta associated with the coordinates are now

p_{x}={\frac {\partial L}{\partial {\dot {x}}}}=-{\frac {qB}{2c}}y

p_{y}={\frac {\partial L}{\partial {\dot {y}}}}={\frac {qB}{2c}}x~,

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 overcomplete.

A Legendre transformation then produces the Hamiltonian,

H(x,y,p_{x},p_{y})={\dot {x}}p_{x}+{\dot {y}}p_{y}-L=V(x,y).

Note that this "naive" Hamiltonian has no dependence on the momenta, which means that equations of motion (Hamilton's equations) are inconsistent.

The Hamiltonian procedure has broken down. One might try to fix the problem by eliminating two of the components of the 4d phase space, say y and p_y, down to a reduced phase space of 2d, 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 constraint on phase space (between momenta and coordinates) that was never taken into account.

Generalized Hamiltonian procedure

In Lagrangian mechanics, if the system has non-holonomic constraints, then one generally adds Lagrange multipliers 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 phase space in Hamiltonian mechanics, but the solution is similar.

Before proceeding, it is useful to understand the notions of weak equality and strong equality. Two functions on phase space, f and g, are weakly equal if they are equal when the equations of motion are satisfied or on shell, denoted $f\approx g$ . If f and g are equal on and off shell, then they are called strongly equal, written f=g. It is important to note that, in order to get the right answer, no weak equations may be used before evaluating derivatives or Poisson brackets.

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 called primary constraints. The constraints, labeled $\phi _{j}$ , must weakly vanish, $\phi _{j}(q,p)\approx 0$ .

Next, one finds the naive Hamiltonian, H, 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 qs and ps only, even if the velocities cannot be inverted into functions of the momenta.

Generalizing the Hamiltonian

Dirac argues that we should generalize the Hamiltonian (somewhat analogously to the method of Lagrange multipliers) to

H^{*}=H+\sum _{j}c_{j}\phi _{j}\approx H,

where the c_j 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, H^* is the broadest generalization of the Hamiltonian possible.

To further illuminate the c_j, 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):

\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~,

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

\left({\frac {\partial H}{\partial q}}+{\dot {p}}\right)\delta q+\left({\frac {\partial H}{\partial p}}-{\dot {q}}\right)\delta p=0~,

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 δq and δp separately to zero, since the variations are somewhat restricted by the constraints. In particular, the variations must be tangent to the constraint surface.

One can demonstrate the solution to

\sum _{n}A_{n}\delta q_{n}+\sum _{n}B_{n}\delta p_{n}=0,

for the variations δq_n and δp_n restricted by the constraints $\phi _{j}\approx 0$ (assuming the constraints satisfy some regularity conditions) is generally^[5]

A_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial q_{n}}}

B_{n}=\sum _{m}u_{m}{\frac {\partial \phi _{m}}{\partial p_{n}}},

where the u_m are arbitrary functions.

Using this result, the equations of motion become

{\dot {p}}_{j}=-{\frac {\partial H}{\partial q_{j}}}-\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial q_{j}}}

{\dot {q}}_{j}={\frac {\partial H}{\partial p_{j}}}+\sum _{k}u_{k}{\frac {\partial \phi _{k}}{\partial p_{j}}}

\phi _{j}(q,p)=0,

where the u_k are functions of coordinates and velocities that can be determined, in principle, from the second equation of motion above.

The Legendre transform between the Lagrangian formalism and the Hamiltonian formalism has been saved at the cost of adding new variables.

Consistency conditions

The equations of motion become more compact when using the Poisson bracket, since if $f$ is some function of the coordinates and momenta then

{\dot {f}}\approx \{f,H^{*}\}_{PB}\approx \{f,H\}_{PB}+\sum _{k}u_{k}\{f,\phi _{k}\}_{PB},

if one assumes that the Poisson bracket with the u_k (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

{\dot {\phi _{j}}}\approx \{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.

There are four different types of conditions that can result from the above:

An equation that is inherently false, such as 1=0 .
An equation that is identically true, possibly after using one of our primary constraints.
An equation that places new constraints on our coordinates and momenta, but is independent of the $u_{k}$ .
An equation that serves to specify the u_k.

The first case indicates that the starting Lagrangian gives inconsistent equations of motion, such as L=q. The second case does not contribute anything new.

The third case gives new constraints in phase space. A constraint derived in this manner is called a secondary constraint. 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 $\phi _{j}$ 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.

Finally, the last case helps fix the u_k. If, at the end of this process, the u_k 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 u_k are plugged in, the result is called the total Hamiltonian.

Fixing the u_k

The u_k must solve a set of inhomogeneous linear equations of the form

\{\phi _{j},H\}_{PB}+\sum _{k}u_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.

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

u_{k}=U_{k}+V_{k},

where $U_{k}$ is a particular solution and $V_{k}$ is the most general solution to the homogeneous equation

\sum _{k}V_{k}\{\phi _{j},\phi _{k}\}_{PB}\approx 0.

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 u_k (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 $V_{k}^{a}$ where the index a runs from 1 to the number of unphysical degrees of freedom, the general solution to the consistency conditions is of the form

u_{k}\approx U_{k}+\sum _{a}v_{a}V_{k}^{a},

where the v_a are completely arbitrary functions of time. A different choice of the v_a corresponds to a gauge transformation, and should leave the physical state of the system unchanged.^[6]

The total Hamiltonian

At this point, it is natural to introduce the total Hamiltonian

H_{T}=H+\sum _{k}U_{k}\phi _{k}+\sum _{a,k}v_{a}V_{k}^{a}\phi _{k}

and what is denoted

H'=H+\sum _{k}U_{k}\phi _{k}.

The time evolution of a function on the phase space, $f$ is governed by

{\dot {f}}\approx \{f,H_{T}\}_{PB}.

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 nongauge-invariant quantities that the distinction becomes important.

The Dirac bracket

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 Dirac brackets, first-class and second-class constraints need to be introduced.

We call a function Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. of coordinates and momenta first class if its Poisson bracket with all of the constraints weakly vanishes, that is,

\{f,\phi _{j}\}_{PB}\approx 0,

for all Template:Mvar. Note that the only quantities that weakly vanish are the constraints $\phi _{j}$ , 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.^[7]

When the first-class secondary constraints are added into the Hamiltonian with arbitrary $v_{a}$ as the first class primary constraints are added to arrive at the total Hamiltonian, then one obtains the extended Hamiltonian. 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.

For the purposes of introducing the Dirac bracket, of more immediate interest are the second class constraints. Second class constraints are constraints that have nonvanishing Poisson bracket with at least one other constraint.

For instance, consider constraints Template:Mvar₁ and Template:Mvar₂ whose Poisson bracket is simply a constant, Template:Mvar,

\{\phi _{1},\phi _{2}\}_{PB}=c~.

Now, suppose one wishes to employ canonical quantization, then the phase-space coordinates become operators whose commutators become Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. times their classical Poisson bracket. Assuming there are no ordering issues that give rise to new quantum corrections, this implies that

[{\hat {\phi }}_{1},{\hat {\phi }}_{2}]=i\hbar ~c,

where the hats emphasize the fact that the constraints are on operators.

On the one hand, canonical quantization gives the above commutation relation, but on the other hand Template:Mvar₁ and Template:Mvar₂ 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 bilinear, antisymmetric, satisfy the Jacobi identity as does the Poisson bracket, reduce to the Poisson bracket for unconstrained systems, and, additionally, the bracket of any constraint with any other quantity must vanish.

At this point, the second class constraints will be labeled ${\tilde {\phi }}_{a}$ . Define a matrix with entries

M_{ab}=\{{\tilde {\phi }}_{a},{\tilde {\phi }}_{b}\}_{PB}.

In this case, the Dirac bracket of two functions on phase space, f and g, is defined as The Aharonov–Casher effect is a quantum mechanical phenomenon in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It is used for example to observe interference of neutrons or fluxons.

In both effects the particle acquires some phase shift ( $\varphi$ ) while traveling along some path P. In the Aharonov–Bohm effect it is

\varphi _{AB}={\frac {q}{\hbar }}\int _{P}\mathbf {A} \cdot d\mathbf {x}

While for the Aharonov–Casher effect it is

\varphi _{AC}={\frac {1}{\hbar c^{2}}}\int _{P}(\mathbf {E} \times {\boldsymbol {\mu }})\cdot d\mathbf {x}

where $q$ is its charge and ${\boldsymbol {\mu }}$ is the magnetic moment.

Bibliography

One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Template:Cite paper

Illustration on the example provided

Returning to the above example, the naive Hamiltonian and the two primary constraints are

H=V(x,y)

\phi _{1}=p_{x}+{\tfrac {qB}{2c}}y,\qquad \phi _{2}=p_{y}-{\tfrac {qB}{2c}}x.

Therefore the extended Hamiltonian can be written

H^{*}=V(x,y)+u_{1}\left(p_{x}+{\tfrac {qB}{2c}}y\right)+u_{2}\left(p_{y}-{\tfrac {qB}{2c}}x\right).

The next step is to apply the consistency conditions $\{\phi _{j},H^{*}\}_{PB}\approx 0$ , which in this case become

\{\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

\{\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.

These are not secondary constraints, but conditions that fix $u_{1}$ and $u_{2}$ . Therefore, there are no secondary constraints and the arbitrary coefficients are completely determined, indicating that there are no unphysical degrees of freedom.

If one plugs in with the values of $u_{1}$ and $u_{2}$ , then one can see that the equations of motion are

{\dot {x}}=\{x,H\}_{PB}+u_{1}\{x,\phi _{1}\}_{PB}+u_{2}\{x,\phi _{2}\}=-{\frac {c}{qB}}{\frac {\partial V}{\partial y}}

{\dot {y}}={\frac {c}{qB}}{\frac {\partial V}{\partial x}}

{\dot {p}}_{x}=-{\frac {1}{2}}{\frac {\partial V}{\partial x}}

{\dot {p}}_{y}=-{\frac {1}{2}}{\frac {\partial V}{\partial y}},

which are self-consistent and coincide with the Lagrangian equations of motion.

A simple calculation confirms that $\phi _{1}$ and $\phi _{2}$ are second class constraints since

\{\phi _{1},\phi _{2}\}_{PB}=-\{\phi _{2},\phi _{1}\}_{PB}={\frac {qB}{c}},

hence the matrix looks like

M={\frac {qB}{c}}\left({\begin{matrix}0&1\\-1&0\end{matrix}}\right),

which is easily inverted to

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}}}\epsilon _{ab},

where $\epsilon _{ab}$ is the Levi-Civita symbol. Thus, the Dirac brackets are defined to be

\{f,g\}_{DB}=\{f,g\}_{PB}+{\frac {c\epsilon _{ab}}{qB}}\{f,\phi _{a}\}_{PB}\{\phi _{b},g\}_{PB}.

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 brackets, and get the correct equations of motion, which one can easily confirm.

To quantize the system, the Dirac brackets between all of the phase space variables are needed. The nonvanishing Dirac brackets for this system are

\{x,y\}_{DB}=-{\tfrac {c}{qB}}

\{x,p_{x}\}_{DB}=\{y,p_{y}\}_{DB}={\frac {1}{2}}

while the cross-terms vanish, and

\{p_{x},p_{y}\}_{DB}=-{\tfrac {qB}{4c}}.

Therefore, the correct implementation of canonical quantization dictates the commutation relations,

[{\hat {x}},{\hat {y}}]=-i{\tfrac {\hbar c}{qB}}

[{\hat {x}},{\hat {p}}_{x}]=[{\hat {y}},{\hat {p}}_{y}]=i{\frac {\hbar }{2}}

with the cross terms vanishing, and

[{\hat {p}}_{x},{\hat {p}}_{y}]=-i{\tfrac {\hbar qB}{4c}}~.

Interestingly, this example has a nonvanishing commutator between ${\hat {x}}$ and ${\hat {y}}$ , which means this structure specifies a noncommutative geometry. (Since the two coordinates do not commute, there will be an uncertainty principle for the x and y positions.)

Similarly, for free motion on a hypersphere Sⁿ, the n+1 coordinates are constrained $x_{i}x^{i}=1$ . From a plain kinetic Lagrangian, it is evident that their momenta are perpendicular to them, $x_{i}p^{i}=0$ . Thus the corresponding Dirac Brackets are likewise simple to work out,^[8]

\{x_{i},x_{j}\}_{DB}=0,

\{x_{i},p_{j}\}_{DB}=\delta _{ij}-x_{i}x_{j},

\{p_{i},p_{j}\}_{DB}=x_{j}p_{i}-x_{i}p_{j}~.

The 2(n+1) constrained phase-space variables (x_i, p_i) obey much simpler Dirac brackets than the 2n unconstrained variables, had one eliminated one of the xs and one of the ps through the two constraints ab initio, would obey plain Poisson brackets. The Dirac brackets add simplicity and elegance, at the cost of excessive (constrained) phase-space variables.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Gamma-ray burst progenitors

Contents

Inadequacy of the standard Hamiltonian procedure

Example of a Lagrangian linear in velocity