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In the case of the four-dimensional bosonic higher-spin theory the relevant higher-spin algebra is very simple thanks to formula_2 and can be built upon two-dimensional quantum Harmonic oscillator. In the latter case two pairs of creation/annihilation operators formula_3 are needed. These can be packed into the quartet |
formula_4 of operators obeying the canonical commutation relations |
where formula_6 is the formula_7 invariant tensor, i.e. it is anti-symmetric. As is well known, the bilinears provide an oscillator realization of formula_7: |
The higher-spin algebra is defined as the algebra of all even functions formula_10 in formula_11. That the functions are even is in accordance with the bosonic content of the higher-spin theory as formula_11 will be shown to be related to the Majorana spinors from the space-time point of view and even powers of formula_11 correspond to tensors. It is an associative algebra and the product is conveniently realised by the Moyal star product: |
with the meaning that the algebra of operators formula_15 can be replaced with the algebra of function formula_16 in ordinary commuting variables formula_17 (hats off) and the product needs to be replaced with the non-commutative star-product. For example, one finds |
and therefore formula_19 as it would be the case for the operators. Another representation of the same star-product is more useful in practice: |
The exponential formula can be derived by integrating by parts and dropping the boundary terms. The prefactor is chosen as to ensure formula_21. In the Lorentz-covariant base we can split formula_22 and we also split formula_23. Then the Lorentz generators are formula_24, formula_25 and the translation generators are formula_26. The formula_27-automorphism can be realized in two equivalent ways: either as formula_28 or as formula_29. In both the cases it leaves the Lorentz generators untouched and flips the sign of translations. |
The higher-spin algebra constructed above can be shown to be the symmetry algebra of the three-dimensional Klein-Gordon equation formula_30. Considering more general free CFT's, e.g. a number of scalars plus a number of fermions, the Maxwell field and other, one can construct more examples of higher-spin algebras. |
The Vasiliev equations are equations in certain bigger space endowed with auxiliary directions to be solved for. The additional directions are given by the doubles of formula_17, called formula_32, |
which are furthermore entangled with Y. The star-product on the algebra of functions in formula_33 in formula_34-variables is |
The integral formula here-above is a particular star-product that corresponds to the Weyl ordering among Y's and among Z's, with the opposite signs for the commutator: |
Moreover, the Y-Z star product is normal ordered with respect to Y-Z and Y+Z as is seen from |
The higher-spin algebra is an associative subalgebra in the extended algebra. In accordance with the bosonic projection is given by formula_38. |
The essential part of the Vasiliev equations relies on an interesting deformation of the Quantum harmonic oscillator, known as deformed oscillators. First of all, let us pack the usual creation and annihilation operators formula_39 in a doublet formula_40. The canonical commutation relations (the formula_41-factors are introduced to facilitate comparison with Vasiliev's equations) |
can be used to prove that the bilinears in formula_43 form formula_44 generators |
In particular, formula_46 rotates formula_43 as an formula_48-vector with formula_49 playing the role of the formula_48-invariant metric. The deformed oscillators are defined by appending the set of generators with an additional generating element formula_51 and postulating |
Again, one can see that formula_46, as defined above, form formula_48-generators and rotate properly formula_43. At formula_56 we get back to the undeformed oscillators. In fact, formula_43 and formula_46 form the generators of the Lie superalgebra formula_59, where formula_43 should be viewed as odd generators. Then, formula_61 is the part of the defining relations of formula_59. |
One (or two) copies of the deformed oscillator relations form a part of the Vasiliev equations where the generators are replaced with fields and the commutation relations are imposed as field equations. |
The equations for higher-spin fields originate from the Vasiliev equations in the unfolded form. |
Any set of differential equations can be put in the first order form by introducing auxiliary fields to denote derivatives. Unfolded approach is an advanced reformulation of this idea that takes into account gauge symmetries and diffeomorphisms. Instead of just formula_63 the unfolded equations are written in the language of differential forms as |
where the variables are differential forms formula_65 of various degrees, enumerated by an abstract index formula_66; formula_67 is the exterior derivative formula_68. The structure function formula_69 is assumed to be expandable in exterior product Taylor series as |
where formula_71 has form degree formula_72 and the sum is over all forms whose form degrees add up to formula_73. The simplest example of unfolded equations are the zero curvature equations formula_74 for a one-form connection formula_75 of any Lie algebra formula_76. Here formula_66 runs over the base of the Lie algebra, and the structure function formula_78 encodes the structure constants of the Lie algebra. |
Since formula_79 the consistency of the unfolded equations requires |
which is the Frobenius integrability condition. In the case of the zero curvature equation this is just the Jacobi identity. Once the system is integrable it can be shown to have certain gauge symmetries. Every field formula_71 that is a form of non-zero degree formula_72 possesses a gauge parameter formula_83 that is a form of degree formula_84 and the gauge transformations are |
The Vasiliev equations generate the unfolded equations for a specific field content, which consists of a one-form formula_75 and a zero-form formula_87, both taking values in the higher-spin algebra. Therefore, formula_88 and formula_89, formula_90. The unfolded equations that describe interactions of higher-spin fields are |
where formula_92 are the interaction vertices that are of higher and higher order in the formula_87-field. The product in the higher-spin algebra is denoted by formula_94. The explicit form of the vertices can be extracted from the Vasiliev equations. The vertices that are bilinear in the fields are determined by the higher-spin algebra. Automorphism formula_27 is induced by the automorphism of the anti-de Sitter algebra that flips the sign of translations, see below. |
If we truncate away higher orders in the formula_87-expansion, the equations are just the zero-curvature condition for a connection formula_75 of the higher-spin algebra and the covariant constancy equation for a zero-form formula_87 that takes values in the twisted-adjoint representation (twist is by the automorphism formula_27). |
The field content of the Vasiliev equations is given by three fields all taking values in the extended algebra of functions in Y and Z: |
As to avoid any confusion caused by the differential forms in the auxiliary Z-space and to reveal the relation to the deformed oscillators the Vasiliev equations are written below in the component form. |
The Vasiliev equations can be split into two parts. The first part contains only zero-curvature or covariant constancy equations: |
where the higher-spin algebra automorphism formula_112 is extended to the full algebra as |
the latter two forms being equivalent because of the bosonic projection imposed on formula_114. |
Therefore, the first part of the equations implies that there is no nontrivial curvature in the x-space since formula_115 is flat. The second part makes the system nontrivial and determines the curvature of the auxiliary connection formula_108: |
The existence of the Klein operators is of utter importance for the system. They realise the formula_112 automorphism as an inner one |
In other words, the Klein operator formula_121 behave as formula_122, i.e. it anti-commutes to odd functions and commute to even functions in y,z. |
These 3+2 equations are the Vasiliev equations for the four-dimensional bosonic higher-spin theory. Several comments are in order. |
To prove that the linearized Vasiliev equations do describe free massless higher-spin fields we need to consider the linearised fluctuations over the anti-de Sitter vacuum. First of all we take the exact solution where formula_150 is a flat connection of the anti-de Sitter algebra, formula_149 and formula_152 and add fluctuations |
Above it was used several times that formula_155, i.e. the vacuum value of the S-field acts as the derivative under the commutator. It is convenient to split the four-component Y,Z into two-component variables as formula_156. Another trick that was used in the fourth equation is the invertibility of the Klein operators: |
The fifth of the Vasiliev equations is now split into the last three equation above. |
The analysis of the linearized fluctuations is in solving the equations one by one in the right order. Recall that one expects to find unfolded equations for two fields: one-form formula_158 and zero-form formula_104. From the fourth equation it follows that |
formula_160 does not depend on the auxiliary Z-direction. Therefore, one can identify formula_161. |
The second equation then immediately leads to |
where formula_163 is the Lorentz covariant derivative |
where ... denote the term with formula_165 that is similar to the first one. The Lorentz covariant derivative comes from the usual commutator action of the spin-connection part of formula_143. The term with the vierbein results from the formula_112-automorphism that flips the sign of the AdS-translations and produces anti-commutator formula_168. |
To read off the content of the C-equation one needs to expand it in Y and analyze the C-equation component-wise |
Then various components can be seen to have the following interpretation: |
The last three equations can be recognized to be the equations of the form formula_179 where formula_180 is the exterior derivative on the space of differential forms in the Z-space. Such equations can be solved with the help of the Poincare Lemma. In addition one needs to know how to multiply by the Klein operator from the right, which is easy to derive from the integral formula for the star-product: |
I.e. the result is to exchange the half of the Y and Z variables and to flip the sign. The solution to the last three equations can be written as |
where a similar formula exists for formula_183. |
Here the last term is the gauge ambiguity, i.e. the freedom to add exact forms in the Z-space, and formula_184. |
One can gauge fix it to have formula_185. Then, one plugs the solution to the third equation, which of the same type, i.e. a differential equation of the first order in the Z-space. Its general solution is again given by the Poincare Lemma |
where formula_187 is the integration constant in the Z-space, i.e. the de-Rham cohomology. It is this integration constant that is to be identified with the one-form formula_188 as the name suggests. After some algebra one finds |
where we again dropped a term with dotted and undotted indices exchanged. The last step is to plug the solution into the first equation to find |
and again the second term on the right is omitted. It is important that formula_191 is not a flat connection, while formula_192 is a flat connection. To analyze the formula_191-equations it is useful to expand formula_191 in Y |
The content of the formula_191-equation is as follows: |
To conclude, anti-de Sitter space is an exact solution of the Vasiliev equations and upon linearization over it one finds unfolded equations that are equivalent to the Fronsdal equations for fields with s=0,1,2,3... . |
so that the fields are now function of formula_208 and space-time coordinates. The components of the fields are required to have the right spin-statistic. The equations need to be slightly modified. |
There also exist Vasiliev's equations in other dimensions: |
The equations are very similar to the four-dimensional ones, but there are some important modifications in the definition of the algebra that the fields take values in and there are further constraints in the d-dimensional case. |
Discrepancies between Vasiliev equations and Higher Spin Theories. |
Most of the studies concern with the four-dimensional Vasiliev equations. |
The correction to the free spin-2 equations due to the scalar field stress-tensor was extracted out of the four-dimensional Vasiliev equations and found to be |
where formula_211 are symmetrized derivatives with traces subtracted. The most important information is in the coefficients formula_212 and in the prefactor formula_213, where formula_214 is a free parameter that the equations have, see Other dimensions, extensions, and generalisations. It is important to note that the usual stress-tensor has no more than two derivative and the terms formula_215 are not independent (for example, they contribute to the same formula_216 AdS/CFT three-point function). This is a general property of field theories that one can perform nonlinear (and also higher derivative) field redefinitions and therefore there exist infinitely many ways to write the same interaction vertex at the classical level. The canonical stress-tensor has two derivatives and the terms with contracted derivatives can be related to it via such redefinitions. |
A surprising fact that had been noticed before its inconsistency with the AdS/CFT was realized is that the stress-tensor can change sign and, in particular, vanishes for formula_217. This would imply that the corresponding correlation function in the Chern-Simons matter theories vanishes, formula_218, which is not the case. |
The most important and detailed tests were performed much later. It was first shown that some of the three-point AdS/CFT functions, as obtained from the Vasiliev equations, turn out to be infinite or inconsistent with AdS/CFT, while some other do agree. Those that agree, in the language of Unfolded equations correspond to formula_219 and the infinities/inconsistencies resulted from formula_220. The terms of the first type are local and are fixed by the higher spin algebra. The terms of the second type can be non-local (when solved perturbatively the master field formula_221 is a generating functions of infinitely many derivatives of higher spin fields). These non-localities are not present in higher spin theories as can be seen from the explicit cubic action. |
As is briefly mentioned in Other dimensions, extensions, and generalisations there is an option to introduce infinitely many additional coupling constants that enter via phase factor formula_223. As was noted, the second such coefficient formula_224 will affect five-point AdS/CFT correlation functions, but not the three-point ones, which seems to be in tension with the results obtained directly from imposing higher spin symmetry on the correlation functions. Later, it was shown that the terms in the equations that result from |
formula_225 are too non-local and lead to an infinite result for the AdS/CFT correlation functions. |
In three dimensions the Prokushkin-Vasiliev equations, which are supposed to describe interactions of matter fields with higher spin fields in three dimensions, are also affected by the aforementioned locality problem. For example, the perturbative corrections at the second order to the stress-tensors of the matter fields lead to infinite correlation functions. There is, however, another discrepancy: the spectrum of the Prokushkin-Vasiliev equations has, in addition to the matter fields (scalar and spinor) and higher spin fields, a set of unphysical fields that do not have any field theory interpretation, but interact with the physical fields. |
Since the Vasiliev equations are quite complicated there are few exact solutions known |
In general relativity, the quadrupole formula describes the rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment. The formula reads |
where formula_2 is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. formula_3 is the gravitational constant, formula_4 the speed of light in vacuum, and formula_5 is the mass quadrupole moment. |
The formula was first obtained by Albert Einstein in 1918. After a long history of debate on its physical correctness, observations of energy loss due to gravitational radiation in the Hulse–Taylor binary discovered in 1974 confirmed the result, with agreement up to 0.2 percent (by 2005). |
In physics, Born reciprocity, also called reciprocal relativity or Born–Green reciprocity, is a principle set up by theoretical physicist Max Born that calls for a duality-symmetry among space and momentum. Born and his co-workers expanded his principle to a framework that is also known as reciprocity theory. |
Born noticed a symmetry among configuration space and momentum space representations of a free particle, in that its wave function description is invariant to a change of variables "x" → "p" and "p" → −"x". (It can also be worded such as to include scale factors, e.g. invariance to "x" → "ap" and "p" → −"bx" where "a", "b" are constants.) Born hypothesized that such symmetry should apply to the four-vectors of special relativity, that is, to the four-vector space coordinates |
Both in classical and in quantum mechanics, the Born reciprocity conjecture postulates that the transformation "x" → "p" and "p" → −"x" leaves invariant the Hamilton equations: |
From his reciprocity approach, Max Born conjectured the invariance of a space-time-momentum-energy line element. Born and H.S. Green similarly introduced the notion an invariant (quantum) metric operator formula_5 as extension of the Minkowski metric of special relativity to an invariant metric on phase space coordinates. The metric is invariant under the group of quaplectic transformations. |
Such a reciprocity as called for by Born can be observed in much, but not all, of the formalism of classical and quantum physics. Born's reciprocity theory was not developed much further for reason of difficulties in the mathematical foundations of the theory. |
However Born's idea of a quantum metric operator was later taken up by Hideki Yukawa when developing his nonlocal quantum theory in the 1950s. In 1981, Eduardo R. Caianiello proposed a "maximal acceleration", similarly as there is a minimal length at Planck scale, and this concept of maximal acceleration has been expanded upon by others. It has also been suggested that Born reciprocity may be the underlying physical reason for the T-duality symmetry in string theory, and that Born reciprocity may be of relevance to developing a quantum geometry. |
Born chose the term "reciprocity" for the reason that in a crystal lattice, the motion of a particle can be described in "p"-space by means of the reciprocal lattice. |
This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry. |
There are different forms of the Poynting vector, the most common are in terms of the E and B or E and H fields. |
For spectral quantities two definitions are in use to refer to the same quantity, in terms of frequency or wavelength. |
In astrophysics, "L" is used for "luminosity" (energy per unit time, equivalent to "power") and "F" is used for "energy flux" (energy per unit time per unit area, equivalent to "intensity" in terms of area, not solid angle). They are not new quantities, simply different names. |
The capstan equation or belt friction equation, also known as Eytelwein's formula (after Johann Albert Eytelwein), relates the hold-force to the load-force if a flexible line is wound around a cylinder (a bollard, a winch or a capstan). |
Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small "holding" force exerted on one side can carry a much larger "loading" force on the other side; this is the principle by which a capstan-type device operates. |
A holding capstan is a ratchet device that can turn only in one direction; once a load is pulled into place in that direction, it can be held with a much smaller force. A powered capstan, also called a winch, rotates so that the applied tension is multiplied by the friction between rope and capstan. On a tall ship a holding capstan and a powered capstan are used in tandem so that a small force can be used to raise a heavy sail and then the rope can be easily removed from the powered capstan and tied off. |
In rock climbing with so-called top-roping, a lighter person can hold (belay) a heavier person due to this effect. |
where formula_2 is the applied tension on the line, formula_3 is the resulting force exerted at the other side of the capstan, formula_4 is the coefficient of friction between the rope and capstan materials, and formula_5 is the total angle swept by all turns of the rope, measured in radians (i.e., with one full turn the angle formula_6). |
Several assumptions must be true for the formula to be valid: |
It can be observed that the force gain increases exponentially with the coefficient of friction, the number of turns around the cylinder, and the angle of contact. Note that "the radius of the cylinder has no influence on the force gain". |
The table below lists values of the factor formula_9 based on the number of turns and coefficient of friction "μ". |
From the table it is evident why one seldom sees a sheet (a rope to the loose side of a sail) wound more than three turns around a winch. The force gain would be extreme besides being counter-productive since there is risk of a riding turn, result being that the sheet will foul, form a knot and not run out when eased (by slacking grip on the tail (free end)). |
It is both ancient and modern practice for anchor capstans and jib winches to be slightly flared out at the base, rather than cylindrical, to prevent the rope (anchor warp or sail sheet) from sliding down. The rope wound several times around the winch can slip upwards gradually, with little risk of a riding turn, provided it is tailed (loose end is pulled clear), by hand or a self-tailer. |
For instance, the factor "153,552,935" (5 turns around a capstan with a coefficient of friction of 0.6) means, in theory, that a newborn baby would be capable of holding (not moving) the weight of two supercarriers (97,000 tons each, but for the baby it would be only a little more than 1 kg). The large number of turns around the capstan combined with such a high friction coefficient mean that very little additional force is necessary to hold such heavy weight in place. The cables necessary to support this weight, as well as the capstan's ability to withstand the crushing force of those cables, are separate considerations. |
Generalization of the capstan equation for a V-belt. |
The belt friction equation for a v-belt is: |
where formula_11 is the angle (in radians) between the two flat sides of the pulley that the v-belt presses against. A flat belt has an effective angle of formula_12. |
The material of a V-belt or multi-V serpentine belt tends to wedge into the mating groove in a pulley as the load increases, improving torque transmission. |
For the same power transmission, a V-belt requires less tension than a flat belt, increasing bearing life. |
Generalization of the capstan equation for a rope lying on an arbitrary orthotropic surface. |
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