Re: h bar and Lorentz transformation

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Date: Mon, 28 Mar 2005 01:46:37 +0000 (UTC)

Non Ame <noname@nospam.net> writes:

>"Ken S. Tucker" <dynamics@vianet.on.ca> writes:

>>See that Bilges B(r',t') differs from Non Ame's
>>B_y(r',t), substantially, where prime's are
>>concerned. Which of you is correct?
>>Need clarification.

>E(r,t) is the electric field at position r and time t. B(r',t) is the
>magnetic field at position r' and time t. B(r',t') is the magnetic field
>at position r' and time t'.

>E_x(r,t) B_y(r',t') - B_y(r',t') E_x(r,t) is the commutator of E_x(r,t)
>and B_y(r',t'). E_x(r,t) B_y(r',t) - B_y(r',t) E_x(r,t) is a special case
>when E_x and B_y are evaluated at exactly the same time.

>Schiff gives the formulae

> E_s(r,t) E_{s'}(r',t') - E_{s'}(r',t') E_s(r,t)

> = B_s(r,t) B_{s'}(r',t') - B_{s'}(r',t') B_s(r,t)

> = 4 pi i hbar ((delta_{ss'}/c^2) d/dt d/dt' - d/dr_s d/dr'_{s'})

> D(r - r', t - t'),

> E_s(r,t) B_{s'}(r',t') - B_{s'}(r',t') E_s(r,t)

> = - 4 pi i hbar epsilon_{ss's"} d/dr_{s"} d/dt' D(r - r', t - t'),

>for s, s' = x, y, z, where delta_{ss'} is the Kronecker delta,
>epsilon_{ss's"} is the Levi-Civita symbol determined by epsilon_{xyz} = 1,
>summation is taken over all values of s" in the last line above, and

>D(rho, tau) = (4 pi |rho|)^{-1} [delta(|rho|+c tau) - delta(|rho|-c tau),

>where delta(r) is the Dirac delta function on R.

Another thing of interest is that Ken S. Tucker has claimed to be able to
derive Quantum Theory as a consequence of General Relativity.

One book that Ken seems to enjoy quoting (and misapplying) is Steven
Weinberg's "Gravitation and Cosmology", so it is interesting to see what
Weinberg has to say about the connection between Quantum Mechanics and
General Relativity.

Weinberg devotes Section 10.8 of his book to a "Quantum Theory of
Gravitation". At the beginning of the Section, Weinberg states that
"At present there does not exist any complete and self-consistent quantum
theory of gravitation". It is interesting to note that, far from
suggesting that Quantum Theory is a derivable of General Relativity,
Weinberg actually asserts that the two theories had not yet been united.
He writes:

        The preceding remarks describe what may be called a semiclassical
        theory of gravitation. The development of a true quantum theory
        of gravitation is unfortunately much more difficult. One approach
        is to construct an interaction Hamiltonian that can create and
        destroy gravitons, and then calculate transition probabilities as
        a power series in this interaction. Usually the Hamiltonian would
        be built up out of quantum fields, of the form

        h_{rho nu}(x) = sum_mu int d^3k {a(k,mu) e_{rho nu}(k,mu)

        exp(i k_lambda x^lambda) + a^{dag}(k,mu) e*_{rho nu}(k,mu)

        exp(- i k_lambda x^lambda)}, (10.8.14)

        where e_{rho nu}(k,mu) is a polarization tensor for a graviton of
        momentum hbar k and helicity mu, and a(k,mu) and a^{dag}(k,mu)
        are the corresponding annihilation and creation operators,
        characterized by the commutation relation

        [a(k,mu),a^{dag}(k',mu')] = delta^3(k-k') delta_{mu mu'} (10.8.15)

        [a(k,mu),a(k',mu')] = [a^{dag}(k,mu),a^{dag}(k',mu')]= 0 (10.8.16)

        The difficulty with this approach comes from the fact that the
        operator (10.8.15) cannot be a Lorentz tensor as long as the
        helicity sum is limited to the physical values mu = +/- 2; as we
        saw in Section 10.2, a true tensor would have helicities 0 and
        +/- 1 as well as +/- 2. It is true that we can start with a true
        tensor and then subject e_{mu nu} to a gauge transformation that
        will eliminate the unphysical helicities 0 and +/- 1, but once we
        choose a gauge in this way, h_{mu nu} is no longer a tensor. To
        put this another way, a gauge condition, such as the statement
        that e_{13}, e_{23}, e_{10}, e_{20}, e_{00}, e_{03} and e_{33}
        vanish for k in the 3-direction, is not Lorentz invariant, as if
        we define these components to vanish, then under a Lorentz
        transformation Lambda^mu_nu, h_{mu nu} will not simply transform
        into Lambda_mu^rho Lambda_nu^sigma h_{rho sigma}, but will be
        subjected to an additional gauge transformation:

        h_{mu nu} -> Lambda_mu^rho Lambda_nu^sigma h_{rho sigma}

        + d epsilon_mu/dx^nu + d epsilon_nu/dx^mu

        It is no easy task to construct a Hamiltonian out of such an
        object in such a way as to obtain Lorentz-invariant transition
        probabilities.

        There are two possible ways out of this difficulty. One
        possibility is to accept the nontensor character of h_{mu nu}, and
        use the noncovariant Hamiltonian formalism to derive Lorentz-
        invariant rules for calculation of transition amplitudes. This
        works fairly easily in electrodynamics, but the self-interaction
        of the gravitational field has so far prevented the completion of
        this program in general relativity. A different method, pioneered
        by Feynman, is to start out with manifestly Lorentz-invariant
        calculational rules, and then tinker with them to prevent the
        appearance of unphysical particles with helicities 0 and +/- 1 in
        physical states. This program has been successfully carried
        through to completion in the work of Fadeev, Mandelstam, and
        DeWitt.

        Unfortunately, the formulation of general rules for the
        calculation of transition probabilities in the quantum theory of
        gravitation has only confirmed the presence of another difficulty:
        The theory contains infinities, arising from integrals over large
        virtual momenta. Quantum electrodynamics contains similar
        infinities, but only in three or four special places, where they
        can be dealt with by a renormalization of mass, charge and wave
        functions. In contrast, the quantum theory of gravitation
        contains an infinite variety of infinities. as can be seen by an
        elementary dimensional argument. The gravitational constant has
        dimensions hbar/m^2, so a term in a dimensionless probability
        amplitude of order G^n will diverge like a momentum-space integral
        int p^{2n-1} dp. In this respect, the theory of gravitation is
        more like other nonrenormalizable theories, such as the Fermi
        theory of beta decay, than it is like quantum electrodynamics.

        Despite these difficulties, there is one very important conclusion
        that can already be drawn from the quantum theory of gravitation:
        It is quite impossible to construct a Lorentz invariant quantum
        theory of particles of mass zero and helicity +/- 2 without
        building some sort of gauge invariance into the theory, because
        only in this way can the interaction of the nontensor field
        h_{mu nu} generate Lorentz-invariant transition amplitudes.
        However, we saw in Section 10.2 that the theory of gravitational
        radiation is gauge-invariant because general relativity is
        generally covariant, and, as argued in Section 4.1, general
        covariance is but the mathematical expression of the Principle of
        Equivalence. It therefore appears that the Principle of
        Equivalence, on which the whole of classical general relativity is
        based, is itself a consequence of the requirement that the quantum
        theory of gravitation should be Lorentz invariant.

Since Weinberg is discussing Lorentz invariance and Lorentz covariance,
the metric is a weak quantum field (Chapter 10 deals with the weak-field
approximation, anyway, so there is no surprise, there), and it also means
that spacetime is contractible (here, "contractible" is being used in its
mathematical meaning), or, in other words, homotopic to a point. It
follows that the quantum theory that Weinberg discusses would not be a
full theory of quantum gravity, but only a weak-field approximation.

Note also that for the annihilation and creation operators introduced in
the passage quoted above,

          a(k,mu) a^{dag}(k',mu') - a^{dag}(k',mu') a(k,mu)

        = delta^3(k-k') delta_{mu mu'},

as a consequence of (10.8.15), where delta^3 is the Dirac delta function
on R^3 and delta_{mu mu'} is the Kronecker delta for helicities mu and
mu', and

        a(k,mu) a(k',mu') = a(k',mu') a(k,mu),

        a^{dag}(k,mu) a^{dag}(k',mu') = a^{dag}(k',mu') a^{dag}(k,mu),

for all values of k, k', mu, mu', as a consequence of (10.8.16).

The fact that Weinberg explicitly discusses a quantum theory of
gravitation in Section 10.8, and made no mention of quantum theory until
that point, demonstrates that he has presented general relativity as a
non-quantum theory up to that point, and, in that section, quantization is
discussed from the point of view of quantum theory as already developed
independently of general relativity. The fact that Weinberg points out
that "at present there does not exist any complete and self-consistent
quantum theory of gravitation", also demonstrates that quantum theory
cannot be developed from general relativity, since, if it could be,
quantum theory would itself be a complete and self-consistent theory of
gravitation (or, alternatively, the consistency of a quantum theory of
gravitation would be a consequence of the consistency of general
relativity). In other words, Weinberg does not present quantum theory as
a consequence of general relativity. Weinberg presents quantum theory and
general relativity as two independent theories that require to be
reconciled.

Weinberg also points out about the problems of renormalization in the case
of weak quantum fields over an almost Minkowski space, and does not even
venture to discuss quantum theory for the general case in general
relativity.

• Previous message: Josef Matz: "Re: magnetic propeties from spinning electric field"
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