From: bergervo@iae.nl   
      
   On 5/18/2018 9:17 AM, Steven Carlip wrote:   
   > On 5/8/18 7:21 AM, Jos Bergervoet wrote:   
   >> On 5/6/2018 10:33 PM, Steven Carlip wrote:   
   >>> On 5/5/18 11:06 PM, Phillip Helbig (undress to reply) wrote:   
   >   
   >>>> Two questions:   
   >>>> First, if negative-mass black holes don't exist, can one by the same   
   >>>> argument rule out negative-mass hammers?   
   >   
   >>> With very high probability.  In principle, a quantum fluctuation   
   >>> could create a negative mass hammer (with a *very* short lifetime)   
   >> Why do you think that is possible? How would the field look like   
   >> (at some point in time during that short lifetime?)   
   >   
   >> Can you give the wave functional for any negative energy "fluctuation"   
   >> (No need to make it a hammer, something at the complexity level of a   
   >> single particle state suffices! Feel free to simplify things to 1+1   
   >> dimensions and/or with just a scalar field. But please show how it's   
   >> possible..)   
   >   
   > The fact that a generic quantum field theory contains states with   
   > regions of negative energy has been known for decades.   
      
   If it's only a 'region' then at least it doesn't contradict the   
   total positive energy requirement (not even for a tiny interval   
   of time!) so my main objection is gone. Still, my question stands,   
   how the field would look in those situations..   
      
   >   ..  The earliest   
   > paper I know (there may be earlier ones) is by Epstein, Glaser, and   
   > Jaffe, Nuovo Cimento 36 (1965) 1016, whose abstract states, "It is   
   > shown that a positive definite local energy density is incompatible   
   > with the usual postulates of local field theory."   
      
   Unfortunately I can't chase paywall articles in this holiday weekend   
   here, but clearly this comes down to how exactly to split the energy   
   into local parts. Which is easy enough to analyze:   
      
   As field theory is basically just the description of our universe by   
   harmonic oscillators in each point of space, with coupling to their   
   neighbors, the simplest toy model is that of 2 coupled oscillators   
   (effectively giving space 1 dimension, of finite size, discretized   
   on - admittedly - a very course lattice!) The Hamiltonian would be:   
      
      H = -d^2/du^2 -d^2/dv^2 + m u^2 + m v^2 + g (u-v)^2   
      
   for the field amplitudes u and v in the two space points of our toy   
   universe. And we could *choose* to split this over the 2 points like:   
      
      H1 = -d^2/du^2 + (m+g) u^2 - g u v   
      H2 = -d^2/dv^2 + (m+g) v^2 - g u v   
      
   If we take for our wave functional a kind of Rydberg state, i.e. a   
   wave packet moving around in the uv-plane in a quasi-classical way,   
   at a time when it is momentarily stationary (so the energy is mainly   
   in the potential, not the momentum) and located in the uv-plane at a   
   position in the lower half of the 1st quadrant, e.g. u=10, v=2, then   
   it can have negative energy for the H2 part while H1 is at the same   
   time positive, provided g is large enough compared to m.   
      
   > Squeezed states of light can have regions of negative energy -- see,   
   > for example, section III.B of Kuo and Ford, gr-qc/9304008   
   > for a proof.   
      
   Yes, probably a somewhat lower excitation than the Rydberg state   
   will suffice. :-)   
      
   > Section III.A of the same paper has an extremely   
   > simple example of a superposition of the vacuum and a two-particle   
   > state that has regions of negative energy.   
      
   Hmm.. that would be the (1 + a u^2 + b v^2 * c u v) exp(-u^2-v^2)   
   wave functional or something similar for the two-point universe..   
   Of course some (u^2+u v) exp(-...) combination comes close to making   
   a bump at the position I mentioned above..   
      
   > A "single particle state" in the usual sense doesn't have negative   
   > energy.   
      
   Indeed with even simpler states (than 2-particle) it definitely   
   won't work.   
      
   >  But the standard definition of a single particle state is   
   > completely nonlocalized -- a momentum eigenstate has a completely   
   > undetermined position   
      
   This is a bit far-stretching. I would state that any superposition   
   of single-particle states is also a single-particle state. So any   
   Gaussian wave packet or otherwise localized shape can be allowed!   
   In fact, even the most basic scattering theory already requires the   
   in- and out-states to be wave packets (albeit sometimes hidden in   
   the derivations..)   
      
     ...   
   >>> provided there is a larger positive energy fluctuation elsewhere.   
   >>   
   >> Why larger and not equal?   
   >   
   > This is the "quantum interest conjecture," due to Ford and Roman,   
   > gr-qc/9901074.  There are also bounds on the amount of negative   
   > energy related to entanglement (the quantum null energy conjecture,   
   > which has recently been proven for a large class of theories --   
   > see Leichenauer, Levine, and Shahbazi-Moghaddam, arXiv:1802.02584).   
      
   Thanks, but that seems to go much deeper! And it certainly is based   
   on less trivial theories than the harmonic oscillator example (which   
   does, however, also answer the "why larger and not equal" question.)   
      
   >> Anyhow, I'm merely interested in what the   
   >> negative fluctuation conceptually is in your picture. (If you can't   
   >> give it's wave functional, then please show in occupation number   
   >> representation how it can be defined!)   
   >   
   >    A negative fluctuation is a state for which there is a region in   
   > which the time-time component of the quantum stress-energy tensor   
   > has a negative expectation value.  The simplest example I know is   
   > the one I mentioned above, a superposition of the vacuum and a   
   > two-particle state of a massless scalar field.   
      
   This leaves one question: is it just a mathematical artifact?   
   After all, the toy Hamiltonian could also have been split as   
      
      Ha = -d^2/du^2 + m u^2 + 1/2 g (u-v)^2   
      Hb = -d^2/dv^2 + m v^2 + 1/2 g (u-v)^2   
      
   in which case of course both parts are manifestly non-negative for   
   any parameter choice where the total H starts out non-negative!   
      
   > You can actually do better in some cases, and find an actual exact   
   > probability distribution of fluctuations of energy in a region   
   > specified by a "smearing function."  For the case of a two-dimensional   
   > conformal field theory, this is done by Fewster, Ford, and Roman,   
   > arXiv:1004.0179, with a generalization by Fewster and Hollands in   
   > arXiv:1805.04281.  For four dimensions, I think the best we have   
   > at the moment is a set of estimates and bounds, in Fewster, Ford,   
   > and Roman, arXiv:1204.3570, and Fewster and Ford, arXiv:1508.02359.   
      
   I'll need time to study this, but as I said my question remains   
   whether the coupled harmonic oscillators in QFT *really need*   
   negative-energy-allowing local Hamiltonian terms?! The examples   
   all seem to be based on 'standard assumptions' about doing things.   
   Why not keep everything manifestly positive? At least for the Ha+Hb   
   splitting above, that is possible, also for N larger than 2.   
      
   --   
   Jos   
      
   --- SoupGate-Win32 v1.05   
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