From: carlip@physics.ucdavis.edu   
      
   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.  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."  States with   
   regions of negative energy can be created by moving mirrors (the   
   shape of a negative energy pulse can be controlled by the acceleration)   
   -- see Fulling and Davies, Proc. R. Soc. Lond. A348 (1976) 393.   
   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.  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.   
      
   A "single particle state" in the usual sense doesn't have negative   
   energy.  But the standard definition of a single particle state is   
   completely nonlocalized -- a momentum eigenstate has a completely   
   undetermined position -- while negative energy can exist only in a   
   finite region (the total energy must still be positive).  But a   
   compact particle-like pulse is easy to construct from a moving   
   mirror.   
      
   >> 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).   
      
   > 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.   
      
   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.   
      
   Steve Carlip   
      
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