Subject: Re: Macavity
Date: Tue, 20 Aug 2002 17:02:51 +0300 From: Dimi Chakalov <dchakalov@surfeu.at> To: Adam Helfer <adam@math.missouri.edu> CC: Larry Ford <ford@cosmos.phy.tufts.edu>, Thomas Roman <roman@ccsu.edu>, Christopher J Fewster <cjf3@york.ac.uk>, William G Unruh <unruh@physics.ubc.ca>, Ken D Olum <kdo@cosmos5.phy.tufts.edu>, Michael John Pfenning <mitchel@neptune.physics.uoguelph.ca>, Eric Poisson <poisson@physics.uoguelph.ca>, Rainer Verch <verch@theorie.physik.unigoettingen.de>, Noah Graham <graham@physics.ucla.edu>, Xavier Siemens <siemens@cosmos2.phy.tufts.edu>, Ralf Schutzhold <schuetz@theory.phy.tudresden.de>, Don N Page <don@phys.ualberta.ca>, Carlos Barcelo <carlos.barcelo@port.ac.uk>, questions@niac.usra.edu BCC: [snip] On Tue, 26 Mar 2002 10:18:45 0600, Adam Helfer wrote: > Macavity was created by T. S. Eliot, in the poem "Macavity,
the Thank you, once more, for the reference. I liked the story of this invisible cat which shows up only when no one is looking at it, just like the negative energy density, http://members.aon.at/chakalov/Ford.html#2 http://members.aon.at/chakalov/Wald.html#3 I'm a bit puzzled by your recent grqc/0208045 [Ref. 1]. You wrote: "Thus there appear to be two approaches for better understanding the allowed spatial distributions of negative energy. One is the search for spacetime averaged quantum inequalities, and the other is the continuation of the program begun in Ref. [19], which seeks to extract as much information as possible from the worldline inequalities. Both of these approaches are currently under investigation." It seems to me that Ref. [19] is grqc/0109061 which I mentioned in my email from Thu, 20 Sep 2001 21:28:18 +0200, http://members.aon.at/chakalov/Ford.html#20_Sep_2001 I'm curious, have you decided to drop the quantum measurement issues, contrary to what you stated in grqc/9709047 http://members.aon.at/chakalov/Ford.html#2 ? What you and your colleagues choose to investigate is your business, I would just like to thank you for your beautiful paper [Ref. 1]. I believe it tallies exactly to the hypothesis about two modes of time, http://members.aon.at/chakalov/dimi.html and does shed light on some very important issues, such as gravity control, http://members.aon.at/chakalov/Tajmar.html http://members.aon.at/chakalov/right.html#UFOs http://members.aon.at/chakalov/Tumulka.html Going back to the real life, it seems to me that there is no instant "now", http://members.aon.at/chakalov/Baez.html#now which is yet another reason to pursue new ideas about time, http://members.aon.at/chakalov/Straumann.html http://members.aon.at/chakalov/Krasnikov.html The situation is very puzzling and exiting, just like the one with the ultraviolet catastrophe back in 1900, isn't it? Kindest regards, Dimi Chakalov
References [Ref. 1] L.H. Ford, Adam D. Helfer, Thomas
A. Roman. Spatially Averaged Quantum Inequalities Do Not Exist in FourDimensional
Spacetime. Fri, 16 Aug 2002 20:02:45 GMT,
"It is well known that quantum fields can produce local
renormalized negative energy densities. This is despite the fact that the
classical expression for the energy density appears to be positive definite.
The negative energy density is possible because renormalization involves
an infinite subtraction, and is needed to make the stress tensor operator
welldefined. If there were no restrictions on the sorts of negative energy
densities attainable, one could have a number of bizarre possibilities,
including traversable wormholes [1, 2], fasterthanlight travel [3, 4,
6] time travel [1, 2, 5], and violations of the second law of thermodynamics
[7]. Such phenomena are, at best, rare.
"The physical interpretation of this constraint is that
there is an inverse relation between the magnitude of the negative energy
and its duration.
"It is natural to ask if there is a generalization of Eq. (4) to fourdimensional spacetime, especially one which gives a nontrivial bound when the sampling is over space only. One of us [22] has given an argument that this is not the case, and that the spatially sampled energy density can be unbounded below. Actually, the argument was written out there for a more generic situation, that of evolution from a Cauchy surface in a general curved spacetime. While explicit constructions were given in this case, they were by pseudodifferential operator techniques, and they were not translated into more conventional quantum fieldtheoretic terms. Also, the details of the more special case of a surface of constant time in Minkowski space were not written out. "So no explicit version of the argument in conventional
specialrelativistic quantum fieldtheoretic terms has yet been given.
The purpose of the present paper is to provide such an example and to draw
as many physical insights as possible from it. In the following section,
we will provide the explicit construction of a class of quantum states
for the massless, minimally coupled scalar field, with negative energy
density, as well as give the twopoint function and the energy density
in this class of states. We then show that although this state satisfies
Eq. (1), the spatially sampled energy density can be arbitrarily negative.
Further implications of this example are discussed in Sect. 3.
"However, the key point here is that we can consider a
sequence of states, each with a finite value of [lambda] . Therefore
each state in this sequence will have the Hadamard form. By progressively
increasing the values of [lambda] as we vary over the states
in the sequence, we can make the energy density in our spatially sampled
region as negative as we like. In the next section, we will discuss some
of the insights which may be drawn from this example.
"One can argue from energy conservation that physically
what is happening is that there must be large fluxes of positive energy
entering the region, which damp out the negative energy sufficiently for
the temporal inequality to hold.
"The current results also shed light on an earlier proposed
explanation of the "Garfinkle box", given in Ref. [19]. This refers to
an unpublished result of Garfinkle [24], who showed that the total energy
of a scalar field, E , contained within an imaginary box in Minkowski
spacetime, at fixed time, is unbounded below. The box is "imaginary" in
the sense that there are no physical boundaries at the walls of the box.
The Garfinkle result is that there exist quantum states for which
E is arbitrarily negative.
"Thus there appear to be two
approaches for better understanding the allowed spatial distributions of
negative energy. One is the search for spacetime averaged quantum inequalities,
and the other is the continuation of the program begun in Ref. [19], which
seeks to extract as much information as possible from the worldline inequalities.
Both of these approaches are currently under investigation."
