This session opened the second half of the conference, devoted to discussion of the problems of *quantizing* the gravitational field, previous sessions having been restricted to the classical domain. The first contribution was an introduction by P. G. BERGMANN

BERGMANN *both* the positions and velocities of the Schwarzschild

These aims have not yet been achieved, but BERGMANN*just* the gravitational and electromagnetic fields.

BERGMANN

In the quantum theory the state vector of a generally covariant system will be subject to various constraints

As remaining problems which must eventually be looked at, BERGMANN

**1**The hyperquantized particle field vs. the treatment of particles as singularities in a quantized gravitational field.

**2**The interaction of the gravitational field with fermions.

**3**The interaction of the gravitational field with other quantized fields.

**4**The relation of elementary particle theory to unitary field theories.

**5**The relation of the law of conservation of energy and momentum arising from the coordinate transformation invariance of general relativity to other *strong* conservative laws of physics with their associated invariance
groups.

Discussion then turned to the problems of measurement of the gravitational field. This item was placed first on the agenda in an attempt to keep physical concepts as much as possible in the foreground in a subject which can otherwise be quickly flooded by masses of detail and which suffers from lack of experimental
guideposts. The question was asked: What are the

Consider just a classical test particle of mass which is initially at rest at the point (we restrict ourselves to one dimension) in a gravitational potential . At the time the position of the particle will be

The gravitational field strength is given by
and can immediately be determined by a measurement of
and
. If however the particle is subject to quantum laws its initial position and velocity are subject to (root mean square) uncertainties related by

leading to an uncertainty in position at time of amount

The initial position measurement may be made by a photon of momentum uncertainty
(or, in principle, by a graviton having the same momentum uncertainty if
one prefers to deal with absolute electrically neutral test particles!). The resulting uncertainty in the initial time may be ignored since it is given by
(assuming
). The final position and time measurements may be made with arbitrarily high precision, using energetic photons, since the experiment is then over.

The gravitational field can now be determined from the classical equation if

However, one must be able to choose
small enough so that
does not change appreciably during the course of the motion. This imposes a condition on the gravitational field, namely

or

If the gravitational field is produced by a point mass M then
,
,
, and the required condition becomes
which can always be satisfied. Evidently the gravitational field of any mass is measurable in principle because of the long “tail” in the Newtonian force law. In fact, the long tail is what permits us to “measure” the

It is, however, still a matter of basic interest to determine whether or

If both the test particle and source has protonic mass then

and all conditions may seemingly be satisfied by choosing
to have the not unreasonable value
sec. Moreover, no complications should arise due to the nonlinear character of the gravitational field (the linear Newtonian approximation should be
masses as small as a proton).

Nevertheless, a number of subtleties enter the problem at this point, chiefly concerning the nature of the recording apparatus, or clock, which measures the requisite time intervals (independently of the photons which interact with the test particle) and how the presence of the clock's mass in the neighborhood may influence results of the experiment.)