# Hixgrid

### Oliver Thewalt

Theoretical Physics | Quantum Biology | Dark Matter Research | Energy Consulting | Creation of Hydrogen ATOM in the Higgs Field >> Vote for Nobel Prize

## What is time?

by unknown author.

It is commonly asserted that in general relativity there is no absolute
simultaneity. On the other hand, it is asserted that, based on the traveling
time of light, we see the Sun as it was 8 minutes ago and the Andromeda nebula
as it was 2.5 million years ago. This seems to conflict with each other -
apparently we have no diffeomorphism invariant way of assigning a relative time
to a distant object. Let us take a closer look at the issues involved.

The invariant way of defining present is to say that x and y are present if the
two points are in a spacelike relation, and to say y was earlier (or later) than
x if y lies in or on the past (or future) light cone. Thus the present is
well-defined as the complement of the closed light cone.

Now suppose that you look at the sun. If one is really pedantic, one would have
to say that you see the sun in your eye, as a 2D object, and not out there in
3D. But we are accustomed to interpret our sensations in 3D and hence put the
sun far away but into the here.

In general relativity, one goes a step further. One thinks in terms of the 4D
spacetime manifold and places the sun there. Calculating the length of the
geodesic gives a value of 0, so the sun is not in your present. Consideration of
the sign of the time component in an arbitrary proper Lorentz frame, one finds
that the sun is in your past, as everything you observe.

But the amount of invariant time passed, as measured by the metric, is zero.
This looks like a paradox. What happened with the claimed 8 minutes?

The answer is that the metric time is not the right way to measure time. It is
the only time available in a Poincare-invariant flat universe, or in a
diffeomorphism invariant curved universe. An empty universe where only
noninteracting observers move has no notion of simultaneity.

But a matter-filled, homogeneous and isotropic universe generally has one,
defined by the rest frame of the galactic fluid with which general relativity
models cosmology. Since the fluid breaks Lorentz symmetry (except in very
special cases, which are ruled out by experiment) it creates a preferred
foliation of spacetime. This foliation gives a well-defined cosmic time, when
scaled to make the expansion of the universe uniform. (Actually there are
several natural scalings = monotone transformations of the time parameter; see
Section 27.9 in Misner/Thorne/Wheeler, so cosmic time without a reference to the
scale used is ambiguous.)

This cosmic time figures in all models of cosmology. The values commonly talked
about when quoting times for cosmological events, such as the date of the big
bang or the time a photon seen now left the Andromeda nebula, refer to this
cosmological time. Concept of Now : Time is passing. This means that what is
''now'' in our subjective experience changes. But there is no concept of ''now''
in physics.

Classical nonrelativistic mechanics does not know the concept of now. One
declares some time to be ''now'' - but which time one declares to be ''now'' is
completely subjective (i.e., in different situations it will be declared
differently). Similarly, one declares some position to be ''here'', but which
position you declare to be ''here'' is completely subjective, in the same sense.

Since different people assign at different times a different meaning to the time
called ''now'' and the position called ''here'', these words have - from an
objective point of view - simply the meaning of a variable denoting time and
position, respectively. Classical relativistic mechanics does not know the
concept of now, either, but things change a little: Here one declares some event
(= spacetime point) to be ''here and now'' - but which event one declares to be
''here and now'' is completely subjective. Nonrelativistic quantum mechanics
treats time completely differently from space (time is a parameter, space
coordinates are operators), and introduces stochastic elements into the
dynamics. but with respect to ''here'' and ''now'', the situation is identical
with that in the classical nonrelativistic case.

Relativistic quantum mechanics restores the treatment of space and time on equal
footing (space annd time coordinates are parameters), and introduces stochastic
elements into the dynamics. But with respect to ''here and now'', the situation
is identical with that in the classical relativistic case.

Once one has chosen ''here'' and ''now'', respectively ''here and now'', it
serves as origin of the tangent hyperplane, in which localized, flat physics can
be done, reflecting faithfully what happens in a neigborhood of the spacetime
point. This is the domain of relativistic quantum field theory. Time in Q.M Time
in Quantum Mechanics : In the traditional formulation of quantum mechanics, time
is not an observable. Nevertheless it can be observed...

In the Schroedinger picture, the state is defined at fixed times, which
distinguishes the time. In this picture, time measurement is difficult to
discuss since the time at which a state is considered is always sharp.

In the Heisenberg picture, time is simply a parameter in the observables, and
therefore also distinguished, but in a different way. Parameters are in fact
just continuous indices and not observables. As 3 is not an observable while p_3
is one, so t is not an observable but H(t) is one. Observables have at _each_
time an expected value; the moment of time (''now'') is not modelled as
observable.

But what can be modelled is a clock, i.e., a system with an observable which
changes with time in a predictable way. If the observable u(t) of a system
satisfies ubar(t) := <u(t)> = u_0 + v (t - t_0)     (v nonzero)    (*) with
sufficient accuracy, one has a clock and can find out by means of <u(t)> how
much time T = Delta t passed between two observed data sets. This is also the
usual way we measure time in classical physics.

Of course, to be a meaningful time measurement, T must be large enough compared
with the intrinsic uncertainty Sigma_T := |v^{-1}| sigma(u(t)). Here sigma(u(t))
= sqrt(<(u(t)-ubar(t))^2>) is the standard deviation in the properly calibrated
(quantum mechanical) state <.>. If (*) has significant errors then Sigma_T is of
course correspondingly larger.

In relativistic quantum field theory (which in its covariant Version can only be
formulated in the Heisenberg picture), the 1-dimensional time t turns into the
4-dimensional space-time position x. Now x is a vector parameter in the
observables (fields), and hence is not an observable. Space and time are now on
the same level (allowing a covariant point of view), but both as
non-observables. The observables are fields; positions and times of particles
are modelled by unsharp 1-dimensional world lines characterized by a high
density of the expectations of the corresponding fields. (Think of the trace of
a particle in a bubble chamber.)

For position and time measurement, one now needs a 4-vector field u(x) with
<u(x)> = u_0 + V (x - x_0) and a nonsingular 4x4 matrix V, and the intrinsic
uncertainty takes the form Sigma_T := sigma(V^{-1}u(x)) with sigma(a(x)) =
sqrt(<(a(x)-abar(x))^*(a(x)-abar(x))>), abar(x)=<a(x)>.

Conclusion: In nonrelativistic quantum mechanics, time is always measured
indirectly via the expectations of distinguished observables of clocks in
calibrated quantum mechanical states. In relativistic quantum field theory, the
same holds for both position and time.

However, this analysis works only when one assigns to single clocks a
well-defined state, hence assumes a version of the Copenhagen interpretation.

From the point of view of the minimal statistical interpretation, one needs in
contrast a whole ensemble of identically prepared clocks to measure time...

Note that in relativistic quantum mechanics, a single particle is described (in
the absence of an external field) by an irreducible representation of the
Poincare group. Here only the components of 4-momentum and the 4-angular
momentum are observables. From these, one can reconstruct observer-dependent
3-dimensional (Newton-Wigner) position operators satisfying canonical
commutation rules, but not a time operator.

Time and Space: In QED (the most accurate theory we have), space and time are
parameters ranging continuously in R^4, coordinatizing the fields that contain
the physical information. These coordinates have no absolute meaning since
changing them by means of a Poincare transformation (a combination of
translation + rotation + Lorentz boost) does not alter the physics.

The resulting affine pseudo-metric space, called Minkowski space, is (in QED)
absolute and physically meaningful: All Poincare invariants expressible in terms
of the fields can (in principle) be determined objectively. In particular, this
holds for the Minkowski distance between space-time points that can be defined
in terms of the fields. Such space-time points include for example all positions
of stars, which are local maxima of field intensities in the backward light cone
of an observer at a particular time, singled out objectively by appropriate
observables.

According to established physics, a real observer is a macroscopic object with
the capacity to record information. The recording process is described by means
of irreversible thermodynamics. In particular, observers can be described to
good accuracy classically, in terms of their associated macroscopic observables.
These are expectation values of corresponding aggregated microscopic variables,
behaving essentially classically according to Ehrenfest's theorem. Large objects
such as stars can similarly be described by their associated macroscopic
observables. The position of an observer and the objects it observes changes in
time, defining their trajectories = world lines (apart from a global Poincare
transformation). This change is (on the macroscopic description level
appropriate for observers) continuous. (The world lines get fuzzy as one
focusses on smaller and smaller details, and become undetermined in principle
when the scale is reached where quantum effects dominate. Indeed, the Heisenberg
uncertainty principle forbids well-defined trajectories of arbitrary accuracy.)

For example: The observer might be the Mount Palomar observatory, at a fixed
time t. (This may be defined locally, say, ''one year after it was built'', a
property that may be encoded in terms of QED using known physics.) The
observer's past light cone cuts out from 4-space a 3-dimensional manifold, which
intersects the world lines of the objects observed at definite points (within
the accuracy of the whole construction) - the positions x(t) of the visible
stars at time t. This is consistent with how astronomical positions are
determined.

On the level of QED - and even one level below, in the standard model - space
and time are not quantized in any sense. What is quantized are the observable
fields, in dependence on time and position.

The situation changes slightly if we consider quantum gravity, thought to be
relevant on the smallest significant scales of our universe. It is tentatively
explored by current physicists, but without any definitive results so far.
Judging from general relativity (which must be the classical limit of any
meaningful quantum gravity), Minkowski space is now replaced by a
pseudo-Riemannian manifold, the translation subgroup of the Poincare group is
extended to the diffeomorphism group (or, because of anomalies, perhaps only to
the volume-preserving subgroup). Therefore, only the invariants under this
bigger group are physical. This still includes geodesic distance, so that the
picture painted above remains valid in Riemann normal coordinates for any
particular observer.

There is an intrinsic asymmetry between observed position and time - even in the
classical relativistic case!

Whether measured or not, a state is _always_ a state at a particular time t.
Thats why we write state densities as rho(t) and wave functions as psi(t), and
have a dynamical equation to tell how the state changes with time.

Note that in a relativistic theory, position becomes like time rather than time
like position: Instead of trajectories depending on time we have fields
depending on space and time. Note that absolutely precise particle positions for
multiparticle systems don't make relativistic sense - position becomes an
intrinsically smeared concept, even classically!

What one can have consistently in relativity is only relative positions of one
particle with respect to a frame attached to a particular particle and its world
line - in the case of the GPS this ''particle'' is the earth. These positions
are again time-dependent. In nonrelativistic quantum mechanics, a unique
position operator is defined only for a system consisting of a single particle
alone in the universe, in an observer-dependent coordinate system. (The observer
must be outside this mini universe.)

For an N-particle system, one has N position operators. If time were like
position, each particle would have its own time, which would make the concept of
time meaningless. But time has meaning; so this is not an option. operators. If
time were like position, each particle would have its own time, which would make
the concept of time meaningless. But time has meaning; so this is not an option.