Oliver Thewalt

    Oliver Thewalt

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    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.