Quantum Fields as having infinite degrees of freedom?

[evac-q8r]

The theory of quantum fields is very strange, indeed, I must admit. Usually in books they introduce a quantum field from the standpoint of a vibrating string in one dimension. Along the string are discrete points or masses that when one of which are disturbed a disturbance is created along the length of the string in the from of a wave. As a result, each mass or point is displaced by a certain amount at a specific time, which makes sense. Even when you pass this case into the case of the continuum in which the points are ever so close together it still makes sense.

But the problem comes when we try to insist that a point particle can be described by a field which has a value at every space-time point. To make matters worse we are now using Minkowski space which is 4-dimensional and the time and space are mixtures of one another. There is no analogy between the original displacement in the string and a displacement of the field in Minkowski space because each space-time point may not move or be displaced. So how can a particle be described by a field which has infinitely many degrees of freedom? Even if we tried to describe this in the one-dimensional case it still makes no sense.

 

[selfAdjoint]

I have seen attempts to visualize this (in the euclidean case) by thinking of a mesh of springs, interconnected in a three dimensional littice with some definite spacing, and all vibrating in SHM. Once you have that in mind, take the lattice spacing to zero, and voila!

The Lorentzian case is harder, and careful physicists point out that the transition from Euclid to Lorentz is not as simple as Wick continuation makes it seem.

 

[Haelfix]

Thats correct Wick rotation is often abused.

The fact that fields need to have infinite degrees of freedom ultimately boils down to a Unitarity problem, and is inescapable if you believe in relativistic quantum mechanics.

 

[evac-q8r]

I just finished reading in a book entitled How is Quantum Field Theory Possible? by Sunny Y. Auyang that the fields are local fields whose Lagrangians depend only on the properties in the infinitesimal neighborhood of the point x. If this is the case, then I might have an easier time understanding what a field is. If I understand correctly a particle field does not take up the entire space but only a localized region.