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The value of a quantum field at a point in spacetime is not precisely determined.

I haven’t studied quantum physics but came across this. Please explain it to me in simple term what it physically means.

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A quantum field at a point relates to the probability of finding a particle at this point. However, according to the Heisenberg Uncertainty Principle, a particle does not have the exact position and exact momentum (and therefore energy) at the same time. The smaller the uncertainty is in the position, the larger the uncertainty is in the energy and momentum. For this reason, if we wanted to measure a particle at a point of spacetime, we would need an infinite amount of energy, which is impossible. Consequently the value of the quantum field at a point cannot be determined precisely.

The idea of quantum field values at points of spacetime is one of the problems with the standard quantum field theory. The mathematical representation of this problem is that a field at a point is not an operator in a Hilbert space, an infinite-dimensional functional space used to describe a state of the quantum system.

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  • $\begingroup$ What about in the case of a scalar field, does the field have a precise value at a given point? $\endgroup$ – parker Aug 14 '18 at 23:49
  • $\begingroup$ Uncertainty means you cannot measure the exact value. This is a consequence of the wave properties of matter. If you have a wave with the wavelength of, say, 1m, and try to measure the value of the wavelength in a very small area, say, 1mm, then you would see the wave shape over such a short distance appears nearly straight instead of sinusoidal. So the precision of your measurement would be limited. Your error margin would be high and you wouldn't know if the wavelength is 1 m or 2m or 0.5m. $\endgroup$ – safesphere Aug 14 '18 at 23:57
  • $\begingroup$ No, it is the same for the scalar field that relates to particles with a zero spin (like the Higgs boson). The value of spin is unrelated to the uncertainty between position and momentum. $\endgroup$ – safesphere Aug 15 '18 at 0:28
  • $\begingroup$ a)We just can’t measure the precise value of a field at a point. or b)The field doesn’t have a precise value at a point. - Which one is correct? Thanks. $\endgroup$ – parker Aug 15 '18 at 17:18
  • $\begingroup$ The latter "b" is correct. However this not that the field value is "blurred" to a certain degree. It is a balance betwen two things. If you measure the exact value of the field, then you don't know where exactly this value is. And if you measure the field at a very specific infinitely small point, then you don't know at all what the field value is. And finally, if you measure the field in the approximate position, then you get the approximate value. The better you know the position, the more uncertain the value is and vice versa. There also other views like the pilot wave theory. $\endgroup$ – safesphere Aug 15 '18 at 17:44

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