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Measuring the position and momentum of a particle is not simultaneously possible according to Heisenberg's uncertainty principle. Heisenberg's uncertainty principle gives us the uncertainty in measuring the position and momentum of a particle simultaneously. We can think of this as the highest possible resolution in measuring distance as $$\Delta{x}$$, and highest possible resolution in measuring momentum as $$\Delta{p}$$. So momentum and position can be thought of as varying in units $$\Delta{p}$$ and $$\Delta{x}$$ respectively, so they can't take on continuous values.

This is my interpretation of Heisenberg's uncertainty principle. My interpretation is not only we cannot measure position and momentum simultaneously, but that momentum and position take on discrete values. For position, it would be $$x+\Delta{x}$$, $$x-\Delta{x}$$, $$x+2\Delta{x}$$, ..., $$x + n\Delta{x}$$, where $$n$$ is an integer. This implies discretization. I thinkIs my interpretation may not be right. If so, what is wrong with my interpretation?

Measuring the position and momentum of a particle is not simultaneously possible according to Heisenberg's uncertainty principle. Heisenberg's uncertainty principle gives us the uncertainty in measuring the position and momentum of a particle simultaneously. We can think of this as the highest possible resolution in measuring distance as $$\Delta{x}$$, and highest possible resolution in measuring momentum as $$\Delta{p}$$. So momentum and position can be thought of as varying in units $$\Delta{p}$$ and $$\Delta{x}$$ respectively, so they can't take on continuous values.

This is my interpretation of Heisenberg's uncertainty principle. My interpretation is not only we cannot measure position and momentum simultaneously, but that momentum and position take on discrete values. For position, it would be $$x+\Delta{x}$$, $$x-\Delta{x}$$, $$x+2\Delta{x}$$, ..., $$x + n\Delta{x}$$, where $$n$$ is an integer. This implies discretization. I think my interpretation may not be right. If so, what is wrong with my interpretation?

I got this question after reading this Forbes article: https://www.forbes.com/sites/startswithabang/2019/04/25/this-is-why-quantum-field-theory-is-more-fundamental-than-quantum-mechanics/?sh=4e4573e2083e

The author, Ethan Siegel, says

The big problem was that quantum mechanics, even relativistic quantum mechanics, wasn't quantum enough to describe everything in our Universe.

Even in General Relativity, where mass and energy curve space, that curved space is continuous, just like any other field.

The problem with this type of formulation is that the fields are on the same footing as position and momentum are under a classical treatment. Fields push on particles located at certain positions and change their momenta. But in a Universe where positions and momenta are uncertain, and need to be treated like operators rather than a physical quantity with a value, we're short-changing ourselves by allowing our treatment of fields to remain classical.

That was the big advance of the idea of quantum field theory, or its related theoretical advance: second quantization. If we treat the field itself as being quantum, it also becomes a quantum mechanical operator.

I interpreted these paragraphs as Heisenberg's uncertainty principle necessitating the treatment of fields as quantized, and thus space as discrete.

Later, however, the author has written articles like: Even In A Quantum Universe, Space And Time Might Be Continuous, Not Discrete and This Is Why Space Needs To Be Continuous, Not Discrete

Does Heisenberg's uncertainty principle imply discretization of position and momentum?

Measuring the position and momentum of a particle is not simultaneously possible according to Heisenberg's uncertainty principle. Heisenberg's uncertainty principle gives us the uncertainty in measuring the position and momentum of a particle simultaneously. We can think of this as the highest possible resolution in measuring distance as $$\Delta{x}$$, and highest possible resolution in measuring momentum as $$\Delta{p}$$. So momentum and position can be thought of as varying in units $$\Delta{p}$$ and $$\Delta{x}$$ respectively, so they can't take on continuous values.

This is my interpretation of Heisenberg's uncertainty principle. My interpretation is not only we cannot measure position and momentum simultaneously, but that momentum and position take on discrete values. For position, it would be $$x+\Delta{x}$$, $$x-\Delta{x}$$, $$x+2\Delta{x}$$, ..., $$x + n\Delta{x}$$, where $$n$$ is an integer. This implies discretization. I think my interpretation may not be right. If so, what is wrong with my interpretation?

I got this question after reading this Forbes article: https://www.forbes.com/sites/startswithabang/2019/04/25/this-is-why-quantum-field-theory-is-more-fundamental-than-quantum-mechanics/?sh=4e4573e2083e

The author, Ethan Siegel, says

The big problem was that quantum mechanics, even relativistic quantum mechanics, wasn't quantum enough to describe everything in our Universe.

Even in General Relativity, where mass and energy curve space, that curved space is continuous, just like any other field.

The problem with this type of formulation is that the fields are on the same footing as position and momentum are under a classical treatment. Fields push on particles located at certain positions and change their momenta. But in a Universe where positions and momenta are uncertain, and need to be treated like operators rather than a physical quantity with a value, we're short-changing ourselves by allowing our treatment of fields to remain classical.

That was the big advance of the idea of quantum field theory, or its related theoretical advance: second quantization. If we treat the field itself as being quantum, it also becomes a quantum mechanical operator.

I interpreted these paragraphs as Heisenberg's uncertainty principle necessitating the treatment of fields as quantized, and thus space as discrete.

Later, however, the author has written articles like: Even In A Quantum Universe, Space And Time Might Be Continuous, Not Discrete and This Is Why Space Needs To Be Continuous, Not Discrete