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This was a very good explanation, thank you. One question: the Standard Model describes a finite number of "fundamental" particles/waveforms, correct? What ma

This was a very good explanation, thank you. One question: the Standard Model describes a finite number of "fundamental" particles/waveforms, correct? What makes them "fundamental"? Presumably they are orthogonal (or at least span a complete vector space), but there must be some restriction on the (otherwise infinite) domain? (e.g. "waveforms with total energy = 1" and "wavelength is kX where k < 4" or some such.)

nilaykumar 14y ago
Fundamental particles enter the Standard Model through the Lagrangian. Each particle is given its own field in the action: [http://en.wikipedia.org/wiki/Standard_Model#Construction_of_the_Standard_Model_Lagrangian http://en.wikipedia.org/wiki/Standard_Model#Construction_of_...]. I'm not quite sure what you mean by "orthogonal" as the mathematics involved is not simply linear algebra.
- colanderman 14y ago
  kmm stated that "we have our quantum field and we write it as the sum of our elementary wavefunctions", so I assumed that any quantum field can be described by a superposition of the elementary wavefunctions convolved with some sort of particle-position function P(p, x) (where, roughly, P(p, x) = E if particle p exists at position x with energy E, otherwise P(p, x) = 0). Am I wrong to assume this? Edit: from what I can understand about the formation of the Standard Model from the Lagrangian, the domain restriction which makes the elementary wavefunctions elementary is wavefunctions which evolve over time (according to EM theory) such that when viewed as particles, their motion is compatible with QM (or relativity?), and that there are only a finite number of distinct wavefunctions which are needed to span this space. Is this roughly correct?
kmm 14y ago
Are you under the impression that all particles live in one All-Embracing Majesty? That is not true, although all particles of the same type are excitations of a single field, different types of particles have their own fields. For example, the electron and the tau have two separate fields. The weak interaction makes for some mixing, but that is not really important. If you're wondering why there's exactly 12 (+ 1 for the Higgs?) fields, I cannot answer that question and it's one of the open questions in current theoretical physics.
- colanderman 14y ago
  I understand that there are different fields. More so my question is, why are there a finite (as opposed to infinite) number of fundamental waveforms for any given field? My intuition is that this is because the domain of "candidate" waveforms is restricted to those which exhibit particle-like behavior (i.e. compatible with a particle physics). I'm not sure if this intuition is correct however.
  - kmm 14y ago
    We chose our basis because it is convenient. It's called a Fock space[1]. You can choose any basis you want, the equations of motion derived from the Lagrangian will tell you how any wavefunction evolves, independently from your choice of basis. The number of waveforms isn't finite by the way. There's a fundamental waveform for every momentum (which can be any real number) and for every number of particles (which must be a positive integer). [1]: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Fock_space
    - colanderman 14y ago
      Yes, I realized by "finite number of waveforms" I actually meant "finite number of waveform-generating functions parameterized over momentum/energy and number of particles". But I think you and lmm have answered my question, thanks!
  - lmm 14y ago
    There are an infinite number of possible waveforms; pretty much any waveform in the electron field is a valid one. What we see as a single electron is simply a "spike" waveform localised in a particular position. There's no reason you couldn't have a more smeared-out waveform that was "an electron somewhere in the universe", though entanglement comes into play at some point. When we have particular constraints (e.g. known energy) that constrains the space of possible waveforms. E.g. when we talk about there being an electron in an orbital around an atomic nucleus, what we actually mean is there's a waveform. of a particular shape around the nucleus. Are you asking why we only ever see waveforms corresponding to whole numbers of electrons? That's the "quantum" part of quantum mechanics; certain values are quantized (e.g. electric charged). I don't have a good intuition for why that's so though, except to observe that the time evolution of a system preserves this quantization, so there's no way to ever go from having one electron to having half an electron (for example).

nilaykumar 14y ago

Fundamental particles enter the Standard Model through the Lagrangian. Each particle is given its own field in the action: [http://en.wikipedia.org/wiki/Standard_Model#Construction_of_the_Standard_Model_Lagrangian http://en.wikipedia.org/wiki/Standard_Model#Construction_of_...]. I'm not quite sure what you mean by "orthogonal" as the mathematics involved is not simply linear algebra.

colanderman 14y ago
kmm stated that "we have our quantum field and we write it as the sum of our elementary wavefunctions", so I assumed that any quantum field can be described by a superposition of the elementary wavefunctions convolved with some sort of particle-position function P(p, x) (where, roughly, P(p, x) = E if particle p exists at position x with energy E, otherwise P(p, x) = 0). Am I wrong to assume this? Edit: from what I can understand about the formation of the Standard Model from the Lagrangian, the domain restriction which makes the elementary wavefunctions elementary is wavefunctions which evolve over time (according to EM theory) such that when viewed as particles, their motion is compatible with QM (or relativity?), and that there are only a finite number of distinct wavefunctions which are needed to span this space. Is this roughly correct?

kmm 14y ago

Are you under the impression that all particles live in one All-Embracing Majesty? That is not true, although all particles of the same type are excitations of a single field, different types of particles have their own fields. For example, the electron and the tau have two separate fields. The weak interaction makes for some mixing, but that is not really important. If you're wondering why there's exactly 12 (+ 1 for the Higgs?) fields, I cannot answer that question and it's one of the open questions in current theoretical physics.

colanderman 14y ago
I understand that there are different fields. More so my question is, why are there a finite (as opposed to infinite) number of fundamental waveforms for any given field? My intuition is that this is because the domain of "candidate" waveforms is restricted to those which exhibit particle-like behavior (i.e. compatible with a particle physics). I'm not sure if this intuition is correct however.
- kmm 14y ago
  We chose our basis because it is convenient. It's called a Fock space[1]. You can choose any basis you want, the equations of motion derived from the Lagrangian will tell you how any wavefunction evolves, independently from your choice of basis. The number of waveforms isn't finite by the way. There's a fundamental waveform for every momentum (which can be any real number) and for every number of particles (which must be a positive integer). [1]: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Fock_space
  - colanderman 14y ago
    Yes, I realized by "finite number of waveforms" I actually meant "finite number of waveform-generating functions parameterized over momentum/energy and number of particles". But I think you and lmm have answered my question, thanks!
- lmm 14y ago
  There are an infinite number of possible waveforms; pretty much any waveform in the electron field is a valid one. What we see as a single electron is simply a "spike" waveform localised in a particular position. There's no reason you couldn't have a more smeared-out waveform that was "an electron somewhere in the universe", though entanglement comes into play at some point. When we have particular constraints (e.g. known energy) that constrains the space of possible waveforms. E.g. when we talk about there being an electron in an orbital around an atomic nucleus, what we actually mean is there's a waveform. of a particular shape around the nucleus. Are you asking why we only ever see waveforms corresponding to whole numbers of electrons? That's the "quantum" part of quantum mechanics; certain values are quantized (e.g. electric charged). I don't have a good intuition for why that's so though, except to observe that the time evolution of a system preserves this quantization, so there's no way to ever go from having one electron to having half an electron (for example).