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I hope that I am using appropriate terminology. My confusion about Quantum Theory (beyond my obvious unfamiliarity with its terminology) is basically twofold: 1) I lack an adequate understanding of how the mathematics of Quantum Theory is supposed to correspond to phenomena in the physical world, and 2) I still have an incomplete picture in my mind of how cause and effect relationships occur at the Quantum level of reality. This is why phenomena such as "entanglement" make absolutely no sense to me. So, in an attempt to come to some understanding of all of this, I would like to know that if what we conceptualize as a "field" is merely an interaction among particles (bosons and fermions in the case of Quantum fields), and particles (themselves) are actually fluctuations in "fields", then which comes first in the hierarchy of cause and effect relationships, particles or "fields"?

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I would say quantum fields are more fundamental. Numbers of particles, what kind (i.e. which field they belong to), where and when they are and their state of motion, are merely ways of describing states of fields. –  WetSavannaAnimal aka Rod Vance Jul 1 at 3:46
Firstly, fields are more fundamental. More helpfully, perhaps something that can help lead your intuition and learning about how the mathematics fit into the physical world is that the mathematics best describes what information can be known about physical phenomenon and how they interact, not necessarily the physical thing itself. The wavefunction is an incredibly useful tool for encapsulating what we know about what something is and how it behaves, but it's not 'real' physical phenomena in the way I think you're conceptualizing it. –  sakanojo Jul 1 at 4:02
I think fields are more fundamental. You can have fields without particles, but you cannot have particles without fields. For example you have the Higgs field in your apartment, but I'm sure there is no Higgs boson at your premises. –  mpv Jul 1 at 8:58
This reminds me of an intresting article I recently read in the german version of scientific american, maybe its in the english version too. They ask about the same question, and come up with the idea that neither is fundamental, but all are just "projections of properties" and that those properties are fundamental. They had a term based on "ontology" to describe it, maybe you would find this intresting... –  PlasmaHH Jul 1 at 9:51
A related article: There are no particles, there are only fields, Am. J. Phys. 81, 211 (2013) and the comment on it and other citations –  Tarek Jul 1 at 11:14
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4 Answers 4

This is a tricky question because it asks about the meaning of words. People use the word "particle" to refer to various, not always well defined, notions in physics.

In the end, I think the simplest and more correct single way to categorize the terms is to interpret "particle" as "excitation of a field". For example, if someone says

"There are two electrons in this box."

I would mentally translate that to

"The electron field in this box has two units of excitation."

This is all much easier to think about if you're familiar with the so-called "second quantization" $^{[1]}$.

Second quantization

Consider a one-dimensional infinte wall potential (ie "particle in a box"). The system has a set of discrete energy levels, which we can index as

$$\left\{ A, B, C, D, \ldots \right\}$$

If we have only one particle, we can denote it's state as eg. $|\Psi \rangle_1 = |B\rangle + |D\rangle$ $^{[2]}$. This is the so-called first quantization. If we have two particles, the situation is significantly more complex because, as you have probably learned, quantum particles are indistinguishable. You probably learned that you have to symmetrize (bosons) or antisymmetrize (fermions) the state vector to account for the fact that the particles are indistinguishable. For example, if you say that particle #1 is in state $|\Psi\rangle_1$ as written above, and particle #2 is in state $|\Psi\rangle_2=|C\rangle$, then the total system state is (assuming boson particles):

$$(|B\rangle_1 + |D\rangle_1)|C\rangle_2 + |C\rangle_1 (|B\rangle_2 + |D\rangle_2)$$ $$= |B\rangle_1 |C\rangle_2 + |D\rangle_1 |C\rangle_2 + |C\rangle_1 |B\rangle_2 + |C\rangle_1 |D\rangle_2.$$

This notation is horrible. In symmetrization/antisymmetrization you are basically saying:

"My notation contains information that it shouldn't, namely the independent states of particles which are actually indistinguishable, so let me add more terms to my notation to effectively remove the unwanted information."

This should seem really awkward and undesirable, and it is.

Let me give an analogy for why this is so bad. Consider a violin string which has a set of vibrational modes. If you want to specify the state of the string, you enumerate the modes and specify the amplitude of each one, eg with a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With that analogy, the first quantization notation, where you index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. All units of excitation are the same because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is. In quantum mechanics, excitations come in discrete units $^{[3]}$, so we could specify a state like this:

$$|n_A\rangle_A |n_B\rangle_B |n_C\rangle_C |n_D\rangle_D$$

where $n_i$ is an integer. In this notation, the state $|\Psi\rangle$ from before is written

$$|\Psi\rangle_1 = |0\rangle_A |1\rangle_B |0\rangle_C |1\rangle_D.$$

For compactness this would often be written $|\Psi\rangle_1=|0101\rangle$. The more complex two particle state would be

$$|0\rangle_A |1\rangle_B |1\rangle_C |0\rangle_D + |0\rangle_A |0\rangle_B |1\rangle_C |1\rangle_D$$

or, more compactly,

$$|0110\rangle + |0011\rangle.$$

This is the so-called second quantization notation. Note that it has less terms than the first quantized version. This is because it doesn't need to undo information that it's not supposed to have.

Back to fields vs. particles

The second quantized notation is far better because it naturally accounts for the "indistinguishable" particles. But, what we really learned, is that particles are actually units of excitation of quantum states. In the field theory language, we'd say that the particle is a unit of excitation of the various modes of the field. I won't say that either fields or particles are more fundamental because one has little meaning without the other, but now that we understand what "particle" really means, the whole situation is hopefully much clearer to you.

P.S. Since I obviously spent a long time writing this, I do hope you'll ask for clarification as needed.

[1] The term "second quantization" is stupid, so don't try to interpret it.

[2] We ignore normalization.

[3] Hence the term "quantum".

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+1 for such good explanations –  HolgerFiedler Jul 1 at 4:51
In this picture (IInd Quantized) bosons and fermions are self explanatory (no need to specify them separately) –  lavkush Jul 1 at 9:16
This seems like a really great explanation of "2nd quantisation", which until now had been nothing more than a deliciously mysterious term for me. I don't feel enlightened about which is more fundamental though! Still, +1 anyway :-) –  Benjohn Jul 1 at 13:49
@Benjohn: In the section titled "Back to fields vs. particles" I tried to explain that, in my opinion and from the perspective of 2nd quantization, the meaning of "particle" is "excitation of field". I don't know that either "particle" or "field" is more "fundamental". The point was to just explain the meaning of the words. –  DanielSank Jul 1 at 15:00
@Benjohn: What's more fundamental, a violin string or the shape that it's in? –  DanielSank Jul 2 at 0:11
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What follows is an answer from an experimental particle physicist, i.e. one who has more knowledge of theoretical physics than the average educated person, but not in a position to teach it :). I can use theoretical results and study data and validate or falsify a theory.

I would like to know that if what we conceptualize as a "field" is merely an interaction among particles (bosons and fermions in the case of Quantum fields),

The concept of "field" in physics is general and mathematical.

A field is a physical quantity that has a value for each point in space and time. ....A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or a tensor, respectively........a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively.

So the quantum mechanical field has operators sitting there at every spacetime point, which when acting will give a measurement value for that field. It does not carry interactions unless put in by hand/experiment.

and particles (themselves) are actually fluctuations in "fields",

Particles are described as excitations of an all permeating quantum field.

then which comes first in the hierarchy of cause and effect relationships, particles or "fields"?

Well, if you did not have the field there would not be a possibility for the particle to manifest itself , in the same way if you do not have space there is noplace to measure a classical electric field. It is like an underlying coordinate system. A field is more than a cause, it is a framework where cause and effect (interactions) can be described.

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This is a good answer. OP, you can take the one I posted as an expansion of the idea of particles as excitations of fields. –  DanielSank Jul 1 at 4:34
@DanielSank Thanks, your answer is good for somebody who has some grounding in formal physics courses. –  anna v Jul 1 at 5:28
Indeed. It's good when multiple people post answers targeted toward different background levels. Much more useful for future readers :) –  DanielSank Jul 1 at 5:33
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When I studied quantum mechanics, my professor advised that I avoid the question "which is more fundamental?" and replace it with "which is more useful?". The problem is that our brains are programmed to think classically, so many concepts in QM have no classical analogue. For that reason, we usually discuss them mathematically in order to avoid ambiguity. One the one hand, we could say that fields are more fundamental and that particles are just excitations of the underlying fields. This accounts for some of the odd behaviours of particles (e.g. why particles of the same type are indistinguishable). However, when we do experiments, we tend to observe discrete objects rather than continuous fields. Ultimately, if I had to give an answer, I'd say that symmetries and commutation relations are fundamental, and representing them as fields or particles or whatever you want is just a matter of preference.

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"which is more useful?" Nice. When I realized this was the right way to think about mathematical objects, physics became a lot easier. –  DanielSank Jul 1 at 4:35
Definitely. We tend to perceive reality this way automatically without even thinking about it. For example, which is more fundamental: the guy standing there about to hit me in the head with a stick or a bundle of organic molecules arranged so that their large scale motions will produce a net force on my head? Now, which one of these interpretations is more useful? –  Ryan L Jul 1 at 4:49
Heh. While I might argue that the bundle of organic molecules is more fundamental in the scientific sense, I think I understand your meaning. –  DanielSank Jul 1 at 4:54
Also, I should add that in Haag's axiomatization of local QFT, neither fields nor particles are necessary. The fundamental object is a way of assigning algebras of observables to open sets of space-time in a causal and Lorentz invariant way (an isotonic co-presheaf of C* algebras for the mathematically minded.) I like axiomatizations like this because you can use whatever kind of ontology you want as long as it satisfies the axioms. Either way you get the same observables and the same predictions, which is what science is really about. (Unfortunately AQFT is in its early days.) –  Ryan L Jul 2 at 19:45
I understood some of those words. –  DanielSank Jul 2 at 20:18
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In my opinion, at this moment of time fields are more fundamental than particles because are more useful. It is easier to describe e.g. processes of creation and anihilation of pairs electron-positron using the notion of field. Vector potential A plays dual role, it represents photons in these processes and describes an interaction in Dirac equation which creates and anihilates these particles. In Young-Mills theories of weak and strong interactions some massive particles behave just like photon. Keeping in mind only particle image it would be impossible to understand these 'particle' processes. However, in the future someone maybe will discover a new more fundamental notion! One must not forget about wave-particle dualism. In my opinion, it is still alive.

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