Could almost nothing really be something? Fourier series.

In summary, according to this conversation, almost nothing could be something, and the Dirac-delta function could be used to describe a three-dimensional vector.
  • #1
Spinnor
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Could almost nothing really be something? Fourier series.

Say we have a large box. Say we have some function defined in this box that is square integrable. Say this function is small except for some small region in the box. This function could be represented as an infinite Fourier series, an infinite sum of functions that add to nearly zero for most of the box but constructively sum for some small region.

In a similar way we can add waves in quantum mechanics such that probability is small but for some localized region in a box. Could the Universe be such that when we have a region with low probability of finding a particle that there really exist this infinite set of waves that just happen to sum to zero at that spot at that time?

Could almost nothing "really" be something?
 
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  • #2
Spinnor said:
Could the Universe be such that when we have a region with low probability of finding a particle that there really exist this infinite set of waves that just happen to sum to zero at that spot at that time?

Keep on studying QM, since what you are describing is basically a Fourier transform. The part about the basis waves being "really there" deserves a word of caution: this is identical to the question of whether the components of a 3D vector in a particular basis are "really there", and in any case science does not answer these questions. Having said that, my answer is that the vector is more real than its components, and so I think that the state of nothing is real and the fact that you choose to express nothing in a basis with infinitely many non-zero components does not make those non-zero components real.
 
  • #3
I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).
 
  • #4
Matterwave said:
I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).

That's pretty interesting, I didn't even know that. Does that mean that a 3-dimensional vector is/can be described by 3 Dirac-delta functions? I've glanced at the delta function but I'm not too familiar with its applicability. I see no reason that this wouldn't also work for abstract vectors and 'states', but I must ask if this is the case.
 
  • #5
Matterwave said:
I'm not sure what you're saying, but what you describes sounds to be something like the Dirac-delta function, which is zero everywhere but at the origin where it is infinite and integrating over a region epsilon on both sides of the origin gets you 1(this is admittedly a "fake" function). This function happens to be the eigenfunction of the coordinate operator (x, y, or z).


I'm thinking of the Gaussian function that can be near zero but for some small region.:

f(x) = a*e^(-((x-b)^2 /2*c^2)))


By the definition of the Gaussian we must let the box get very large?
 
  • #6
It's interesting to note that the Feynman Path Integral of Quantum Mechanics for at least a free particle can easily be derived from the Dirac Delta function as noted here:

http://hook.sirus.com/users/mjake/delta_physics.htm

This makes me wonder if all of physics could be derived from the Dirac Delta function.
 
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  • #7
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Please review the https://www.physicsforums.com/showthread.php?t=5374", especially our policy on speculative, personal theory, and the IR forum. Unless you can cite peer-reviewed publication or standard physics sources, personal websites are NOT valid references to be used here.

Zz.
 
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FAQ: Could almost nothing really be something? Fourier series.

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a combination of sine and cosine waves. It is named after the mathematician Joseph Fourier and is used to analyze and approximate functions in various fields such as physics, engineering, and signal processing.

How does a Fourier series relate to "almost nothing"?

In mathematics, "almost nothing" refers to a set with measure zero, meaning it has no length, area, or volume. A Fourier series can be used to represent these types of sets as a combination of infinitely many sine and cosine waves, which can help us understand and analyze their properties.

Can a Fourier series represent any function?

No, a Fourier series can only represent periodic functions, meaning they repeat themselves over a certain interval. If a function is not periodic, then it cannot be represented by a Fourier series. However, many non-periodic functions can be approximated by a Fourier series.

What is the significance of "almost nothing" in mathematics?

The concept of "almost nothing" is important in mathematics as it allows us to define and understand sets with unusual properties. These sets can help us study and solve problems in various fields, such as geometry, topology, and analysis.

Are there any practical applications of Fourier series?

Yes, Fourier series have numerous practical applications in fields such as physics, engineering, and signal processing. They are used to analyze and approximate functions, solve differential equations, and compress data. They are also the foundation of many modern technologies, such as digital cameras, MRI machines, and audio and video compression algorithms.

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